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\title[Period relations as period terms]{Period relations as period terms: linear independence at CM and Beta periods}
\author{Shawn Krisman}
\subjclass[2020]{Primary 11J81; Secondary 11J89, 11J91, 11J95, 11G15}
\keywords{transcendence, linear independence, complex multiplication, Beta values, periods, algebraic subgroup theorem,
four exponentials conjecture, elliptic functions}
\date{9 November 2026}

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\begin{abstract}
We prove linear independence results for periods of CM abelian varieties and Beta values by using a known relation between
periods as the period term in the algebraic subgroup theorem. For example, $1$, $B(\tfrac27,\tfrac27)$, $B(\tfrac37,\tfrac37)$
and $B(\tfrac37,\tfrac37)/B(\tfrac17,\tfrac17)$ are linearly independent over $\Qb$, and families of such relations attached to
the CM types of $\Q(\zeta_N)$ give the linear independence of $1$, a common ratio and $\kappa\ge2$ periods. The same method
proves Waldschmidt's Conjecture~42 at six classes of CM ratios (for instance, $\wp(\log2/\vp)$ and $\wp(\log3/\vp)$ are not both
algebraic on the lemniscatic curve) and several four exponentials statements at CM periods. All results are explicit
consequences of theorems of Waldschmidt and of Roy and Waldschmidt.
\end{abstract}

\maketitle
\section{Introduction}\label{sec:intro}

Waldschmidt's algebraic subgroup theorem with period term \cite[Thm.~4.1]{W88} bounds the rank of a group of logarithms of
algebraic points lying on a subspace $V$, with a saving $\kappa$ for the periods that lie on $V$; the theorem of Roy and
Waldschmidt \cite{RW93} puts it in a form with number-field coefficients. A period on $V$ comes from a relation among periods.
In the companion note \cite{Note} the period data come from the choice of the slope (for instance the period $(2\pi i,2ic\Omega)$
on the line of slope $c\Omega/\pi$) and from Legendre's relation and its analogue $P_\sigma P_{\bar\sigma}\in\pi\Qb^\times$ for CM
abelian varieties. Here we use quadratic relations among first-kind periods of four different eigen-slots, which exist
on CM abelian varieties of dimension $\ge3$ by Deligne's theorem in the form of Koblitz and Ogus, and, in the other results, the
quotients of periods of two different eigen-slots. Each relation, placed on the line, is the period term of the count.

\subsection*{Notation}
Let $\Qb$ be the algebraic closure of $\Q$ in $\C$, $\Lg=\{z\in\C: e^z\in\Qb^\times\}$ and $\Lt=\Qb+\Qb\Lg$. For an
elliptic curve $E\colon y^2=4x^3-g_2x-g_3$ with $g_2,g_3\in\Qb$ we write $\wp,\zeta$ for its Weierstrass functions,
$\Lambda_E$ for its period lattice, $\eta(\lambda)=\zeta(z+\lambda)-\zeta(z)$ for the quasi-periods, and
$\Lg_E=\{u\in\C: u\in\Lambda_E \text{ or } \wp(u)\in\Qb\}$. We use two curves throughout:
\begin{align*}
E_i&\colon y^2=4x^3-4x, & \Lambda&=\vp\,\Z[i], & \vp&=\tfrac{\Gamma(1/4)^2}{2\sqrt{2\pi}}, & \eta(\vp)&=\tfrac{\pi}{\vp};\\
E_\rho&\colon y^2=4x^3-4, & \Lambda&=\Omega_\rho\Z[\rho], & \Omega_\rho&=\tfrac{\Gamma(1/3)^3}{2^{4/3}\pi}, & \eta(\Omega_\rho)&=\tfrac{2\pi}{\sqrt3\,\Omega_\rho}.
\end{align*}
(The quasi-periods follow from the Legendre relation and $\zeta(iz)=-i\zeta(z)$, $\zeta(\rho z)=\bar\rho\zeta(z)$. The real
periods are Beta integrals, $\vp=\int_1^\infty(x^3-x)^{-1/2}dx=\tfrac12B(\tfrac14,\tfrac12)$ and
$\Omega_\rho=\int_1^\infty(x^3-1)^{-1/2}dx=\tfrac13B(\tfrac16,\tfrac12)$, and the $\Gamma$-forms follow from the reflection and
duplication formulas; these are instances of the Chowla--Selberg formula \cite{Gr78}.)

Let $A/\Qb$ be a simple abelian variety with complex multiplication by a CM field $F$ of degree $2g$, with CM type
$\Phi$. For an embedding $\sigma$ of $F$, an \emph{eigen-differential} is a class $\omega_\sigma\in H^1_{\mathrm{dR}}(A/\Qb)$
with $[x]^*\omega_\sigma=\sigma(x)\omega_\sigma$; it is of the first kind (holomorphic) iff $\sigma\in\Phi$. An
\emph{eigen-period} is a nonzero number $P_\sigma=\int_\gamma\omega_\sigma$, $\gamma\in H_1(A,\Z)$. By Riemann's bilinear
relations $P_\sigma P_{\bar\sigma}\in\pi\Qb^\times$. For a CM abelian variety that is not simple, every nonzero
eigen-period is an algebraic multiple of an eigen-period of a simple factor, of the same kind (\S\ref{sec:setting}). For $N\ge3$ and $0<a,b<N$ with $a+b\ne N$, the Beta value
$B(a/N,b/N)=\Gamma(a/N)\Gamma(b/N)/\Gamma((a+b)/N)$ is, up to $\Qb^\times$, an eigen-period of a CM abelian variety
(a factor of the Jacobian of the Fermat curve of degree $N$), of the first kind iff $a+b<N$ (\S\ref{sec:setting}).

\subsection*{The level-seven example}
Products of CM abelian varieties involving a simple factor of dimension $\ge3$ can have homogeneous quadratic relations among
first-kind periods of four different eigen-slots (Remark~\ref{rem:where}). On the Jacobian $A$ of $C\colon y^7=x(1-x)$ (genus $3$, CM by $\Q(\zeta_7)$), with $E$ an elliptic curve
with CM by $\Q(\sqrt{-7})$, the Beta cocycle and one reflection give
\[
B(\tfrac17,\tfrac17)\,B(\tfrac27,\tfrac27)=\frac{\sin(3\pi/7)}{\sin(\pi/7)}\,B(\tfrac37,\tfrac37)\,B(\tfrac17,\tfrac27),
\]
where $B(j/7,j/7)$ ($j=1,2,3$) are the periods of the holomorphic eigen-differentials $dx/y^{7-j}$ on $C$ and
$B(1/7,2/7)\in\Qb^\times\Omega_E$ (Chowla--Selberg \cite{Gr78}). Theorem~\ref{thm:X} (\S\ref{sec:X}) uses such a relation as the
period term. On $y^7=x(1-x)$ it gives:

\begin{corollary}[Corollary X7]\label{cor:X7intro}
\begin{itemize}
\item[(a)] For $O,Q\in C(\Qb)$ with $Q-O$ of infinite order in $A$,
\[
\int_O^Q\frac{dx}{y^5}\Big/\int_O^Q\frac{dx}{y^6}\ \notin\ \Qb\cdot\frac{B(3/7,3/7)}{B(1/7,1/7)} .
\]
\item[(b)] The numbers $1$, $B(\tfrac27,\tfrac27)$, $B(\tfrac37,\tfrac37)$, $B(\tfrac37,\tfrac37)/B(\tfrac17,\tfrac17)$
are linearly independent over $\Qb$; equivalently, so are $\pi$, $\Gamma(\tfrac27)^2\Gamma(\tfrac37)$,
$\Gamma(\tfrac17)\Gamma(\tfrac37)^2$, $\Gamma(\tfrac27)\Gamma(\tfrac37)^2/\Gamma(\tfrac17)$.
\end{itemize}
\end{corollary}

\noindent Corollary~\ref{cor:X7intro} is Corollary~\ref{cor:X7}(a), (c). The relation is the period term in the following
sense. Let $z_1,z_2,z_3$ be the eigen-coordinates on $\mathrm{Lie}(A)$ dual to $dx/y^6,dx/y^5,dx/y^4$, with periods
$P_j\in\Qb^\times B(\tfrac j7,\tfrac j7)$. On $G=A\times E\times\Ga^2$ read the map
$\Psi=(z_1+z_E+q_1,\ z_3+z_2+q_2)\colon\mathrm{Lie}(G)\to\C^2$ and the line $V=\C(P_1,P_3)$. The relation puts two
elements of the $K$-span of $\Psi(\Omega(G))$ on $V$, linearly independent over a number field $K$, and the theorem of Roy and Waldschmidt then allows no further logarithm of an algebraic point
on $V$ (Theorem~\ref{thm:X7}); Corollary~\ref{cor:X7intro} follows by specialising the logarithms. The general statement,
for four distinct first-kind eigen-slots with $P_aP_b\in\Qb^\times P_cP_d$, is Theorem~\ref{thm:X}.

\subsection*{Several parallel pairs}
At level $N$, let $\tau\in(\Z/N)^\times\setminus\{\pm1\}$ and let $T(\tau)$ be the set of CM types $\Phi$ of $\Q(\zeta_N)$
with $1,\tau\in\Phi$; it has $\kappa=2^{g-2}$ elements, where $2g=\varphi(N)$. Replacing $\tau$ by $-\tau$ in each $\Phi$ gives
a partner type $\Phi'$, and by Deligne's theorem the ratios $P_{\Phi'}/P_\Phi$ of the corresponding first-kind periods all lie in
one class $\Qb^\times r$. Theorem~\ref{thm:Xk} places all $\kappa$ pairs on one line and gives:
\emph{$1$, $r$ and the $\kappa$ periods $P_{\Phi'}$ are linearly independent over $\Qb$}, and the same for every subfamily.
For $\kappa=1$ this follows from W\"ustholz's analytic subgroup theorem (multiply by $\pi/P_{\Phi'}$); the point is
$\kappa\ge2$, i.e.\ $g\ge3$. For instance, at $N=15$, $\tau=2$ ($\kappa=4$), the numbers
\[
1,\quad \frac{B(\frac4{15},\frac4{15})}{B(\frac1{15},\frac2{15})},\quad B(\tfrac4{15},\tfrac4{15}),\quad
B(\tfrac1{15},\tfrac7{15}),\quad B(\tfrac2{15},\tfrac6{15}),\quad B(\tfrac13,\tfrac13)
\]
are linearly independent over $\Qb$. Levels $9$ and $24$ give analogues of Corollary~\ref{cor:X7intro} with elementary proofs
of the relations (Theorem~\ref{thm:X924}): for instance $1$, $\vp$, $B(\frac1{24},\frac1{12})$,
$B(\frac1{24},\frac1{12})/B(\frac1{24},\frac1{24})$ are linearly independent over $\Qb$.

Products of periods are in general not $1$-periods, so these statements are outside the period theorem of Huber and W\"ustholz
\cite{HW}, which describes the $\Qb$-linear relations among $1$-periods. They would reduce to it if a single Gamma monomial
multiplied every number of the list into $\Qb^\times$ times a $1$-period. In the Koblitz--Ogus dictionary there is no such
multiplier for the lists of Corollary~\ref{cor:X7intro}, Corollary~\ref{cor:X924} and the example above
(Remark~\ref{rem:multiplier}, \lab{Computed}).

\subsection*{Other results}
The same method, with the quotient of the periods of two different eigen-slots as the slope of the line, gives the following
(\S\S\ref{sec:SE}--\ref{sec:H2}).
\begin{itemize}
\item \emph{Waldschmidt's Conjecture~42 at CM ratios} (Theorem~\ref{thm:C42}). For instance $\wp_i(\log2/\vp)$ and
$\wp_i(\log3/\vp)$ are not both algebraic, nor are $\wp_i(\Omega_\rho\log2/\pi)$ and $\wp_i(\Omega_\rho\log3/\pi)$.
\item \emph{Strong elliptic four exponentials} at $t\in\Qb^\times(\Omega'/\Omega)^{\pm1}\cup\Qb^\times(\pi/\Omega\Omega')^{\pm1}$
for two non-isogenous CM curves (Corollary~\ref{cor:SE4}), and on eigen-coordinates of CM abelian varieties at
$t\in\Qb^\times(P'/P_\sigma)^{\pm1}$ (Theorem~\ref{thm:EA}). For instance $\wp_i(\Omega_\rho/\vp)$ and $\wp_i(\Omega_\rho u/\vp)$
are not both algebraic for every $u\in\Lg_{E_i}\setminus\{0\}$.
\item \emph{Four exponentials for quasi-logarithms} (Theorem~\ref{thm:QL}). For instance, if $u_1,u_2\in\Lg_{E_i}$ and $\vp$ are
linearly independent over $\Q(i)$, then $(\vp/\Omega_\rho)\zeta_i(u_1)$ and $(\vp/\Omega_\rho)\zeta_i(u_2)$ are not both
$\Qb$-linear combinations of $1$ and quasi-logarithms $\zeta_i(u)$.
\item \emph{Collinearity of period points} of three CM curves with distinct CM fields (Theorem~\ref{thm:CL4}), a
codimension-two application.
\item \emph{Imaginary quadratic multiplication} of signature $(r,s)$ (Theorem~\ref{thm:U}): on a simple
Picard Jacobian, the abelian integrals $\int^P dx/y^2$, $\int^Px\,dx/y^2$ are proportional, with factor $\log\alpha/2\pi i$, to the
two periods along one cycle only if $P$ is torsion and $\alpha$ a root of unity.
\end{itemize}

\subsection*{Position}
All of this is an explicit application of published theorems: Waldschmidt's subgroup theorem with period term
\cite[Thm.~4.1]{W88}, its hyperplane case \cite[\S5d]{W88}, and the theorem of Roy and Waldschmidt \cite[Th\'eor\`eme]{RW93},
used with the weight of the quotient group that appears inside its proof (\S\ref{sec:method}). We claim no new method. Several
results are direct instances of the theorem of Roy and Waldschmidt as printed: Theorem~\ref{thm:SE}(1), Theorem~\ref{thm:EA} for
elliptic curves, Theorems~\ref{thm:X7}, \ref{thm:X924}(1), the families of Theorem~\ref{thm:Xk} (and through them
Theorem~\ref{thm:X924}(2)), and rows (iii), (iv) of the line form of Theorem~\ref{thm:C42} (Remark~\ref{rem:printed}). What we
contribute is the choice of the period relation placed on the line, the explicit statements and examples, and the proofs
written out at these periods: the group, the map, the line, the period datum, the quotient, and either the exclusion of every
algebraic subgroup or the computation of the weight. Theorems~\ref{thm:CL4} and~\ref{thm:U} use Waldschmidt's theorem on
subspaces of codimension $\ge2$, subgroup by subgroup.

As far as we found, none of these statements is in print, with the following exceptions and partial overlaps (\S\ref{sec:lit}).
For Theorem~\ref{thm:Xk} with $\kappa=1$ the independence statement is W\"ustholz's theorem in substance. Theorem~\ref{thm:QL} in
the class $\Qb^\times\Omega^{\pm1}$ is of degree one by Laurent's identity, and we expect it to follow from the analytic subgroup
theorem \lab{Heuristic}. The cases $t\in\Qb$ of
(U1), (U3) follow easily from W\"ustholz's theorem. Corollary~\ref{cor:C42}(b) follows from known results and complex
conjugation when $\log\alpha,\log\beta$ and their complex conjugates span a $\Q$-space of dimension $\ge3$; what is new is the
conjugation-stable case, such as $(\log2,\log3)$. Likewise the conclusion of Theorem~\ref{thm:EA}, and that of
Corollary~\ref{cor:SE4}, follow from the elliptic and abelian strong six exponentials theorems of Roy and Waldschmidt and complex
conjugation when $\Qb y_1+\Qb y_2$ is not stable under complex conjugation (for Theorem~\ref{thm:EA} we checked this only at the level
of a proof sketch). Our examples lie in the conjugation-stable cases.

\medskip\noindent\emph{Conventions.} Unless labelled otherwise, every statement is proved here given the published theorems
cited; we use the labels \lab{Lit}, \lab{Proof sketch}, \lab{Heuristic} (depending on the period conjecture or the
Rohrlich--Lang conjecture), \lab{Numerical} and \lab{Computed} (exact combinatorics by a script). Every result relies on
Waldschmidt's Theorem~4.1 as printed, including the one step of its proof that \cite{W88} indicates by reference to other papers
(\S\ref{sec:W88}). The results were found with substantial use of AI tools, and all review so far is by AI referees; no human
expert has checked the proofs. \S\ref{sec:formal} says which statements are checked in Lean.

\medskip\noindent\emph{Organisation.} \S\ref{sec:method} states the published theorems and the shape of the arguments,
\S\ref{sec:CMsetting} the linear algebra of CM abelian varieties and of universal vector extensions. \S\ref{sec:X} proves
Theorem~\ref{thm:X} and its instances at levels $7$, $9$, $24$ and Theorem~\ref{thm:Xk}; \S\ref{sec:SE} the strong elliptic and
eigen-slot statements, \S\ref{sec:QL} four exponentials for quasi-logarithms, \S\ref{sec:C42} Conjecture~42 at CM ratios and
\S\ref{sec:H2} the codimension-two applications. \S\ref{sec:formal} is on formal verification and the literature.

\section{The published theorems}\label{sec:method}

\subsection{Waldschmidt's subgroup theorem}\label{sec:W88}
\begin{theorem}[{\cite[Thm.~4.1, pp.~385--386]{W88}}]\label{thm:W88}
Let $G=\Ga^{d_0}\times\Gm^{d_1}\times G_2$ be a connected commutative algebraic group over $\Qb$ of dimension
$d=d_0+d_1+d_2\ge1$, with projections $\pi_0,\pi_1$ onto the first two factors ($d_0,d_1$ need not be maximal). Let
$V\subset T_G(\C)$ be a subspace of dimension $N<d$, $W\subset V$ a subspace defined over $\Qb$, and $Y\subset V$ a
finitely generated subgroup with $\Gamma=\exp_GY\subset G(\Qb)$; put $\kappa=\mathrm{rk}_\Z(Y\cap\ker\exp_G)$. Then there
is a connected algebraic subgroup $G'\neq G$ defined over $\Qb$ such that, with $\delta=\dim G/G'$,
$\delta_0=\dim\Ga^{d_0}/\pi_0(G')$, $\delta_1=\dim\Gm^{d_1}/\pi_1(G')$, $\delta_2=\delta-\delta_0-\delta_1$,
$\lambda=\mathrm{rk}_\Z\,\Gamma/(\Gamma\cap G')$ and $\tau=\dim W/(W\cap T_{G'})$, one has $\delta>\tau$ and
\begin{equation}\label{eq:41}
(\lambda+\delta_1+2\delta_2)(d-N)\le(\delta-\tau)(d_1+2d_2-\kappa).
\end{equation}
\end{theorem}
\noindent ($G'=0$ is allowed.) The proof \cite[\S\S6--7]{W88} combines an auxiliary function \cite[Prop.~6.1]{W88} with
Philippon's zero estimate \cite{Ph86}. Prop.~6.1 asks for a polynomial that vanishes with multiplicity $\ge T$ along $W$ at the
points of $\exp_G(Y(S))$, with the saving $\kappa$ in the parameters. Waldschmidt says \cite[p.~396]{W88} that this is proved
in \cite[Prop.~2.4]{W83} when $W=0$ and the exponent $b$ is $1$, and that the estimates for the derivatives are provided by
Bertrand's Lemma~7 \cite[App.~I]{W79}, and he says to compare with Bertrand \cite{Be85} and Philippon--Waldschmidt
\cite{PW88}. So for $W\ne0$ the printed proof only indicates this step. All our direct applications of
Theorem~\ref{thm:W88} have $W\ne0$ and $\kappa>0$, and so do the instances of it behind Theorem~\ref{thm:RW} below. We use
Theorem~\ref{thm:W88} as printed. Put
\begin{equation}\label{eq:NR}
2N\ge d_1+2d_0+\kappa .
\end{equation}
This is not a hypothesis of Theorem~\ref{thm:W88}; it is the range in which this step has been checked outside print. For
$G=\Ga^{d_0}\times\Gm^{d_1}\times E_1\times\dots\times E_m$, with $E_j$ elliptic curves, Theorem~\ref{thm:W88} under
(\ref{eq:NR}) is proved in Lean \cite{LSG,KrL}. For other groups, in particular for CM abelian varieties of dimension $\ge2$ and
for universal vector extensions, we rely on the printed theorem; no human expert has checked this step. Every direct
application of Theorem~\ref{thm:W88} below satisfies (\ref{eq:NR}), and each such proof says so. Theorem~\ref{thm:W88} is
stated for any $d_0$ (p.~385); Waldschmidt himself applies its consequence \cite[Thm.~1.1]{W88} with $d_0=2$, $W\ne0$,
$\kappa=0$ in the proof of \cite[Cor.~2.1]{W88}. When $d_0\ge1$ the printed proof uses an exponent $b>1$ in the case
$\delta=\tau$ (p.~397), which belongs to the same step. Several of our applications (Theorem~\ref{thm:SE}(1),
Proposition~\ref{prop:X}, Theorem~\ref{thm:Xk}) have $d_0=2$.

\subsection{The hyperplane case}\label{sec:stmtR}
\begin{theorem}[{\cite[Thm.~1.1, pp.~375--376, under (5.4) of \S5d, p.~394]{W88}}]\label{thm:W88k}
Let $G=\Ga^{d_0}\times\Gm^{d_1}\times G_2$ be a connected commutative algebraic group over $\Qb$ of dimension $d\ge1$, with
$d_0$, $d_1$ maximal and $d_2=d-d_0-d_1$. Let $V$ be a hyperplane of $T_G(\C)$, $W\subset V$ a subspace of dimension
$t\ge0$ defined over $\Qb$, and $Y=\Z y_1+\dots+\Z y_m\subset V$ a subgroup of rank $m$ with $\exp_GY\subset G(\Qb)$. Put
$\kappa=\mathrm{rk}_\Z(Y\cap\ker\exp_G)$. If
\[
m>(d_1+2d_2-\kappa)(d-1-t),\tag{5.4}
\]
then $V$ contains a nonzero algebraic Lie subalgebra of $T_G(\C)$ defined over $\Qb$, that is, $T_{G'}\subset V$ for some
connected algebraic subgroup $G'\ne0$ of $G$ defined over $\Qb$.
\end{theorem}

\noindent The printed Theorem~1.1 has the hypothesis $m>(d_1+2d_2)(d-1-t)$, without $\kappa$; \cite[\S5d]{W88} proves it under
the weaker hypothesis (5.4) (p.~394 prints $\kappa=\ell-m$, a misprint for $m-\ell$). That proof applies Theorem~\ref{thm:W88} to
$G$ and then again to an auxiliary subgroup \cite[Lemma~5.1]{W88}. Since $\alpha(G)=d_1+2d_2$ for a maximal decomposition
(\S\ref{sec:RW}), Theorem~\ref{thm:W88k} is, in contrapositive form, the case $K=\Q$, $\Psi=\mathrm{id}$ of Theorem~\ref{thm:RW} below
for hyperplanes $V$ (stated for a finitely generated $Y$, with $\kappa$ the rank of its periods). Theorems~\ref{thm:W88}
and~\ref{thm:W88k} are stated for literal products. Every connected commutative $G$ over $\Qb$ is isogenous to
$\Ga^{d_0(G)}\times\Gm^{d_1(G)}\times G_2$ (a surjection onto $\Ga$, resp.\ $\Gm$, restricts onto a $\Ga$, resp.\ a subtorus, of the
linear part, which splits off up to isogeny), and both statements are invariant under isogeny (as \cite[p.~389]{W88} notes), so we
apply them to groups that are only isogenous to such a product.

\subsection{The theorem of Roy and Waldschmidt}\label{sec:RW}
Let $G$ be a connected commutative algebraic group over $\Qb$, $T_G$ its tangent space at the origin with its
$\Qb$-structure, $\Omega(G)=\ker\exp_G$ its periods and $\Lambda(G)=\exp_G^{-1}(G(\Qb))$ the logarithms of its algebraic
points. For a number field $K$ of degree $n$, $\Omega_K(G)$ and $\Lambda_K(G)$ are the $K$-spans of $\Omega(G)$ and
$\Lambda(G)$ in $T_G(\C)$. Fix a basis $\theta=(\theta_1,\dots,\theta_n)$ of $K/\Q$ and let
$p_\theta\colon T_{G^n}=T_G^n\to T_G$, $(z_j)\mapsto\sum_j\theta_jz_j$. Let $d_0(G)$ (resp.\ $d_1(G)$) be the largest $d$ with a
surjection $G\to\Ga^d$ (resp.\ $G\to\Gm^d$) over $\Qb$, and $\alpha(G)=2\dim G-2d_0(G)-d_1(G)$. Let $\cE$ be a
finite-dimensional complex vector space with a $\Qb$-structure, $\Psi\colon T_G(\C)\to\cE$ a $\Qb$-rational linear map, and
$\Psi_\theta=\Psi\circ p_\theta$. Let $H_0$ (resp.\ $H_\Psi$) be the largest connected algebraic subgroup of $G^n$ with
$p_\theta(T_{H_0})=0$ (resp.\ $T_{H_\Psi}\subset\ker\Psi_\theta$), and put
\[
\alpha_K(G)=\alpha(G^n/H_0)/n\ \ \text{\cite[p.~360]{RW93}},\qquad \alpha_{K,\Psi}(G)=\alpha(G^n/H_\Psi)/n\ \ \text{(the
\emph{inner weight})};
\]
neither depends on $\theta$ (a change of basis induces an isogeny of $G^n$), and $\alpha_{K,\Psi}(G)\le\alpha_K(G)$
\cite[p.~364]{RW93}. Finally $S_K(G)$ is the set of
subspaces $p_\theta(T_{G'})$ of $T_G$, for $G'$ an algebraic subgroup of $G^n$.

\begin{theorem}[{\cite[Th\'eor\`eme, (i)$\Rightarrow$(iii), p.~363, with the inequality of step (b), p.~364]{RW93}}]\label{thm:RW}
Let $V\subset\cE$ be a complex subspace containing no nonzero $\Psi(S)$, $S\in S_K(G)$, and $W\subset V$ a subspace defined over
$\Qb$. Put $\kappa=\dim_K\bigl(V\cap\Psi(\Omega_K(G))\bigr)$. Then $V\cap\Psi(\Lambda_K(G))$ is a finite-dimensional $K$-vector
space, and
\[
\dim_K\bigl(V\cap\Psi(\Lambda_K(G))\bigr)\le\bigl(\alpha_{K,\Psi}(G)-\kappa\bigr)\dim_\C(V/W)\le\bigl(\alpha_K(G)-\kappa\bigr)\dim_\C(V/W).
\]
\end{theorem}

\noindent Roy and Waldschmidt state the second inequality, for every connected commutative $G$ over $\Qb$, every number field $K$
and every $V$. Their proof has two steps. Step (b) (p.~364) reduces to $K=\Q$, $\Psi=\mathrm{id}$ on $G^n/H_\Psi$; it is
written in full, and the inequality obtained before its last line (``On peut aussi majorer $\alpha(G')$ par $n\alpha_K(G)$'')
is the first inequality above. (Step (b) also applies (a) to $\ker\Psi'$, to get equalities in its (4); the inequality needs
only the surjections there, and that instance is in any case W\"ustholz's theorem.) Step (a) (p.~363) is the case $K=\Q$,
$\Psi=\mathrm{id}$. It follows formally from \cite[Thm.~4.1]{W88} by Roy's categorical method \cite[\S6]{R1}: Roy shows that the
statements of his Proposition~2.1 hold with the period-carrying function $a'(X)=\alpha(G)-\mathrm{rk}(Y\cap\Omega(G))$
\cite[p.~266]{R1}, and step (a) outlines how the bound follows from them. In the proofs, Theorem~\ref{thm:RW} is used only with
$\cE=\C^2$, $\Psi$ onto, $V$ a line and $W=0$ (the remarks of \S\ref{sec:H2} also use the case $K=\Q$, $\Psi=\mathrm{id}$, i.e.\
Theorem~\ref{thm:W88k} and \cite[Thm.~4.1]{W88} directly); then the subspace of $T_{G^n/H_\Psi}$ to which step (b) reduces is a hyperplane,
and for hyperplanes \cite[Remarque 2]{RW93} cites step (a) from \cite[Thm.~1.1]{W88}, i.e.\ Theorem~\ref{thm:W88k}. By
\cite[Remarque 1]{RW93}, the condition on $V$ holds whenever $V$ has no nonzero $\Qb$-rational point (and then $W=0$); this is
the only case we use. We call $\alpha_{K,\Psi}(G)$ the \emph{inner weight}.

\emph{Upper bounds for the inner weight.} For a connected algebraic subgroup $H$ of a connected commutative group $G$ one has
$\alpha(G/H)\le\alpha(G)$, since $\alpha(H)+\alpha(G/H)\le\alpha(G)$ \cite[Lemme~4.3]{R1} and $\alpha(H)\ge0$. If $U\subset G^n$ is a
connected algebraic subgroup with $T_U\subset\ker\Psi_\theta$, then $U\subset H_\Psi$, so $G^n/H_\Psi$ is a quotient of $G^n/U$ and
\[
\alpha_{K,\Psi}(G)\le\alpha(G^n/U)/n .
\]
Moreover $\alpha(G'')\le2\dim G''-2a-b$ whenever $G''$ admits surjections onto $\Ga^a$ and onto $\Gm^b$. So an explicit $U$ and
explicit surjections bound the inner weight from above, and an upper bound is all that the applications need.

\subsection{The shape of the arguments}\label{sec:RtoApp}
Every result of \S\S\ref{sec:X}--\ref{sec:C42} has the following shape. There are a group $G$ (a product of copies of $\Ga$,
$\Gm$, CM elliptic curves, simple CM abelian varieties and their universal vector extensions), a $\Qb$-rational map
$\Psi\colon T_G\to\C^2$ (onto), a line $V=\C(1,s)$ (or $\C w$) whose slope is transcendental, and $W=0$. So the condition of
Theorem~\ref{thm:RW} holds by \cite[Remarque 1]{RW93}, and $\dim V/W=1$. A period relation gives a period datum on $V$, so
$\kappa\ge1$ (for Theorem~\ref{thm:X} two periods on $V$ give $\kappa\ge2$, and $\kappa$ pairs give $\kappa$ for
Theorem~\ref{thm:Xk}), and a counterexample to the statement gives further vectors on $V$, $K$-independent of the period data.
Their total number is bounded by $\alpha_{K,\Psi}(G)-\kappa\le\alpha(G'')/n-\kappa$ for an explicit quotient $G''=G^n/U$. The
table lists the numbers (for $n=[K:\Q]$ large enough; ``data'' counts the period data together with the vectors a counterexample
would give).

\begin{center}
{\small
\begin{tabular}{>{\raggedright\arraybackslash}p{5.0cm}>{\raggedright\arraybackslash}p{4.2cm}llll}
result & $G''$ & $\alpha(G'')/n$ & $\kappa$ & data & bound\\\hline
Prop.~\ref{prop:X} (Thms.~\ref{thm:X}, \ref{thm:X7}, \ref{thm:X924}) & $\Ga^2\times\prod_jA_j''$ & $4$ & $2$ & $3$ & $2$\\
Thm.~\ref{thm:Xk} ($\kappa$ pairs) & $\Ga^2\times\prod_jA_j''$ & $2\kappa$ & $\kappa$ & $\kappa+1$ & $\kappa$\\
Thm.~\ref{thm:SE}(1), Thm.~\ref{thm:EA} (EA) & $\Ga^2\times A_1''\times A_2''\times A'''$ & $3$ & $1$ & $3$ & $2$\\
Thm.~\ref{thm:SE}(2), Thm.~\ref{thm:EA} (EA$^\natural$) & $\Ga\times A_1''\times A_2''\times Q''$ & $3+2/n$ & $1$ & $3$ & $2+2/n$\\
Thm.~\ref{thm:QL} ($X=\Gm$) & $Q_1\times\Gm^n\times Q_3$ & $3+4/n$ & $1$ & $3$ & $2+4/n$\\
Prop.~\ref{prop:C42line}, rows (iii), (iv) & $\Gm^n\times A''\times A'''$ & $3$ & $1$ & $3$ & $2$\\
Prop.~\ref{prop:C42line}, rows (i), (ii), (v), (vi) & $\Gm^n\times A''\times Q''$ & $3+2/n$ & $1$ & $3$ & $2+2/n$
\end{tabular}
}
\end{center}

\noindent Here the $A''$ are abelian varieties of dimension $n/2$ \emph{for every $g$}: reading one eigen-coordinate kills all
but $n/2$ dimensions of $A^n$ (Lemma~\ref{lem:Kprime}); and $Q''$, $Q_1$, $Q_3$ are extensions of such an abelian variety by
$\Ga$, of dimension $n/2+1$. In each row the bound is below the number of data for $n$ large enough ($n\ge3$, resp.\ $n\ge5$ for
Theorem~\ref{thm:QL}), which is the contradiction.

We give two kinds of proofs. In the \emph{direct} proofs (Proposition~\ref{prop:X}, Theorems~\ref{thm:Xk}, \ref{thm:SE}(1),
\ref{thm:QL}, and row (iv) of Proposition~\ref{prop:C42line} in Remark~\ref{rem:C42direct}) we apply Theorem~\ref{thm:W88} once,
to $G''=G^n/H_\Psi$, and exclude every connected algebraic subgroup by an explicit case check, with (\ref{eq:NR}) checked at
$G''$. In these applications $V''=\Psi''^{-1}(V)$ is a hyperplane, $N=d-1$, and $W''=\ker\Psi''$, so that $\tau=\delta+r'-2$ where
$r'=\dim\Psi''(T_{G'})$; thus only $G'$ with $r'\le1$ can satisfy $\delta>\tau$, and then $\delta-\tau=2-r'$. In the other proofs
(Theorem~\ref{thm:EA}, hence Theorem~\ref{thm:SE}(2), and Proposition~\ref{prop:C42line}) we apply Theorem~\ref{thm:RW} with
the upper bound for the inner weight of \S\ref{sec:RW}, which needs only an explicit $U$, the dimension of $G^n/U$ and $\kappa$. The
direct proofs show which subgroups are tight and by how much; the others are shorter. Theorems~\ref{thm:CL4} and~\ref{thm:U}
(\S\ref{sec:H2}) are not of this shape: there $V''$ has codimension $\ge2$.

\begin{remark}[The printed weight]\label{rem:printed}
For several results the printed weight $\alpha_K$ of Theorem~\ref{thm:RW} already suffices, on the group $G$ itself, with the
weights of \S\ref{sec:weights}: (i) Theorem~\ref{thm:SE}(1), and Theorem~\ref{thm:EA} for elliptic curves:
$\alpha_K(\Ga\times E\times\Ga\times E\times E')=3$, $\kappa\ge1$; (ii) Proposition~\ref{prop:X} when the four slots are all the
first-kind slots of the varieties involved (levels $7$ and $9$): $\alpha_K(G)=\sum_A\dim A=4$, $\kappa=2$ (Remark~\ref{rem:X5d});
(iii) Theorem~\ref{thm:Xk} for a whole family $T(\tau)$: $\alpha_K=2\kappa$; through it, and the slot-splitting step of
\S\ref{sec:XfromP}, Theorem~\ref{thm:X924}(2) and every subfamily of Theorem~\ref{thm:Xk} (the paragraph after the proof of
Theorem~\ref{thm:X924}); (iv) rows (iii), (iv) of Proposition~\ref{prop:C42line}: $\alpha_K(\Ga\times\Gm\times E\times E')=3$,
$\kappa\ge1$. With W\"ustholz's and Masser's theorems for the final deductions, these results are direct instances of
\cite[Th\'eor\`eme (iii)]{RW93} as printed. For Theorem~\ref{thm:EA} with $\max(g,g')\ge2$, Theorem~\ref{thm:SE}(2), the fibre
rows of Proposition~\ref{prop:C42line} and Theorem~\ref{thm:QL}, the printed weight on the groups used here gives no conclusion
(for instance $\alpha_K(\Ga\times A\times\Ga\times A\times A')=2g+g'$, and $\alpha_K(\Gm\times E^\natural\times E)=4+2/n$ by
Lemma~\ref{lem:natural}), and the inner weight is used.
\end{remark}

\section{CM abelian varieties and universal vector extensions}\label{sec:CMsetting}
This section gives the linear algebra of CM abelian varieties of any dimension $g$ and of their universal vector extensions
(\S\S\ref{sec:setting}--\ref{sec:Euniv}), and the weights (\S\ref{sec:weights}). Most of it is also in \cite[\S3]{Note}.

\subsection{CM abelian varieties}\label{sec:setting}
Let $A/\Qb$ be a simple abelian variety of dimension $g$ with CM by the CM field $F$, $[F:\Q]=2g$, with (primitive) CM
type $\Phi$. For $\sigma\in\Phi$ let $z_\sigma\colon\mathrm{Lie}(A)\to\C$ be the $\Qb$-rational eigen-coordinate dual to a
holomorphic eigen-differential $\omega_\sigma$, so that $z_\sigma([x]v)=\sigma(x)z_\sigma(v)$ for $x\in\mathcal O_F$. Since
$H_1(A,\Q)=F\gamma_0$ for any $0\ne\gamma_0\in\Omega(A)$, we have $z_\sigma(x\gamma_0)=\sigma(x)P_\sigma$ with
$P_\sigma=z_\sigma(\gamma_0)$, so $z_\sigma(\Omega(A))\subset\sigma(F)P_\sigma$.

\emph{Reduction to the simple case.} A CM abelian variety over $\Qb$ is isogenous to $\prod_iB_i^{m_i}$ with $B_i$ simple
CM abelian varieties of primitive type, with CM fields $F_i$. By its \emph{CM algebra} we mean any commutative semisimple
$\Q$-subalgebra $\mathcal E\subset\mathrm{End}^0(A)$ of degree $2\dim A$ (so $\mathcal E=\prod_i\mathcal E_i$ with
$\mathcal E_i\supseteq F_i$ maximal commutative in $M_{m_i}(F_i)$), and eigen-differentials of $A$ are taken with respect to
$\mathcal E$; the argument below does not depend on this choice. Being maximal commutative, $\mathcal E$ contains the
centre $\prod_iF_i$ of $\mathrm{End}^0(A)$ and its central idempotents. An eigen-character $\mathcal E\to\Qb$ is nonzero
on exactly one of these, so the eigen-differential lives on one isotypic part $B_i^{m_i}$, and the character restricts to
an embedding $\sigma_0$ of $F_i$, which acts on $B_i^{m_i}$ diagonally. An eigen-differential is therefore, on that part,
a $\sigma_0$-eigen class for $F_i$, i.e.\ a $\Qb$-combination of pull-backs $\mathrm{pr}_k^*\omega_{\sigma_0}$ of one
$\sigma_0$-eigen differential of $B_i$, of the same kind. So every nonzero eigen-period lies in
$\Qb\,\sigma_0(F_i)P_{\sigma_0}(B_i)\setminus\{0\}=\Qb^\times P_{\sigma_0}(B_i)$, as stated in the introduction.

\begin{lemma}[Lemma K$'$]\label{lem:Kprime}
Let $F\subset K$, where $K$ is a number field containing the Galois closure of $F$, $n=[K:\Q]$, and let
$\theta_1,\dots,\theta_n$ be a basis of $K/\Q$. For an embedding $\sigma\colon F\to K$ put
$\psi_\sigma(m)=\sum_j\theta_j\sigma(m_j)$ on $F^n$. Then $(\psi_\sigma)_\sigma\colon F^n\to K^{2g}$ is a $\Q$-linear
bijection, each $\psi_\sigma$ is $\sigma$-semilinear and onto $K$, and for every set $\Sigma$ of embeddings the common
kernel of $(\psi_\sigma)_{\sigma\in\Sigma}$ is an $F$-subspace of $F$-dimension $n-|\Sigma|\,n/(2g)$.
\end{lemma}

\begin{proof}
$F^n\cong F\otimes_\Q K$ via $m\mapsto\sum m_j\otimes\theta_j$, and $F\otimes_\Q K\cong K^{2g}$ via
$x\otimes y\mapsto(\sigma(x)y)_\sigma$, since $K$ contains the Galois closure; the composite is $(\psi_\sigma)_\sigma$.
Semilinearity is clear. The common kernel is the preimage of $\{0\}^{\Sigma}\times K^{2g-|\Sigma|}$, of $\Q$-dimension
$(2g-|\Sigma|)n$, i.e.\ of $F$-dimension $(2g-|\Sigma|)n/(2g)$.
\end{proof}

\noindent For $g=1$, $F=k$ is an imaginary quadratic field, $\psi_{\mathrm{id}}$ and $\psi_{\overline{\phantom{x}}}$ are the
maps $\psi$, $\phi$ below, and Lemma~\ref{lem:Kprime} becomes the following lemma, which we use throughout for CM elliptic
curves (we keep its direct proof).

\begin{lemma}\label{lem:K}
Let $k\subset K$ be an imaginary quadratic field inside a number field of degree $n$, $\theta_1,\dots,\theta_n$ a basis of
$K/\Q$, and on $k^n$ put $\psi(m)=\sum\theta_jm_j$, $\phi(m)=\sum\theta_j\bar m_j$. Then $(\psi,\phi)\colon k^n\to K\times K$
is a $\Q$-linear bijection; $\ker\psi$ and $\ker\phi$ are $k$-subspaces of dimension $n/2$, and $\ker\psi\cap\ker\phi=0$.
\end{lemma}

\begin{proof}
$k^n\cong K\otimes_\Q k$ via $m\mapsto\sum\theta_j\otimes m_j$, and $K\otimes_\Q k\cong K\times K$ via
$x\otimes y\mapsto(xy,x\bar y)$; the composite is $(\psi,\phi)$. $\psi$ is $k$-linear and $\phi$ is $k$-semilinear, and
both are onto $K$, which has $k$-dimension $n/2$.
\end{proof}
\noindent An \emph{eigen-slot} is a pair $(A,\sigma)$ with $A/\Qb$ a simple CM abelian variety and $\sigma$ an embedding
of its CM field; its periods are the eigen-periods $P_\sigma$ of $\omega_\sigma$ (on $A^\natural$ if $\sigma\notin\Phi$). Two
eigen-slots are equal iff the abelian varieties are isogenous and the embeddings correspond under the isogeny. An
eigen-slot is of the first kind iff $\sigma\in\Phi$, and of the second kind otherwise. We use the following standard
facts.
\begin{itemize}
\item[(S1)] \lab{Lit} (standard; see Shimura \cite{Sh98}). $\mathrm{End}^0(A^n)=M_n(F)$, and $\mathrm{Hom}$ between non-isogenous simple
abelian varieties is $0$. Abelian subvarieties of $A^n$ are the $B_S$, $S\subset F^n$ an $F$-subspace, with
$\dim B_S=g\dim_FS$ and $T(B_S)=\{(\sigma(M)u_\sigma)_{\sigma\in\Phi}:u_\sigma\in\C^{\dim S}\}$ in the eigen-coordinates
($M$ a matrix whose columns span $S$). On $T(B_S)$ the form $\theta^{\mathsf T}z_\sigma$ vanishes iff $\psi_\sigma(S)=0$,
and different $\sigma\in\Phi$ carry independent $u_\sigma$.
\item[(P)] \lab{Proved} Every connected algebraic subgroup of $\Ga^a\times\Gm^b\times B$, $B$ an abelian variety, is a
product $(H'\cap\Ga^a)\times(H'\cap\Gm^b)\times B'$; subgroups of $\Ga^a$ are $\Qb$-subspaces and of $\Gm^b$ rational
subtori. (The image of $H'$ in $B$ is an abelian subvariety $B'$, and $H'/(H'\cap M)$, $M$ the linear part, maps to the
linear group $M/(H'\cap M)$ trivially; so $H'=(H'\cap M)^0\times B'$ by dimension, and $H'\cap M$ splits since
$\mathrm{Hom}(\Ga,\Gm)=\mathrm{Hom}(\Gm,\Ga)=0$.)
\item[(Ch)] By Dedekind's independence of characters, the vectors $(\sigma(x))_{\sigma\in\Phi}$, $x\in F$, span $K^g$ over
$K$. Hence $\Lambda_K(A)$, read in the eigen-coordinates, is $\prod_{\sigma\in\Phi}K\,z_\sigma(\Lambda(A))$, and likewise
for $\Omega_K(A)$: the slot coordinates of $K$-combinations of logarithms (or periods) can be prescribed independently.
\item[(W)] \lab{Lit} (W\"ustholz \cite{Wu89}). The numbers $1$, $\pi$ and nonzero periods of pairwise distinct
eigen-slots (first or second kind, on one or several simple CM abelian varieties) are linearly independent
over $\Qb$. In particular $P_\sigma\notin\Qb$ and $P_\sigma/\pi\notin\Qb$. A nonzero eigen-period of $A$ (resp.\ a
second-kind eigen-period of $A^\natural$) is not in $\Qb\Lg$.
\end{itemize}
All of these follow from W\"ustholz's theorem on $\Ga\times\Gm\times\prod_iA_i^\natural$ (resp.\ with $\Gm^r$ added), applied to
the point with coordinates $1$, $2\pi i$ (resp.\ the logarithms) and the cycles, as in the following example. For example, $P/\pi\notin\Qb$: otherwise the point $(\gamma_0,2\pi i)$ of $\mathrm{Lie}(A\times\Gm)$ lies in a
$\Qb$-hyperplane $z_\sigma=a\,x$; the smallest algebraic subgroup whose Lie algebra contains it is $A\times\Gm$ ($A$ is
simple and the subgroup is a product by (P)), whose Lie algebra is not in the hyperplane.

\smallskip
\noindent\emph{Universal vector extension.} For $\sigma\notin\Phi$ the eigen-coordinate $w_\sigma$ lives on
$\mathrm{Lie}(A^\natural)$. We use the following facts. The input from the literature is \lab{Lit} (Mazur--Messing
\cite{MM74}): $\mathrm{Lie}(A^\natural)=H^1_{\mathrm{dR}}(A)^\vee$ functorially, and $A^\natural$ is universal among
extensions of $A$ by vector groups (every such extension of $B$ by $U$ is the push-out of $B^\natural$ along a unique linear
map $W_B\to U$, where $W_B$ is the fibre of $B^\natural$). The rest is \lab{Proved} by the one-line arguments given.
\begin{itemize}
\item[(F1)] $\mathrm{Lie}(A^\natural)=H^1_{\mathrm{dR}}(A)^\vee$ with functorial $\mathcal O_F$-action. In the dual eigenbasis
$(e_\tau)$ the fibre Lie algebra is $\bigoplus_{\tau\notin\Phi}\C e_\tau$, and $w_\sigma$ ($\sigma\notin\Phi$) is nonzero
exactly on $e_\sigma$, which lies in the fibre. \emph{Proof:} the Lie algebra sequence of $0\to V\to A^\natural\to A\to0$
is the dual of $0\to H^0(A,\Omega^1)\to H^1_{\mathrm{dR}}(A)\to H^1(A,\mathcal O)\to0$, so the fibre is the annihilator
of $H^0(A,\Omega^1)=\bigoplus_{\tau\in\Phi}\Qb\omega_\tau$.
\item[(F2)] For $B\subset A^n$, $B^\natural\to(A^n)^\natural$ is injective on fibres and $\pi^{-1}(B)=B^\natural+V^n$ ($V$ the
fibre). \emph{Proof:} on fibres the map is dual to the restriction $H^1(A^n,\mathcal O)\to H^1(B,\mathcal O)$, which is
onto because $B$ is a direct factor of $A^n$ up to isogeny (Poincar\'e reducibility); and $B^\natural$ maps onto $B$.
\item[(F3)] $\mathrm{Ext}(B,\Ga)\cong H^1(B,\mathcal O)$ and $\mathrm{Hom}(B,\Ga)=0$: a connected $H$ over $B$ with given
intersection $W'$ with the fibre exists iff $W'\supseteq W_B$, and then $H=B^\natural+W'$ is unique. \emph{Proof:} such
an $H$ is a splitting over $B$ of $\pi^{-1}(B)/W'$, which by (F2) and universality is the push-out of $B^\natural$ along
$W_B\to V^n/W'$; it splits iff this map is $0$, i.e.\ $W_B\subseteq W'$, and the splitting is unique since
$\mathrm{Hom}(B,V^n/W')=0$.
\item[(F4)] $T(B_S^\natural)=\bigoplus_\tau\tau(S)\otimes\C$ and $W_{B_S}=\bigoplus_{\tau\notin\Phi}\tau(S)\otimes\C$.
\emph{Proof:} under the comparison isomorphism $H^1_{\mathrm{dR}}(B_S)^\vee\cong H_1(B_S,\Q)\otimes\C$, with
$H_1(B_S,\Q)=S\gamma_0\subset F^n\gamma_0$, the $\tau$-eigen component is $\tau(S)\otimes\C$; the fibre part is $\tau\notin\Phi$
by (F1).
\end{itemize}
The periods of $A^\natural$ satisfy $\Omega(A^\natural)\otimes\Q=F\tilde\gamma_0$ with $w_\sigma(x\tilde\gamma_0)=\sigma(x)P_\sigma$,
and the values $w_\sigma(\Lambda(A^\natural))$ (``$\sigma$-fibre values'') are second-kind abelian integrals
$\int\omega_\sigma+\Qb$.

\smallskip
\noindent\emph{Beta values} \lab{Lit} (Weil \cite{We76}; Rohrlich's appendix to \cite{Gr78}; Koblitz--Ogus \cite{KO79}). For $N\ge3$
and $0<a,b<N$ with $a+b\not\equiv0\bmod N$, $B(a/N,b/N)$ is, up to $\Qb^\times$, the period of an eigen-differential on
the Fermat curve of degree $N$, with Koblitz--Ogus label $\Phi_{a,b}=\{u\in(\Z/N)^\times:\langle ua/N\rangle+\langle
ub/N\rangle+\langle u(N-a-b)/N\rangle=1\}$ (after reducing $a/N,b/N$ to lowest common denominator); the corresponding
isotypic factor of the Jacobian is a power of a simple CM abelian variety, and the differential is of the first kind iff
$a+b<N$. Deligne's theorem (stated by Koblitz--Ogus \cite{KO79}; it rests on Deligne's theorem that Hodge cycles on
abelian varieties are absolutely Hodge \cite{De82}) gives: if
$\mathbf 1_{\Phi_a}+\mathbf 1_{\Phi_b}=\mathbf 1_{\Phi_c}+\mathbf 1_{\Phi_d}$ on $(\Z/N)^\times$, then
$P_aP_b\in\Qb^\times P_cP_d$. Together with the rule that $\pi$ corresponds to the constant function $1$, and
$\mathbf 1_\Phi+\mathbf 1_{-\Phi}\equiv1$, the same dictionary contains $P_\sigma P_{\bar\sigma}\in\pi\Qb^\times$.
That these are all multiplicative relations is the Rohrlich--Lang conjecture, used below only in remarks labelled
\lab{Heuristic}.

\subsection{CM elliptic curves and their universal vector extensions}\label{sec:Euniv}
For a CM elliptic curve $E$ let $E^\natural$ be its universal vector extension, with base coordinate $z$ and conjugate-eigen
fibre coordinate $w$: $\mathcal O=\mathrm{End}(E)$ acts on $T(E^\natural)=H^1_{\mathrm{dR}}(E)^\vee$ by its identity
character on $z$ and by the conjugate character on $w$ (the eigenline of $H^1_{\mathrm{dR}}$ for $\bar a$ is spanned by
$x\,dx/y+c\,dx/y$ with $c\in\Qb$, so these coordinates are rational; this is (F1) for $g=1$).
In these coordinates the periods are $(\lambda,\eta^*(\lambda))$,
$\lambda\in\Lambda_E$, where $\eta^*(\lambda)=\eta(\lambda)-\kappa_1\lambda$ satisfies $\eta^*(a\lambda)=\bar a\,\eta^*(\lambda)$;
the logarithms of points of $E^\natural(\Qb)$ are $(u,\zeta^*(u)+t)$ with $u\in\Lg_E$, $t\in\Qb$ and
$\zeta^*=\zeta-\kappa_1\,\mathrm{id}$. On $E_i$ and $E_\rho$, $\kappa_1=0$. Since the characters $a\mapsto a,\bar a$ of $k$
are $K$-independent for $K\supseteq k$,
\begin{equation}\label{eq:split}
\Omega_K(E^\natural)=K\lambda_0\times K\eta^*(\lambda_0)\qquad(\Lambda_E\otimes\Q=k\lambda_0).
\end{equation}

We call \emph{fibre values} the $w$-coordinates of $\Lambda(E^\natural)$, that is, $\zeta^*(u)+\Qb$ for
$u\in\Lg_E\setminus\Lambda_E$ and $\eta^*(\lambda)+\Qb$ at lattice points, and write $\Fib$ for the set of fibre values. By
Legendre and $\eta^*(a\lambda)=\bar a\eta^*(\lambda)$, $\Omega\eta^*(\Omega)\in\Qb^\times\pi$ for every CM period $\Omega$.

\subsection{Weights}\label{sec:weights}
$\alpha_K(\Ga)=0$, $\alpha_K(\Gm)=1$, and for an elliptic curve $\alpha_K(E)=1$ if $E$ has CM by $k$ and $K\supseteq k$, else
$2$ \cite[Lemme, p.~361]{RW93}; more generally $\alpha_K(A)=g$ for a simple CM abelian variety $A$ of dimension $g$ when
$K$ contains the Galois closure of its CM field (\cite[Lemme, p.~361]{RW93}; it is also Lemma~\ref{lem:Kprime} with
$\Sigma=\Phi$); $\alpha_K$ is additive on products \cite[(2)]{RW93}. For the universal vector extension $E^\natural$ of a CM
curve and $K\supseteq k$ (used only to say when the printed weight gives no conclusion) we have:

\begin{lemma}\label{lem:natural}
$\alpha_K(E^\natural)=2+2/n$.
\end{lemma}

\begin{proof}
In the coordinates $(z,w)$ of \S\ref{sec:Euniv}, let $H\subset(E^\natural)^n$ be connected with $p_\theta(T_H)=0$, with
abelian image $B\subset E^n$. Then $B=B_S$ for a $k$-subspace $S\subset k^n$ on which both $\psi$ (base) and $\phi$ (fibre)
of Lemma~\ref{lem:K} vanish; since $\ker\psi\cap\ker\phi=0$, $B=0$. Hence $H$ lies in the fibre $\Ga^n$, and the largest
such $H$ is the rational hyperplane $\sum\theta_jt_j=0$, of dimension $n-1$. The quotient
has dimension $n+1$ and admits no surjection to $\Ga$ or $\Gm$ (it is a non-split extension of $E^n$ by $\Ga$), so
$\alpha=2(n+1)$ and $\alpha_K=2+2/n$.
\end{proof}

\smallskip\noindent\emph{Laurent's identity.} We use it only in remarks, to say which statements are linear among periods.
Let $\Lambda_E=\Omega\,\Z[\alpha]$ (a sublattice of finite index suffices), and write Masser's relation \cite{Ma75} as
$\eta(\alpha\Omega)=\bar\alpha\eta(\Omega)+c\,\Omega$ with $c\in\Qb$. Legendre's relation gives
$(\alpha-\bar\alpha)\eta(\Omega)-c\Omega=2\pi i/\Omega$, and $f(z)=\zeta(\alpha z)-\bar\alpha\zeta(z)-cz$ is an elliptic
function defined over $\Qb$. With the third-kind period $\lambda(u,\Omega)=\Omega\zeta(u)-\eta(\Omega)u$ one obtains, for
$u\in\Lg_E$ not a pole of $f$,
\begin{equation}\label{eq:laurent}
\frac{2\pi i\,u}{\Omega}=\bar\alpha\,\lambda(u,\Omega)-\lambda(\alpha u,\Omega)+\Omega f(u),\qquad f(u)\in\Qb .
\end{equation}
On $E_i$ this reads $2\eta(\vp)u=i\lambda(iu,\vp)-\lambda(u,\vp)$ (cf.\ \cite[\S9, p.~160]{La82}, where the general CM
form is derived; the display above is its case $\mu=i$). So $\Omega\zeta^*(u)$ and $\eta^*(\Omega)u$ are, up to
$\Qb$-linear combinations, periods of $1$-motives; the identity concerns a period and a logarithm of the same curve.

\section{First-kind period relations as period terms}\label{sec:X}

\subsection{Set-up}
A \emph{first-kind slot} is an eigen-slot of the first kind (\S\ref{sec:setting}), written $j=(A_j,\sigma_j)$ with
$A_j/\Qb$ a simple CM abelian variety (CM field $F_j$, primitive type $\Phi_j$) and $\sigma_j\in\Phi_j$; $z_j$ is the
corresponding eigen-coordinate on $\mathrm{Lie}(A_j)$ and $P_j=z_j(\gamma_j)\ne0$ for a fixed $\gamma_j\in\Omega(A_j)$, so
$z_j(\Omega(A_j))\subset\sigma_j(F_j)P_j$. As in \S\ref{sec:setting}, two slots are equal iff the abelian varieties are
isogenous and the embeddings correspond; distinct slots of one $A$ have distinct embeddings. By (W), periods of distinct slots are $\Qb$-linearly independent. Let $a,b,c,d$ be four pairwise distinct
first-kind slots (distinct in the sense just defined, i.e.\ up to isogeny) with a \emph{first-kind relation}
\begin{equation}\label{eq:R}
P_aP_b=\mu\,P_cP_d,\qquad\mu\in\Qb^\times .
\end{equation}
Let $G=\prod_A A\times\Ga^2$, the product over the isogeny classes among $A_a,A_b,A_c,A_d$, each taken once (so the factors $A$ are pairwise non-isogenous; a slot $z_j$ on an $A_j$ isogenous to the chosen representative
$A$ is read on $A$ through an isogeny, which rescales $P_j$ and $z_j(\Lambda(A_j))$ by $\Qb^\times$), let
$\beta\in\Qb^\times$ and
\[
\Psi=\bigl(z_a+z_d+q_1,\;z_c+z_b+q_2\bigr)\colon T_G\to\C^2,\qquad V=\C w,\quad w=(P_a,\beta P_c),
\]
where $q_1,q_2$ are the $\Ga$-coordinates. Then $w':=(P_d,\beta P_b/\mu)=(P_d/P_a)\,w$ by (\ref{eq:R}). Let $K$ be a
number field containing the Galois closures of all $F_j$, $\mu$ and $\beta$, and $n=[K:\Q]$. By (Ch) and
$z_j(\Omega(A_j))\subset\sigma_j(F_j)P_j$,
\[
\Psi(\Omega_K(G))=\mathcal P:=(KP_a+KP_d)\times(KP_c+KP_b),
\]
so $w,w'\in\Psi(\Omega_K(G))$, and they are $K$-independent since $P_d/P_a\notin\Qb$.

\begin{proposition}\label{prop:X}
$V\cap\Psi(\Lambda_K(G))=Kw+Kw'$.
\end{proposition}

\begin{theorem}[Theorem X]\label{thm:X}
In the set-up above, let $v_j\in\Qb\Lambda(A_j)$ for $j=a,b,c,d$ and $q_1,q_2\in\Qb$ satisfy
\begin{equation}\label{eq:X}
\bigl(z_a(v_a)+z_d(v_d)+q_1\bigr)\cdot\beta P_c=\bigl(z_c(v_c)+z_b(v_b)+q_2\bigr)\cdot P_a .
\end{equation}
Then $q_1=q_2=0$ and $z_j(v_j)\in\Qb P_j$ for each $j$. If $v_j\in\Lambda(A_j)$, then $\exp_{A_j}(v_j)$ is a torsion point.
\end{theorem}


\noindent When several of the slots lie on the same $A$ (as in \S\ref{sec:X7}), the $v_j$ may be logarithms of different
points: by (Ch), a single element of $\Lambda_K(A)$ has slot coordinates $z_j(v_j)$ for any prescribed $v_j$.

\subsection{Proof of Proposition~\ref{prop:X}}
Suppose $v\in V\cap\Psi(\Lambda_K(G))$, $v\notin Kw+Kw'$. Let $\theta$, $p_\theta$ and $\Psi_\theta=\Psi\circ p_\theta$ be as before.

\emph{The group.} For each isogeny class $A$ let $\Sigma_A$ be the set of slots on $A$ used by $\Psi$. By (P) and (S1), the
largest connected subgroup with tangent space in $\ker\Psi_\theta$ is
$U=\prod_AB_{\ker(\psi_\sigma)_{\sigma\in\Sigma_A}}\times\{\theta^{\mathsf T}q_1=\theta^{\mathsf T}q_2=0\}$, and
$G''=G^n/U=\Ga^2\times A''$ with $\dim A''=\sum_A|\Sigma_A|\,n/2=2n$ by Lemma~\ref{lem:Kprime}. So $d=2n+2$, $d_0=2$,
$d_1=0$, $d_2=2n$. The induced $\Psi''\colon T_{G''}\to\C^2$ is onto, $V''=\Psi''^{-1}(V)$ is a hyperplane ($N=2n+1$), and
$W''=\ker\Psi''\subset V''$ is $\Qb$-rational of dimension $2n$.

\emph{The data.} Let $Y''$ be generated by the images of lifts of $D\theta_hw$, $D\theta_hw'$ (in $\Omega(G^n)$) and
$\theta_hv$, $h=1,\dots,n$. Since $U$ is connected, $\Omega(G'')$ is the image of $\Omega(G^n)$, so
$\Psi''(\Omega(G''))\subset\mathcal P$, and $V\cap\mathcal P=Kw+Kw'$ (Lemma~\ref{lem:Lprime} with $L=0$). Hence
$\mathrm{rk}\,Y''=3n$, $\kappa=2n$ exactly, and $\Gamma=\exp Y''$ has rank $n$. Since $d_1+2d_2-\kappa=2n$ and $d-N=1$,
(\ref{eq:41}) reads
\begin{equation}\label{eq:Xineq}
\lambda+2\delta_2\le(2-r')\cdot2n .
\end{equation}

\begin{lemma}[Lemma S$'$]\label{lem:Sprime}
Every connected algebraic subgroup of $G''$ is $L\times B$ with $L$ a $\Qb$-subspace of $\Ga^2$ and $B$ an abelian
subvariety of $A''$. $\Psi''(T_B)$ is spanned by the target directions of the slots on which $B$ is not killed: it is $0$
iff $B=0$; it is $\subset\C(1,0)$ iff the slots $c,b$ are killed, and then $\dim B\le n$; similarly for $\C(0,1)$ (slots
$a,d$ killed); otherwise it is $\C^2$.
\end{lemma}

\begin{proof}
The product decomposition is (P). By (S1) the slot forms $\theta^{\mathsf T}z_j$ on $T(B_S)$ involve independent vectors
$u_\sigma$, and each slot feeds exactly one target coordinate, so $\Psi''(T_B)$ is a coordinate subspace; it is $0$ only for
$B=0$ by maximality of $U$. If only the slots of one target survive, $S/\ker$ embeds into $K^{2}$ via the two surviving
$\psi_\sigma$, so $\dim B\le2\cdot n/2=n$ (Lemma~\ref{lem:Kprime}).
\end{proof}

\begin{lemma}[Lemma L$'$]\label{lem:Lprime}
For $L\in\{0,\ \C(1,0),\ \C(0,1)\}$ one has $V\cap(L+\mathcal P)=Kw+Kw'$.
\end{lemma}

\begin{proof}
If $L\ne\C(1,0)$ and $\xi w\in L+\mathcal P$, then $\xi P_a\in KP_a+KP_d$, so $\xi=x+yP_d/P_a$ ($x,y\in K$) and
$\xi w=xw+yw'$. If $L=\C(1,0)$, then $\xi\beta P_c\in KP_c+KP_b$, so $\xi=x+yP_b/P_c=x+y\mu P_d/P_a$ and $\xi w=xw+y\mu w'$.
\end{proof}

\emph{The case check.} Let $G'=L\times B$ with $r'=\dim\Psi''(T_{G'})\le1$, $e=\dim B$. Then $\delta_2=2n-e$. If
$\Psi''(T_{G'})\in\{0,\C(1,0),\C(0,1)\}$, then $\lambda=n$: if $\exp(\sum c_h\tilde y_h)\in G'$ for the lifts $\tilde y_h$
of $\theta_hv$, then $xv\in\Psi''(T_{G'})+\mathcal P$ with $x=\sum c_h\theta_h\in K$, so $x=0$ by Lemma~\ref{lem:Lprime}.
\begin{center}
\begin{tabular}{lllll}
$G'$ & $r'$ & $\lambda$ & (\ref{eq:Xineq}) reads & holds?\\\hline
$0$ & $0$ & $n$ & $5n\le4n$ & never\\
$B\ne0$, image an axis ($e\le n$), $L\in\{0,\text{same axis}\}$ & $1$ & $n$ & $5n-2e\le2n$ & never ($e\le n$)\\
$L$ an axis, $B=0$ & $1$ & $n$ & $5n\le2n$ & never\\
$L$ a non-axis line, $B=0$ & $1$ & $\ge0$ & $\lambda+4n\le2n$ & never\\
$r'=2$ & & & $\delta>\tau$ fails &
\end{tabular}
\end{center}
So no connected $G'\ne G''$ satisfies the conclusion of Theorem~\ref{thm:W88}, for any $n$; the minimum excess
(left side minus right side) is $n$, attained at $G'=0$ and at axis subgroups with $e=n$. Condition (\ref{eq:NR}) reads
$4n+2\ge2n+4$; $d_0=2$ is allowed (\S\ref{sec:W88}). This contradiction proves
Proposition~\ref{prop:X}. \qed

\begin{remark}[Second derivation]\label{rem:X5d}
Waldschmidt's \cite[\S5d]{W88} ($\kappa$-refined Theorem~1.1, hyperplane case) applies to $(G'',V'',W'',Y'')$ directly:
condition (5.4) there reads $m=3n>(d_1+2d_2-\kappa)(d-1-t)=2n$, and $V''$ contains no nonzero Lie algebra of an algebraic
subgroup, by Lemma~\ref{lem:Sprime} and $P_c/P_a\notin\Qb$. Here $m=\mathrm{rk}\,Y''$; p.~394 of \cite{W88} prints
``$\kappa=\ell-m$'', which must be read $m-\ell$ ($\ell=\mathrm{rk}\,\Gamma$, cf.\ pp.~392--393), as noted after Theorem~\ref{thm:W88k}.

\emph{Relation to \cite[Th\'eor\`eme (iii)]{RW93}.} With the weights of \S\ref{sec:weights}, $\alpha_K(G)=\sum_A\dim A\ge4$, with
equality exactly when the four slots are all the first-kind slots of the varieties involved. In that case (levels $7$ and
$9$ below) Theorem~\ref{thm:RW} with $\kappa=2$ gives $\dim_K(V\cap\Psi(\Lambda_K(G)))\le(4-2)\cdot1=2$, so
Proposition~\ref{prop:X} already follows from it (Remark~\ref{rem:printed}); otherwise (level $24$ below) the printed
weight on $G$ itself is larger, and Theorem~\ref{thm:RW} with the inner weight $\alpha_{K,\Psi}=4$ is used
(\S\ref{sec:RtoApp}). But the level-$24$ quadruple consists of two of the four pairs of a
full family of Theorem~\ref{thm:Xk}, so Theorem~\ref{thm:X} at level $24$ follows from that family by the slot-splitting
step of \S\ref{sec:XfromP} (the paragraph after the proof of Theorem~\ref{thm:X924}).
\end{remark}

\subsection{Proof of Theorem~\ref{thm:X} from Proposition~\ref{prop:X}}\label{sec:XfromP}
Enlarge $K$ to contain $q_1,q_2$ and the coefficients of the $v_j$, so that $v_j\in K\Lambda(A_j)$; the proposition holds
for every such $K$. When several slots lie on one $A$, (Ch) gives a single element of $\Lambda_K(A)$ with the prescribed slot
coordinates $z_j(v_j)$. So hypothesis (\ref{eq:X}) says that the point $\Psi(v_a,\dots,v_d,q_1,q_2)$ lies in
$V\cap\Psi(\Lambda_K(G))$, and Proposition~\ref{prop:X} gives $x,y\in K$ with
\[
z_a(v_a-x\gamma_a)+z_d(v_d-y\gamma_d)+q_1=0,
\]
and similarly in the second coordinate. \emph{Slot splitting.} Write $v_a-x\gamma_a$ and $v_d-y\gamma_d$ as
$K$-combinations of finitely many $u_1,\dots,u_m\in\Lambda(A)$ (for each isogeny class $A$ involved). The point
$\xi=(1,(u_i)_i)\in\mathrm{Lie}(\Ga\times\prod_AA^{m})$ has $\exp\xi$ algebraic, and the displayed relation is a $\Qb$-linear
form $\ell$ vanishing at $\xi$. By W\"ustholz's theorem \cite{Wu89}, the smallest algebraic subgroup $H$ whose Lie algebra
contains $\xi$ has $T_H\subset\ker\ell$. By (P) and (S1), $H=L\times\prod_AB_{S_A}$, and $L=\Ga$ because the $\Ga$-coordinate of $\xi$
is $1$; so $(1,0)\in T_H$ and $q_1=0$. On $T(B_S)$ the slot forms involve independent vectors $u_\sigma$ (S1), and the
slots $a$, $d$ are distinct; so each slot form vanishes on $T_H$ separately, and $z_a(v_a)=xP_a$, $z_d(v_d)=yP_d$.
Likewise $q_2=0$, $z_c(v_c)\in KP_c$ and $z_b(v_b)\in KP_b$. \emph{Torsion.} Let $v_j\in\Lambda(A_j)$ with
$z_j(v_j)=x_jP_j$. If $x_j=0$, W\"ustholz on the simple $A_j$ gives $v_j=0$. Otherwise the smallest algebraic subgroup of
$A_j^2$ whose Lie algebra contains $(v_j,\gamma_j)$ is a proper abelian subvariety $B_S$ (its tangent space lies in
$\ker(z_j\oplus-x_jz_j)$) that projects onto the second factor. So $\dim_FS=1$ and $B_S$ is the graph of some
$\alpha\in F_j=\mathrm{End}^0(A_j)$; hence $v_j\in F_j\gamma_j\subset\Q\,\Omega(A_j)$ and $\exp(v_j)$ is torsion. \qed

\subsection{The level-seven instance}\label{sec:X7}
Let $C\colon y^7=x(1-x)$. It has genus $3$ (branch points $0,1,\infty$), and $dx/y^k$ is holomorphic exactly for
$k=4,5,6$ (orders $6-k$ at $0$ and $1$, $2k-8$ at $\infty$). The automorphism $y\mapsto\zeta y$ ($\zeta=e^{2\pi i/7}$)
multiplies $dx/y^6,dx/y^5,dx/y^4$ by $\zeta,\zeta^2,\zeta^3$, so $A=\mathrm{Jac}(C)$ has CM by $\Q(\zeta_7)$ with type
$\{1,2,3\}$, which has trivial stabiliser in $(\Z/7)^\times$; so $A$ is simple. Let $z_1,z_2,z_3$ be the eigen-coordinates
dual to $dx/y^6,dx/y^5,dx/y^4$. On the closed cycle $\gamma$ = (lift of $[0,1]$ to the sheet $y>0$) minus (its image under
$y\mapsto\zeta y$),
\[
\int_\gamma\frac{dx}{y^{7-j}}=(1-\zeta^{j})\,B(\tfrac j7,\tfrac j7)=:P_j\qquad(j=1,2,3).
\]
Let $E=E_{-7}$ be an elliptic curve over $\Qb$ with CM by $\Q(\sqrt{-7})$; by Chowla--Selberg \lab{Lit} \cite{Gr78} it has a period
$\Omega\in\Qb^\times B(\tfrac17,\tfrac27)$. The Beta cocycle $B(x,y)B(x+y,z)=B(y,z)B(x,y+z)$ at $(\tfrac17,\tfrac17,\tfrac27)$
and one reflection give, exactly,
\[
B(\tfrac17,\tfrac17)B(\tfrac27,\tfrac27)=B(\tfrac17,\tfrac27)B(\tfrac17,\tfrac37),\qquad
B(\tfrac17,\tfrac37)=\mu\,B(\tfrac37,\tfrac37),
\]
\[
\mu=\frac{\Gamma(\frac17)\Gamma(\frac67)}{\Gamma(\frac37)\Gamma(\frac47)}=\frac{\sin(3\pi/7)}{\sin(\pi/7)}=1+2\cos\tfrac{2\pi}7,
\]
so $P_1P_2\in\Qb^\times P_3\Omega$, a relation (\ref{eq:R}) with $a=z_1$, $b=z_2$, $c=z_3$, $d=E$. (In Koblitz--Ogus labels:
$\{1,2,3\}+\{1,4,5\}=\{1,3,5\}+\{1,2,4\}$; the label $\{1,2,4\}$ is induced from $\Q(\sqrt{-7})$.) No use of Deligne's theorem
is needed at this level.

\begin{theorem}[Theorem X7]\label{thm:X7}
Let $v_1,v_2,v_3\in\Lambda(A)$, $u\in\Lg_E$, $q_1,q_2\in\Qb$ and $\beta\in\Qb^\times$. If
\[
\bigl(z_1(v_1)+u+q_1\bigr)\cdot\beta P_3=\bigl(z_3(v_3)+z_2(v_2)+q_2\bigr)\cdot P_1,
\]
then $q_1=q_2=0$ and $\exp(v_1),\exp(v_2),\exp(v_3)$, $\exp_E(u)$ are torsion points.
\end{theorem}

\noindent This is Theorem~\ref{thm:X} (with $\Omega$ normalised so that $P_1P_2=\mu'P_3\Omega$ and $\mu'\in K$); here
$n$ is a multiple of $6$ and the margin is $n$. Since $z_1,z_2,z_3$ are all the first-kind slots of $A$,
Proposition~\ref{prop:X} for these slots already follows from Theorem~\ref{thm:RW}
($\alpha_K(A\times E\times\Ga^2)=3+1$, $\kappa=2$; Remark~\ref{rem:X5d}, Remark~\ref{rem:printed}).


\begin{corollary}\label{cor:X7}
\begin{itemize}
\item[(a)] For every non-torsion point $\exp(v)\in A(\Qb)$: $z_2(v)/z_1(v)\notin\Qb\cdot P_3/P_1=\Qb\cdot
B(\tfrac37,\tfrac37)/B(\tfrac17,\tfrac17)$. In particular Corollary~\ref{cor:X7intro}(a) holds.
\item[(b)] For every non-torsion $u\in\Lg_E$ and every $v\in\Lambda(A)$: $z_3(v)/u\notin\Qb^\times\cdot P_3/P_1$.
\item[(c)] $P_1P_2$, $P_1P_3$, $P_1$, $P_3$ are $\Qb$-linearly independent; equivalently, Corollary~\ref{cor:X7intro}(b).
\end{itemize}
\end{corollary}

\noindent That (c) is not a consequence of the analytic subgroup
theorem is \lab{Heuristic} (Remark~\ref{rem:notlin}).

\begin{proof}
(a) Data $v_1=v_2=v$, $v_3=0$, $u=0$, $q_1=q_2=0$, $\beta=\gamma$: if $z_2(v)=\gamma\,z_1(v)P_3/P_1$ then the hypothesis of
Theorem~\ref{thm:X7} holds, so $\exp(v)$ is torsion. The values $0$ and $\infty$ of the ratio do not occur for $v\ne0$:
$z_j(v)=0$ forces $v=0$ by W\"ustholz on the simple $A$. For $O,Q\in C(\Qb)$, $z_j$ of the Abel--Jacobi logarithm of
$Q-O$ is $\int_O^Q$ of the dual differential. (b) Data $v_3=v$, $u$, $\beta=\gamma\in\Qb^\times$, all others $0$.
(c) Substituting (\ref{eq:R}) into $\alpha_1P_1P_2+\alpha_2P_1P_3+\alpha_3P_1+\alpha_4P_3=0$ gives
$(\alpha_2P_1+\alpha_1\mu'\Omega+\alpha_4)\cdot P_3=(-\alpha_3)\cdot P_1$. So, with $\beta=1$ and $K\ni\alpha_i$,
$v=(\alpha_2P_1+\alpha_1\mu'\Omega+\alpha_4,\,-\alpha_3)\in V\cap\Psi(\Lambda_K(G))$, and Proposition~\ref{prop:X} gives
$v=(xP_1+y\Omega,\ xP_3+yP_2/\mu')$ with $x,y\in K$. The $\Qb$-independence of $1,P_2,P_3$ (W) gives $x=y=\alpha_3=0$, and
then that of $1,P_1,\Omega$ gives $\alpha_1=\alpha_2=\alpha_4=0$. Dividing the four numbers by $P_1$ gives
$P_2,P_3,1,P_3/P_1$, which are $B(\tfrac27,\tfrac27),B(\tfrac37,\tfrac37),1,B(\tfrac37,\tfrac37)/B(\tfrac17,\tfrac17)$ up to
factors in $\Q(\zeta_7)^\times$. The Gamma form follows from $B(\tfrac17,\tfrac17)=\Gamma(\tfrac17)^2/\Gamma(\tfrac27)$ (the
definition) and, by the reflection formula, $B(\tfrac27,\tfrac27)=\Gamma(\tfrac27)^2\Gamma(\tfrac37)\sin(\tfrac{3\pi}7)/\pi$
and $B(\tfrac37,\tfrac37)=\Gamma(\tfrac37)^2\Gamma(\tfrac17)\sin(\tfrac\pi7)/\pi$, after multiplying by $\pi$.
\end{proof}

\noindent \emph{No hidden symmetry.} $C$ is hyperelliptic: $(2x-1)^2=1-4y^7$, and $x\mapsto1-x$ is the hyperelliptic
involution, acting as $[-1]$ on $A$ and multiplying $z_1,z_2,z_3$ by $-1$ together. $\mathrm{Aut}(C)\cong\Z/14$ and
$\mathrm{End}^0(A)=\Q(\zeta_7)$ acts diagonally, $z_j([x]v)=\sigma_j(x)z_j(v)$; so no symmetry links $z_1,z_2,z_3$ of one
point in a way that could produce $P_3/P_1$ in (a).

\subsection{Remarks: linearisation and other relations}\label{sec:Xlin}
\begin{remark}[Not linearisations]\label{rem:notlin}
\lab{Heuristic} By an abelian analogue of Laurent's identity (\ref{eq:laurent}), which we do not reproduce, each
$\ell_j:=\pi z_j(v)/P_j$ is a $\Qb$-combination of $1$-periods; it linearises only products $P_{\bar\sigma}z_\sigma$ of a
period with the logarithm of the conjugate slot on the same variety. Dividing (\ref{eq:X}) by $P_aP_c$ and using
(\ref{eq:R}) gives
\[
\ell_c-\beta\ell_a+\frac{P_d}{P_a}\,(\mu\ell_b-\beta\ell_d)+\pi\Bigl(\frac{q_2}{P_c}-\frac{\beta q_1}{P_a}\Bigr)=0,
\]
where the ratio $P_d/P_a$ of periods of two different slots multiplies a combination of $1$-periods. Unless
$\mu\ell_b=\beta\ell_d$ this is not a $\Qb$-linear relation among $1$-periods, so the analytic subgroup theorem does not
apply as it does to (\ref{eq:laurent}). For (c): $P_3/P_1\in\Qb^\times P_3P_{\bar1}/\pi$ has Koblitz--Ogus function
$\mathbf 1_{\{1,3,5\}}-\mathbf 1_{\{1,2,3\}}$, which takes three values, while every slot period and $\pi$ have functions
with at most two values; under the Rohrlich--Lang (or period) conjecture, $P_3/P_1$ is not in the $\Qb$-span of
$1$-periods, and W\"ustholz's theorem does not decide (c). Chudnovsky's theorem gives the algebraic independence of $\pi$
and $\Omega_{-7}\sim\Gamma(\frac17)\Gamma(\frac27)/\Gamma(\frac37)$, which does not suffice, since (a)--(c) involve
$P_1,P_2,P_3$. By the period conjecture (the Mumford--Tate group of $A$ is a $4$-dimensional torus and $\Omega$ adds no
transcendence degree), Theorem~\ref{thm:X7} is a consequence of the period conjecture.
\end{remark}

\begin{remark}[No Gamma-monomial multiplier]\label{rem:multiplier}
\lab{Computed} In the dictionary of Koblitz and Ogus (\S\ref{sec:setting}), a product of values $\Gamma(a/N)$ and of a power of
$\pi$ corresponds to a $\Q$-valued function on $(\Z/N)^\times$, with $\Gamma(a/N)\mapsto(t\mapsto1-\langle ta/N\rangle)$ and
$\pi\mapsto1$, products going to sums; by Deligne's theorem, products with the same function have an algebraic ratio. The numbers
$1$, $\pi$ and the eigen-periods, of either kind, attached to the CM types of $\Q(\zeta_N)$ correspond (up to $\Qb^\times$) to the
functions $0$, $1$ and the indicators of the half-sets $\Phi\subset(\Z/N)^\times$ with $\Phi\sqcup(-\Phi)=(\Z/N)^\times$. An exact
computation (a Python script of about fifty lines) checks, for each of the following lists, that there is no function $g$ on
$(\Z/N)^\times$ such that $g+f$ is one of these functions for every member $f$ of the list:
\begin{itemize}
\item $1$, $B(\frac27,\frac27)$, $B(\frac37,\frac37)$, $B(\frac37,\frac37)/B(\frac17,\frac17)$ (Corollary~\ref{cor:X7}(c));
\item $1$, $B(\frac49,\frac49)$, $B(\frac29,\frac29)$, $B(\frac29,\frac29)/B(\frac19,\frac19)$ (Corollary~\ref{cor:X924}(2));
\item $1$, $\vp$, $B(\frac1{24},\frac1{12})$, $B(\frac1{24},\frac1{12})/B(\frac1{24},\frac1{24})$ (Corollary~\ref{cor:X924}(4), with
$\vp=\frac12B(\frac14,\frac12)$);
\item the six numbers of the example after Theorem~\ref{thm:Xk} ($N=15$, $\kappa=4$).
\end{itemize}
Since each list contains $1$, $g$ itself must be one of these functions, so the search is finite. As controls, the list $1$, $r$,
$P_{\Phi'}$ of a single pair at $N=15$ and the list $1$, $P_3/P_1$ at $N=7$ do admit such a $g$. The classical Gamma identities (reflection, multiplication, the
Beta cocycle) preserve these functions, so no Gamma monomial can be shown by them to turn all the numbers of one of these lists
into $\Qb^\times$-multiples of $1$, $\pi$ or CM eigen-periods at once; under the Rohrlich--Lang conjecture no Gamma monomial does \lab{Heuristic}. So no reduction of this kind to W\"ustholz's theorem or to the period theorem for $1$-periods of Huber and W\"ustholz
\cite{HW} can be obtained from the classical Gamma identities; under the Rohrlich--Lang conjecture none exists \lab{Heuristic}. This is more than Remark~\ref{rem:notlin}, which concerns $P_3/P_1$ alone.
\end{remark}

\begin{remark}[Where first-kind relations occur]\label{rem:where}
\emph{Elliptic slots.} Among CM elliptic slots (any number of non-isogenous curves) the relations of Deligne's theorem
are only the Legendre relations \lab{Proved}: the indicators are $(1\pm\chi_k)/2$ with $\chi_k$ odd quadratic characters,
which are linearly independent. (That there are no other relations is the Rohrlich--Lang conjecture, \lab{Heuristic}.)
So for products of CM elliptic curves, the two-curve statements of \cite[\S6]{Note} (two Legendre relations on a line) are the
only constructions of this kind.

\emph{Labels.} \lab{Proved} (elementary combinatorics). For CM types of $\Q(\zeta_N)$ containing $1$, a first-kind
partner of $(\Phi_a,\Phi_b)$ exists iff $\Phi_a,\Phi_b$ differ on at least two $\pm$-pairs: with
$\Delta=\Phi_a\triangle\Phi_b$, the partners are $(\Phi_a\cap\Phi_b)\cup X$ and $(\Phi_a\cap\Phi_b)\cup(\Delta\setminus X)$, $X$
one element from each $\pm$-pair in $\Delta$, $X\notin\{\Phi_a\setminus\Phi_b,\Phi_b\setminus\Phi_a\}$. Hence first-kind
relations among labels exist at every level $N$ with $\varphi(N)/2\ge3$. The passage from labels to periods is Deligne's
theorem \lab{Lit} (\S\ref{sec:setting} for Beta values; in general \cite{De82}).

\emph{Examples.} The smallest levels are $N=7$ and $N=9$, with one relation each: at $N=7$ the instance above; at $N=9$
one with slots of a simple CM $3$-fold with CM by $\Q(\zeta_9)$ and of $E_\rho$,
$B(\tfrac19,\tfrac19)B(\tfrac19,\tfrac49)=3^{1/3}\frac{\sin(2\pi/9)}{\sin(\pi/9)}B(\tfrac19,\tfrac29)B(\tfrac13,\tfrac13)$
(\lab{Proved}, Lemma~\ref{lem:rel924}, from Gauss's multiplication formula and the reflection formula). At $N=24$ there is one whose slots are two CM elliptic curves (CM by $\Q(\sqrt{-6})$ and $\Q(i)$) and two slots
of a simple CM $4$-fold: $B(\tfrac1{24},\tfrac1{24})B(\tfrac1{24},\tfrac14)\in\Qb^\times
B(\tfrac1{24},\tfrac1{12})B(\tfrac1{24},\tfrac18)$ (\lab{Proved}, Lemma~\ref{lem:rel924}). Both give instances of
Theorem~\ref{thm:X} (\S\ref{sec:X924}).
\end{remark}

\subsection{Levels $9$ and $24$, and several pairs}\label{sec:X924}

\begin{lemma}\label{lem:rel924}
\begin{itemize}
\item[(1)] $B(\tfrac19,\tfrac19)\,B(\tfrac19,\tfrac49)=3^{1/3}\,\frac{\sin(2\pi/9)}{\sin(\pi/9)}\,B(\tfrac19,\tfrac29)\,B(\tfrac13,\tfrac13)$,
and $B(\tfrac19,\tfrac19)\,B(\tfrac49,\tfrac49)=16\sin(\tfrac\pi9)\sin^3(\tfrac{2\pi}9)\,B(\tfrac29,\tfrac29)\,B(\tfrac13,\tfrac13)$.
\item[(2)] $B(\tfrac1{24},\tfrac1{24})\,B(\tfrac1{24},\tfrac14)=\mu_{24}\,B(\tfrac1{24},\tfrac1{12})\,B(\tfrac1{24},\tfrac18)$, where
\[\mu_{24}=2^{-1/12}\,3^{1/8}\,\frac{\sin(7\pi/24)\sin(9\pi/24)}{\sin(10\pi/24)\sin(6\pi/24)}.\]
\end{itemize}
\end{lemma}

\noindent Both sides of each identity also agree numerically to $60$ digits \lab{Numerical}.

\begin{proof}
We use Euler's reflection formula $\Gamma(z)\Gamma(1-z)=\pi/\sin\pi z$ and Gauss's multiplication formulas
$\Gamma(z)\Gamma(z+\frac12)=2^{1-2z}\sqrt\pi\,\Gamma(2z)$ and $\Gamma(z)\Gamma(z+\frac13)\Gamma(z+\frac23)=2\pi\,3^{1/2-3z}\Gamma(3z)$.

(1) With $g_k=\Gamma(k/9)$ and $s_k=\sin(k\pi/9)$, the case $m=3$ at $z=1/9,2/9$ gives $g_1g_4g_7=2\pi\,3^{1/6}g_3$ and
$g_2g_5g_8=2\pi\,3^{-1/6}g_6$, and reflection gives $g_1g_8=\pi/s_1$, $g_2g_7=\pi/s_2$. Hence
$g_4=2\cdot3^{1/6}s_2\,g_2g_3/g_1$ and $g_5=2\cdot3^{-1/6}s_1\,g_1g_6/g_2$, and the first ratio equals
$g_1^2g_4g_6/(g_2^2g_3g_5)=3^{1/3}s_2/s_1$. The second identity follows from the first combined with
$B(\frac19,\frac29)/B(\frac29,\frac29)=2\cdot3^{1/6}s_2$ and $B(\frac19,\frac49)/B(\frac49,\frac49)=s_4/s_1$ (consequences
of the same formulas) and $8s_1s_2s_4=\sqrt3$.

(2) With $g_k=\Gamma(k/24)$ the left side divided by $B(\frac1{24},\frac1{12})B(\frac1{24},\frac18)$ is
$g_1g_4g_6/(g_2^2g_7)$. Write $R_k$ for reflection at $z=k/24$, $D_z$ for duplication at $z$ and $T_z$ for triplication at $z$.
The combination $R_6-R_7-R_9+R_{10}-D_{2/24}+D_{3/24}+T_{1/24}-T_{2/24}$ (exponents of the $g_k$ added) has exponent vector
$e_1=e_4=e_6=1$, $e_2=-2$, $e_7=-1$, all others $0$; its constants multiply to $\mu_{24}$ (the powers of $\pi$ cancel; the
powers of $2$ give $2^{-5/6+3/4}=2^{-1/12}$, those of $3$ give $3^{3/8-1/4}=3^{1/8}$).
\end{proof}

\noindent\emph{The slots} \lab{Lit} (Koblitz--Ogus labels \cite{KO79}; \S\ref{sec:setting}; Chowla--Selberg \cite{Gr78}).
At level $9$, $B(\frac19,\frac19)$, $B(\frac29,\frac29)$, $B(\frac49,\frac49)$ are, up to factors in $\Q(\zeta_9)^\times$,
periods of three distinct first-kind eigen-differentials $dx/y^8$, $dx/y^7$, $dx/y^5$ of the curve $C_9\colon y^9=x(1-x)$
(genus $4$), whose Jacobian is isogenous to $A_9\times E_\rho$ with $A_9$ a simple CM threefold (CM by $\Q(\zeta_9)$, type
$\{1,2,4\}$), and $B(\frac13,\frac13)\in\Qb^\times\Omega_\rho$ is the period of $dx/y^6$, which comes from $E_\rho$. At level
$24$, $B(\frac1{24},\frac1{24})\in\Qb^\times\Omega_{-6}$ and $B(\frac1{24},\frac14)\in\Qb^\times\vp$ are periods of CM curves
$E_{-6}$, $E_i$ with CM by $\Q(\sqrt{-6})$ and $\Q(i)$, and $B(\frac1{24},\frac1{12})$, $B(\frac1{24},\frac18)$ are, up to
$\Qb^\times$, periods $P_c$, $P_d$ of two distinct first-kind eigen-coordinates $z_c,z_d$ of a simple CM fourfold $A_{24}$
(CM by $\Q(\zeta_{24})$). By Lemma~\ref{lem:rel924}, $P_1P_4\in\Qb^\times P_2\Omega_\rho$ and $\Omega_{-6}\vp\in\Qb^\times
P_cP_d$, so Theorem~\ref{thm:X} applies to both quadruples.

\begin{theorem}[Theorems X9 and X24]\label{thm:X924}
\begin{itemize}
\item[(1)] \textup{(Level $9$.)} On $A_9$ let $z_1$, $z_2$, $z_4$ be the coordinates dual to the eigen-differentials $dx/y^8$, $dx/y^7$,
$dx/y^5$, with periods $P_j\in\Qb^\times B(\frac j9,\frac j9)$. If $v_1,v_2,v_4\in\Lambda(A_9)$, $u\in\Lg_{E_\rho}$, $q_1,q_2\in\Qb$,
$\beta\in\Qb^\times$ and $(z_1(v_1)+u+q_1)\beta P_2=(z_2(v_2)+z_4(v_4)+q_2)P_1$, then $q_1=q_2=0$ and all of $\exp(v_j)$,
$\exp_{E_\rho}(u)$ are torsion.
\item[(2)] \textup{(Level $24$.)} If $u\in\Lg_{E_{-6}}$, $u'\in\Lg_{E_i}$, $v_c,v_d\in\Lambda(A_{24})$, $q_1,q_2\in\Qb$,
$\beta\in\Qb^\times$ and $(u+z_d(v_d)+q_1)\beta P_c=(z_c(v_c)+u'+q_2)\Omega_{-6}$, then $q_1=q_2=0$ and all exponentials are
torsion.
\end{itemize}
\end{theorem}

\begin{corollary}\label{cor:X924}
\begin{itemize}
\item[(1)] For every non-torsion point $\exp(v)\in A_9(\Qb)$, the ratio $z_4(v)/z_1(v)$ is not in
$\Qb\,B(\frac29,\frac29)/B(\frac19,\frac19)$.
\item[(2)] The numbers $1$, $B(\frac49,\frac49)$, $B(\frac29,\frac29)$, $B(\frac29,\frac29)/B(\frac19,\frac19)$ are linearly
independent over $\Qb$.
\item[(3)] For non-torsion $u\in\Lg_{E_{-6}}$ and every $u'\in\Lg_{E_i}$: $u'/u\notin\Qb^\times
B(\frac1{24},\frac1{12})/B(\frac1{24},\frac1{24})$.
\item[(4)] The numbers $1$, $\vp$, $B(\frac1{24},\frac1{12})$, $B(\frac1{24},\frac1{12})/B(\frac1{24},\frac1{24})$ are
linearly independent over $\Qb$.
\end{itemize}
\end{corollary}

\noindent That (2) and (4) are not consequences of the analytic subgroup theorem is \lab{Heuristic}, as for
Corollary~\ref{cor:X7}(c) (Remark~\ref{rem:notlin}).

\begin{proof}[Proof of Theorem~\ref{thm:X924} and Corollary~\ref{cor:X924}]
As for Theorem~\ref{thm:X7} and Corollary~\ref{cor:X7}, taking for $(a,b,c,d)$ the slots
$(z_1,z_4,z_2,E_\rho)$, resp.\
$(E_{-6},E_i,z_c,z_d)$ in the relation (\ref{eq:R}) and in (\ref{eq:X}). The case analysis of Proposition~\ref{prop:X} uses only that the four
slots are distinct and of the first kind, the relation (\ref{eq:R}), and $\dim A''=2n$, which holds here ($3n/2+n/2$, resp.\
$n+n/2+n/2$, by Lemma~\ref{lem:Kprime}). In (3), Theorem~\ref{thm:X924}(2) makes $u$ torsion, so no condition on $u'$ is
needed.
\end{proof}

\noindent At level $9$ the slots $z_1,z_2,z_4$ are all the first-kind slots of $A_9$, so, as at level $7$,
Proposition~\ref{prop:X} for them already follows from Theorem~\ref{thm:RW} ($\alpha_K=3+1$, $\kappa=2$;
Remark~\ref{rem:X5d}). At level $24$ only two of the four first-kind slots of $A_{24}$ are used ($\alpha_K=4+1+1$), so the
printed weight on its own group gives the bound $(6-2)\cdot1=4$ and no conclusion; Theorem~\ref{thm:RW} with the inner weight
applies (\S\ref{sec:RtoApp}). Moreover its two
pairs are two of the four pairs of the full family $T(11)$ at $N=24$ of Theorem~\ref{thm:Xk} below: in Koblitz--Ogus labels
\cite{KO79}, $B(\frac1{24},\frac1{24})\leftrightarrow\{1,5,7,11\}\mapsto B(\frac1{24},\frac1{12})\leftrightarrow\{1,5,7,13\}$
($E_{-6}\to A_{24}$) and $B(\frac1{24},\frac18)\leftrightarrow\{1,5,11,17\}\mapsto B(\frac1{24},\frac14)\leftrightarrow
\{1,5,13,17\}$ ($A_{24}\to E_i$); the other two pairs are $\{1,7,11,19\}\mapsto\{1,7,13,19\}$ ($A_{24}$ to a curve with CM by
$\Q(\sqrt{-3})$) and $\{1,11,17,19\}\mapsto\{1,13,17,19\}$ (a curve with CM by $\Q(\sqrt{-2})$ to $A_{24}$). So it follows
from item (iii) of Remark~\ref{rem:printed} by the slot-splitting step of \S\ref{sec:XfromP}: (Ch) moves a point of $\Lambda_K$ of the
level-$24$ group into $\Lambda_K$ of the group of $T(11)$ with the same $\Psi$-image (after rescaling the second target by a
constant in $\Qb^\times$, which keeps it an instance), Theorem~\ref{thm:Xk} applies, and W\"ustholz's theorem splits the
slots, which kills the coefficients of the two unused pairs. This is the same kind of input as the W\"ustholz and Masser
steps used for the other direct instances. Hence Theorem~\ref{thm:X924}(2) and Corollary~\ref{cor:X924}(3) also
follow from the printed form \cite[Th\'eor\`eme (iii)]{RW93} (with W\"ustholz's theorem, and Deligne's theorem for the common slope of
$T(11)$ as in the proof of Theorem~\ref{thm:Xk}), and Corollary~\ref{cor:X924}(4) is part of the
independence statement of Theorem~\ref{thm:Xk} for $T(11)$ (there $r\in\Qb^\times B(\frac1{24},\frac1{12})/B(\frac1{24},
\frac1{24})$, and the $P_{\Phi'}$ include $B(\frac1{24},\frac1{12})$ and $B(\frac1{24},\frac14)\in\Qb^\times\vp$). The same
applies to every proper subfamily of Theorem~\ref{thm:Xk}. What is ours at level $24$ is the elementary proof of the
relation (Lemma~\ref{lem:rel924}(2)), which avoids Deligne's theorem.

\begin{theorem}[Theorem X$^\kappa$: several pairs]\label{thm:Xk}
Let $N\ge3$, $G_N=(\Z/N)^\times$ of order $2g$, $\tau\in G_N\setminus\{\pm1\}$, and let $T(\tau)$ be the set of CM types
$\Phi\subset G_N$ with $1,\tau\in\Phi$ (so $\kappa:=|T(\tau)|=2^{g-2}$). For $\Phi\in T(\tau)$ put
$\Phi'=(\Phi\setminus\{\tau\})\cup\{-\tau\}$, and let $P_\Phi$, $P_{\Phi'}$ be periods of the corresponding first-kind slots
(the eigen-slot at the embedding $1$ of a CM abelian variety with CM by $\Q(\zeta_N)$ of type $\Phi$, resp.\ $\Phi'$). Then
the $2\kappa$ slots are distinct, there are $\beta_\Phi\in\Qb^\times$ and $r$ with $\beta_\Phi P_{\Phi'}=rP_\Phi$ for all
$\Phi$, and with $G=\Ga^2\times\prod A$ (each isogeny class once), $\Psi=(\sum_\Phi z_\Phi+q_1,\
\sum_\Phi\beta_\Phi z_{\Phi'}+q_2)$, $V=\C(1,r)$ and every number field $K\supseteq\Q(\zeta_N)$ containing the $\beta_\Phi$:
\[ V\cap\Psi(\Lambda_K(G))=\bigoplus_{\Phi\in T(\tau)}K\,P_\Phi(1,r). \]
Consequently $1$, $r$ and the $\kappa$ numbers $P_{\Phi'}$ are linearly independent over $\Qb$. The same holds for every
subfamily of $T(\tau)$.
\end{theorem}

\begin{proof}
\emph{Distinctness.} $\Phi\ni\tau$, $\Phi'\not\ni\tau$, and $\Phi\mapsto\Phi'$ is injective, so the $2\kappa$ labels are
distinct; distinct labels give distinct slots, since $A_{\Phi_2}\sim A_\Phi$ forces $\Phi_2=\Phi u$ and then the
$1$-eigenline of $A_{\Phi_2}$ is the $u^{\mp1}$-eigenline of $A_\Phi$, another embedding unless $u\in\mathrm{Stab}\,\Phi$. For an induced type the slot lives on the simple factor $B$ with CM field
$F_0=\Q(\zeta_N)^{\mathrm{Stab}\,\Phi}$ and primitive type $\Phi_0$ (the factors $A$ of $G$ are these simple factors, each isogeny class
once). Isogenous simple factors have equal abelian CM fields $F_0\subset\Q(\zeta_N)$, so $B_{\Phi_0'}\sim B_{\Phi_0}$ forces
$\Phi_0'=\Phi_0u_0$, and the embedding $1|_{F_0}$ of $B_{\Phi_0'}$ is $u_0^{\mp1}|_{F_0}$ on $B_{\Phi_0}$, which is $1|_{F_0}$ only if
$u_0\in\mathrm{Stab}\,\Phi_0=1$; so distinct types give distinct slots in this case too.
\emph{Common slope.} $\mathbf 1_{\Phi'}-\mathbf 1_\Phi=\delta_{-\tau}-\delta_\tau$ for all $\Phi$, so
$P_{\Phi_1'}P_{\Phi_2}\in\Qb^\times P_{\Phi_1}P_{\Phi_2'}$: for Fermat labels by Deligne's theorem \cite{De82} in the form of
\cite{KO79} (\S\ref{sec:setting}), and in general by the linearity in $\Phi$ of Shimura's period symbol, proved by Blasius
\cite{Bl86} (via Deligne's theorem on absolute Hodge cycles) and, for abelian CM fields, by Anderson \cite{An82}.
\emph{The count.} The proof of Proposition~\ref{prop:X} goes through with $\kappa$ pairs: $G''=\Ga^2\times A''$ with
$\dim A''=\kappa n$ (Lemma~\ref{lem:Kprime}); the periods contribute $\kappa n$ to $\mathrm{rk}(Y''\cap\ker\exp)$; one extra
datum gives $m=(\kappa+1)n$; every slot occurs in one target only, and distinct slots (several on one $A$ are distinct
embeddings) carry independent coordinates on $T_B$ (S1), so $\Psi''(T_B)\in\{0,\C(1,0),\C(0,1),\C^2\}$; and for an axis
image $\ell$, $V\cap(\ell+\Psi(\Omega_K(G)))=\bigoplus_\Phi KP_\Phi(1,r)$ (as in Lemma~\ref{lem:Lprime}), whence $\lambda=n$.
Condition (\ref{eq:41}) then reads $n+2\kappa n\le2\kappa n$ for $G'=0$, $n+2(\kappa n-e)\le\kappa n$ with $e\le\kappa n/2$ for axis
images, and $\lambda+2\kappa n\le\kappa n$ for a non-axis line in $\Ga^2$: never (margin $n$). Condition (\ref{eq:NR}):
$2\kappa n+2\ge\kappa n+4$. \emph{Independence.} A relation $a_0+a_1r+\sum\gamma_\Phi P_{\Phi'}=0$ ($K$ containing the
coefficients) gives the point $(a_1,-a_0-\sum\gamma_\Phi P_{\Phi'})=a_1(1,r)$ of $V\cap\Psi(\Lambda_K(G))$, whose first
coordinate must lie in $\sum KP_\Phi$; so $a_1=0$ by (W), and then $\gamma=0$, $a_0=0$ by (W).
\end{proof}

\begin{remark}
For one pair ($\kappa=1$, i.e.\ $g=2$, or a one-pair subfamily) the independence statement follows from W\"ustholz's
theorem alone (multiply by $\pi/P_{\Phi'}$, which makes every member a $1$-period); for $\kappa\ge2$ we know no common
Gamma-monomial multiplier that does this (Remark~\ref{rem:multiplier} excludes one for the example below), and we
do not know a linear proof. (That there is none is \lab{Heuristic}, as in
Remark~\ref{rem:notlin}.) For the whole family $T(\tau)$ the theorem already follows from the printed form \cite[Th\'eor\`eme (iii)]{RW93}
\lab{Proved}: every CM type contains exactly one of $\tau,-\tau$, so the $2\kappa$ types $\Phi,\Phi'$ are all the CM types
of $\Q(\zeta_N)$ containing $1$, and the $2\kappa$ slots are all the first-kind slots of the simple varieties involved;
hence $\alpha_K(G)=\sum_A\dim A=2\kappa$, and Theorem~\ref{thm:RW} gives $\dim_K\le(2\kappa-\kappa)\cdot1=\kappa$
(Remark~\ref{rem:printed}). This covers the example below. For a proper subfamily the printed weight on its own group can be
larger, but the subfamily follows from its full family $T(\tau)$ by the slot-splitting step of \S\ref{sec:XfromP}, and its
independence statement is part of that of the full family (as at level $24$, after the proof of
Theorem~\ref{thm:X924}).
\end{remark}

\begin{example}
$N=15$, $\tau=2$: the four pairs $B(\frac1{15},\frac2{15})\mapsto B(\frac4{15},\frac4{15})$,
$B(\frac1{15},\frac3{15})\mapsto B(\frac1{15},\frac7{15})$, $B(\frac2{15},\frac2{15})\mapsto B(\frac2{15},\frac6{15})$,
$B(\frac1{15},\frac1{15})\mapsto B(\frac13,\frac13)$ form the whole family $T(2)$ ($\kappa=4$; their common slope class was also
checked exactly), so $1$, $B(\frac4{15},\frac4{15})/B(\frac1{15},\frac2{15})$, $B(\frac4{15},\frac4{15})$,
$B(\frac1{15},\frac7{15})$, $B(\frac2{15},\frac6{15})$, $B(\frac13,\frac13)$ are linearly independent over $\Qb$.
\end{example}

\section{Strong elliptic four exponentials on eigen-slots}\label{sec:SE}
In this section the strong four exponentials statement is read on a CM curve or on one eigen-coordinate of a CM abelian
variety instead of on $\Gm$. Let $E,E'$ be elliptic curves over $\Qb$ with complex multiplication by $k,k'$,
$\Omega,\Omega'$ nonzero periods, $\eta=\eta^*(\Omega)$ and $\eta'=\eta^*(\Omega')$ the conjugate-eigen quasi-periods
($\Omega\eta,\Omega'\eta'\in\Qb^\times\pi$). Write $\Lt_E=\Qb+\Qb\Lg_E$, and $\Fib_E$, $\Fib_{E'}$ for the sets of fibre
values (\S\ref{sec:Euniv}; $\Fib_E$ contains $\Qb$, the values $\zeta^*(u)$, $u\in\Lg_E$, and $\eta$). Two CM curves are
isogenous iff $k=k'$.

\begin{theorem}[Theorems SE, SE$^\natural$]\label{thm:SE}
Let $c\in\Qb^\times$ and $y_1,y_2\in\Lt_E$ with $\Omega,y_1,y_2$ linearly independent over $\Qb$.
\begin{itemize}
\item[(1)] \textup{(SE)} If $k\neq k'$ and $t=c\Omega'/\Omega$, then $ty_1,ty_2$ are not both in $\Qb+\Qb\Lg_E+\Qb\Lg_{E'}$.
\item[(2)] \textup{(SE$^\natural$)} If $k\neq k'$ and $t=c\eta'/\Omega$, then $ty_1,ty_2$ are not both in $\Qb\Lg_E+\Qb\Fib_{E'}$.
\end{itemize}
\end{theorem}

\noindent The analogue at $t=c\eta/\Omega\in\Qb^\times\pi/\Omega^2$, with $ty_1,ty_2$ not both in $\Qb\Lg_E+\Qb\Fib_E$, is
Theorem~SE$_0$ of \cite[Thm.~5.7]{Note}, which also gives Corollary~\ref{cor:SE4} below in the classes
$\Qb^\times(\pi/\Omega^2)^{\pm1}$ \cite[Cor.~5.8]{Note}.

\begin{corollary}[Strong elliptic four exponentials]\label{cor:SE4}
Let $k'\neq k$, and let $t\in\Qb^\times(\Omega'/\Omega)^{\pm1}$ or $t\in
\Qb^\times(\pi/\Omega\Omega')^{\pm1}$. If $y_1,y_2\in\Lt_E$ are linearly independent over $\Qb$, then $ty_1$ and $ty_2$ are
not both in $\Lt_E$. In particular, if $u_1,u_2\in\Lg_E$ are linearly independent over $k$, then $\wp_E(tu_1)$ and
$\wp_E(tu_2)$ are not both algebraic.
\end{corollary}

\begin{lemma}\label{lem:SE0}
$\Omega'\notin\Lt_E$ and $\eta'\notin\Lt_E$ if $k'\neq k$; in particular $t\notin\Qb$ in both cases of
Theorem~\ref{thm:SE}.
\end{lemma}

\begin{proof}
\lab{Lit: W\"ustholz \cite{Wu89}} If $\Omega'=\beta_0+\sum\beta_i\ell_i$ with $\ell_i\in\Lg_E$, the vector $(1,\ell,\Omega')$
of $\mathrm{Lie}(\Ga\times E^r\times E')$ lies in a $\Qb$-hyperplane; the smallest algebraic subgroup through it is
$H\times B\times E'$ ($\mathrm{Hom}(E,E')=0$), whose tangent space is not in the hyperplane. For $\eta'$ use
$\Ga\times E^r\times E'^\natural$: the smallest subgroup contains $0\times0\times E'^\natural$ (its image in
$\Ga\times E'^\natural$ maps onto $E'$ and has $\Ga$-coordinate $1$, so it is $\Ga\times E'^\natural$; the kernel part
cannot mix since $\mathrm{Hom}(E',E)=0$ and $\mathrm{Hom}(\Ga,E^r)=0$), and the fibre direction is not in the hyperplane.
\end{proof}

\begin{proof}[Proof of Corollary~\ref{cor:SE4}]
The statement is invariant under $t\mapsto\gamma t$ ($\gamma\in\Qb^\times$) and under $t\mapsto1/t$ (replace $y_j$ by
$ty_j$), and $\Qb^\times\pi/\Omega\Omega'=\Qb^\times\eta'/\Omega$. If
$\Omega\in\Qb y_1+\Qb y_2$ then $t\Omega\in\Lt_E$, i.e.\ $\Omega'$ or $\eta'$ lies in $\Lt_E$; this is excluded by
Lemma~\ref{lem:SE0}. So Theorem~\ref{thm:SE} applies, since $\Lt_E$ lies in each of its targets ($\Qb\subset\Fib_{E'}$). For the last sentence: $\wp(z)\in\Qb\cup\{\infty\}$ means $z\in\Lg_E$, and $k$-independent elliptic logarithms of
a CM curve are $\Qb$-independent (Masser \cite{Ma75}).
\end{proof}


\begin{proof}[Proof of Theorem~\ref{thm:SE}]
Let $K\supseteq kk'(c)$ contain all coefficients; then $n=[K:\Q]\ge4$. Let
$\theta$ be a $\Q$-basis of $K$, and $\psi(m)=\sum\theta_jm_j$, $\phi(m)=\sum\theta_j\bar m_j$ on $k^n$, with kernels of
$k$-dimension $n/2$ (Lemma~\ref{lem:K}). By Lemma~\ref{lem:SE0}, $t\notin K$.

(1) $G=\Ga\times E\times\Ga\times E\times E'$, $\Psi=(a_1+z_1,\,a_2+z_2+z')$, $V=\C(1,t)$. Then
$G''=\Ga^2\times A_1\times A_2\times A'$ with each abelian factor of dimension $n/2$ ($d_0=2$, $d_1=0$, $d_2=3n/2$); the
data are $w=(\Omega,c\Omega')$ and $v_j=(y_j,ty_j)$, so $m=3n$ and $\kappa=n$ (a period in $Y''$ has first coordinate in
$K\Omega$). Connected subgroups are $H\times B_{12}\times B'$ ($\mathrm{Hom}(E,E')=0$); for $B_{12}\subset A_1\times A_2$ the image
$\Psi''(T_{B_{12}})$ is $K$-rational, and $\dim B_{12}\le n/2$ when it is a line. For $r'=\dim\Psi''(T_{G'})=1$,
(\ref{eq:41}) reads $\mathrm{rk}\,Y_{G'}+2\dim B\ge4n$ ($B$ the abelian part of $G'$), and for $G'=0$ it reads
$6n-\kappa\le2(3n-\kappa)$, i.e.\ $n\le0$ since $\kappa=n$: excluded, with margin $n$. Image $\C(0,1)$ gives $\mathrm{rk}\,Y_{G'}\le n$ and $\dim B\le n$; a non-$K$-rational
line from $H$ gives $\mathrm{rk}\,Y_{G'}\le3n$, $B=0$; a $K$-rational line $\ell\neq\C(0,1)$ gives $\dim B\le n/2$ and
$\mathrm{rk}\,Y_{G'}\le2n$, because $v_j\in\ell+\Psi(\Omega_K(G))$ for $j=1,2$ would give $y_j\in K\Omega+K\Omega^2/\Omega'$ if
$\ell=\C(1,0)$, and $y_j(1-pt)\in K\Omega+K\Omega'=\Omega(K+Kt)$ if $\ell=\C(p,1)$ (the period datum is $\Omega'=t\Omega/c$,
and $1-pt\neq0$ as $t\notin K$); either way $\Omega,y_1,y_2$ would be $K$-dependent. No subgroup satisfies
(\ref{eq:41}); the minimum margin is $n$. Condition (\ref{eq:NR}): $3n+2\ge4+n$.

(2) This is the case $g=g'=1$ of Theorem~\ref{thm:EA} (EA$^\natural$) below: take $A=E$, $\sigma=\mathrm{id}$, $A'=E'$, and for
$\tilde\sigma$ the conjugate embedding of $k'$, read on $E'^\natural$ through its conjugate-eigen fibre coordinate $w'$
(\S\ref{sec:Euniv}). Then $P'=\eta'$, $z_\sigma(\Lambda(E))=\Lg_E$, $w_{\tilde\sigma}(\Lambda(E'^\natural))=\Fib_{E'}$ and
$\Lt_\sigma=\Lt_E$. (Likewise (1) is the case $g=g'=1$ of Theorem~\ref{thm:EA} (EA), so the proof of Theorem~\ref{thm:EA} gives a
second proof of (1).)
\end{proof}

\begin{example}\label{ex:SE}
With $E=E_i$ (period $\vp$) and $E'=E_\rho$ (period $\Omega_\rho$): for every $u\in\Lg_{E_i}\setminus\{0\}$, e.g.\ the logarithm of
$(2,\sqrt{24})$, the numbers $\wp_i(\Omega_\rho/\vp)$ and $\wp_i(\Omega_\rho u/\vp)$ are not both algebraic. This is
Corollary~\ref{cor:SE4} in the class $\Omega'/\Omega$, with $(y_1,y_2)=(1,u)$, which are $\Qb$-independent since $u\ne0$ is
transcendental; it is a consequence of Theorem~\ref{thm:SE}(1).
\end{example}

\noindent\emph{Provenance.} Theorem~\ref{thm:SE}(1) is a direct instance of \cite[Th\'eor\`eme (iii)]{RW93} as printed
(Remark~\ref{rem:printed}); the proof above applies Theorem~\ref{thm:W88} directly and shows that the tight subgroups have margin
$n$. For (2) the printed weight gives no conclusion on the group used here, and the proof uses the inner weight. As far as we
found, neither (1) nor (2) is stated in print; for spans $\Qb y_1+\Qb y_2$ that are not stable under complex conjugation,
Corollary~\ref{cor:SE4} also follows from the elliptic strong six exponentials theorem \cite[Cor.~1]{RW93} on $\Ga\times E$ and
complex conjugation, since $\Lt_E$ is conjugation-stable for CM $E$ (\S\ref{sec:lit}).

\smallskip
\noindent\emph{Eigen-slots.} Let $A/\Qb$ be simple with CM by $F$ and primitive type $\Phi$, $\sigma\in\Phi$, $P_\sigma\neq0$ a
period of $z_\sigma$, and $\Lt_\sigma=\Qb+\Qb z_\sigma(\Lambda(A))$.

\begin{theorem}[Theorems EA, EA$^\natural$]\label{thm:EA}
Let $(A',\sigma')$ be another first-kind slot ($A'=A$, $\sigma'\neq\sigma$ allowed), resp.\ $(A',\tilde\sigma)$ a
second-kind slot ($\tilde\sigma\notin\Phi'$, read on $A'^\natural$ through $w_{\tilde\sigma}$; $A'=A$ allowed), where an $A'$
isogenous to $A$ is replaced by $A$ (this rescales $P'$ and the target by $\Qb^\times$; then either $A'=A$ or
$\mathrm{Hom}(A,A')=0$, and we require $\sigma'\neq\sigma$ under the induced identification of the $F$-actions), with
period $P'$, and $t=cP'/P_\sigma$, $c\in\Qb^\times$. If $y_1,y_2\in\Lt_\sigma$ and $P_\sigma,y_1,y_2$ are linearly
independent over $\Qb$, then $ty_1,ty_2$ are not both in $\Lt_\sigma+\Qb z_{\sigma'}(\Lambda(A'))$, resp.\ in
$\Qb z_\sigma(\Lambda(A))+\Qb w_{\tilde\sigma}(\Lambda(A'^\natural))$. Consequently, for $t\in\Qb^\times(P'/P_\sigma)^{\pm1}$
and $\Qb$-independent $y_1,y_2\in\Lt_\sigma$, $ty_1,ty_2$ are not both in $\Lt_\sigma$.
\end{theorem}

\noindent The slot convention matters: if $A'$ is isogenous to $A$ but not equal to it, then without the convention $t$ can be
algebraic and the statement fails.

\begin{proof}
We apply Theorem~\ref{thm:RW} with the upper bound for the inner weight of \S\ref{sec:RW}. Let $K$ be a number field containing
the Galois closures of $F$ and $F'$, the number $c$ and all coefficients needed below, with $n=[K:\Q]\ge4$, and let $\theta$ be a
basis of $K/\Q$. For EA let $G=\Ga\times A\times\Ga\times A\times A'$ and $\Psi=(a_1+z_\sigma,\,a_2+z_\sigma+z_{\sigma'})$, where
the first $z_\sigma$ is read on the first factor $A$, the second on the second factor $A$, and $z_{\sigma'}$ on $A'$ (if $A'=A$,
the last factor is a third copy of $A$). For EA$^\natural$ let $G=\Ga\times A\times A\times A'^\natural$ and
$\Psi=(a+z_\sigma,\,z_\sigma+w_{\tilde\sigma})$, read in the same way. Let $V=\C(1,t)$ and $W=0$.

\emph{The condition on $V$.} $P'$ and $P_\sigma$ are periods of distinct eigen-slots (by the slot convention), so $t\notin\Qb$ by
(W). Hence $V$ has no nonzero $\Qb$-rational point, and Theorem~\ref{thm:RW} applies.

\emph{The data.} $\Lambda_K(G)$ is the product of the $K$-spans of the logarithms of the factors, and $\Lambda_K(\Ga)=\Qb$. So
$\Psi(\Lambda_K(G))$ is $(\Qb+Kz_\sigma(\Lambda(A)))\times(\Qb+Kz_\sigma(\Lambda(A))+Kz_{\sigma'}(\Lambda(A')))$ for EA and
$(\Qb+Kz_\sigma(\Lambda(A)))\times(Kz_\sigma(\Lambda(A))+Kw_{\tilde\sigma}(\Lambda(A'^\natural)))$ for EA$^\natural$. Suppose that
$y_1,y_2$ are as in the statement and that $ty_1,ty_2$ lie in the target. Since $K$ contains the coefficients, $v_j=(y_j,ty_j)$
lies in $V\cap\Psi(\Lambda_K(G))$ for $j=1,2$. The period datum $w=(P_\sigma,cP')$ lies in $V\cap\Psi(\Omega_K(G))$, since
$P_\sigma=z_\sigma(\gamma_0)$ for a period $\gamma_0$ of $A$, $P'=z_{\sigma'}(\gamma_0')$ for a period $\gamma'_0$ of $A'$ (resp.\
$P'=w_{\tilde\sigma}(\tilde\gamma'_0)$ for a period $\tilde\gamma'_0$ of $A'^\natural$, \S\ref{sec:setting}), and $c\in K$. So
$\kappa\ge1$. The first coordinates $P_\sigma,y_1,y_2$ of $w,v_1,v_2$ are linearly independent over $\Qb$, hence over $K$, so
$\dim_K(V\cap\Psi(\Lambda_K(G)))\ge3$.

\emph{The inner weight.} By Lemma~\ref{lem:Kprime}, $\ker\psi_\sigma\subset F^n$ has $F$-dimension $n-n/(2g)$, so
$B_{\ker\psi_\sigma}\subset A^n$ has dimension $gn-n/2$, and by (S1) the form $\theta^{\mathsf T}z_\sigma$ vanishes on its tangent
space; likewise for $\sigma'$ on $A'^n$. For EA put
\[
U=\{\theta\cdot a_1=0\}\times B_{\ker\psi_\sigma}\times\{\theta\cdot a_2=0\}\times B_{\ker\psi_\sigma}\times B_{\ker\psi_{\sigma'}} .
\]
Then $U$ is connected, $T_U\subset\ker\Psi_\theta$, and $G^n/U=\Ga\times A_1''\times\Ga\times A_2''\times A'''$ with three abelian
varieties of dimension $n/2$; it has dimension $3n/2+2$ and surjects onto $\Ga^2$, so $\alpha(G^n/U)\le3n+4-4=3n$. For
EA$^\natural$, let $t_{\tilde\sigma}$ be the $\tilde\sigma$-coordinate of the fibre of $(A'^\natural)^n$ and put
\[
U=\{\theta\cdot a=0\}\times B_{\ker\psi_\sigma}\times B_{\ker\psi_\sigma}\times\bigl(B'^\natural_{\ker\psi_{\tilde\sigma}}
+\{\theta^{\mathsf T}t_{\tilde\sigma}=0\}\bigr).
\]
By (F1) and (F4), $\theta^{\mathsf T}w_{\tilde\sigma}$ vanishes on the tangent space of the last factor, and the quotient $Q''$ of
$(A'^\natural)^n$ by it is an extension of $A'^n/B'_{\ker\psi_{\tilde\sigma}}$ (dimension $n/2$) by $\Ga$, of dimension $n/2+1$ (the
fibre part $W_{B'}$ of $B'^\natural_{\ker\psi_{\tilde\sigma}}$, of dimension $g'n-n/2$, lies in $\{\theta^{\mathsf T}t_{\tilde\sigma}=0\}$;
compare \cite[proof of Thm.~4.6]{Note}). So $G^n/U=\Ga\times A_1''\times A_2''\times Q''$ has dimension $3n/2+2$ and surjects onto
$\Ga$, and $\alpha(G^n/U)\le3n+4-2=3n+2$.

Hence $\alpha_{K,\Psi}(G)\le3$, resp.\ $\le3+2/n$, and Theorem~\ref{thm:RW} gives $\dim_K(V\cap\Psi(\Lambda_K(G)))\le2$, resp.\
$\le2+2/n<3$, a contradiction. For the consequence: the statement is symmetric under $t\mapsto1/t$ (replace $y_j$ by $ty_j$); if
$P_\sigma\in\Qb y_1+\Qb y_2$, then $cP'=tP_\sigma\in\Lt_\sigma$, which W\"ustholz's theorem excludes (as in the proof of
Lemma~\ref{lem:SE0}, on $\Ga\times A^r\times A'$, resp.\ $\Ga\times A^r\times A'^\natural$, using (S1) and, for $A'=A$, that the
slot forms of distinct embeddings involve independent parameters); otherwise the first part applies, since $\Lt_\sigma$ lies in
both targets.
\end{proof}

\begin{example}
On $A=\mathrm{Jac}(y^7=x(1-x))$ with $z_j$ dual to $dx/y^{7-j}$ and $P_j\in\Qb^\times B(\frac j7,\frac j7)$
(\S\ref{sec:X7}), put $t=B(\frac27,\frac27)/B(\frac17,\frac17)$. For every non-torsion $\exp(v)\in A(\Qb)$ (then $P_1$, $1$, $z_1(v)$ are $\Qb$-linearly independent, by W\"ustholz's theorem on
$\Ga\times A$ as in \S\ref{sec:XfromP}), the numbers $t$ and $tz_1(v)$ are not both in $\Qb+\Qb z_1(\Lambda(A))+\Qb z_2(\Lambda(A))$.
\end{example}

\noindent For $g=g'=1$, EA with non-isogenous curves is Theorem~\ref{thm:SE}(1), EA$^\natural$ with $A'=E'$ non-isogenous to $E$
is Theorem~\ref{thm:SE}(2), and EA$^\natural$ with $A'=E$, $\tilde\sigma=\bar\sigma$ is Theorem~SE$_0$ of \cite{Note}. For $g\ge2$
or $g'\ge2$ the proof uses the inner weight. As far as we found, Theorem~\ref{thm:EA} for $\max(g,g')\ge2$ is not in print.


\section{Four exponentials for quasi-logarithms}\label{sec:QL}
Let $E$ have CM by $k$, $\Omega\ne0$ a period,
$\eta=\eta^*(\Omega)$, and $\Fib_E$ the set of fibre values of $E$ (\S\ref{sec:Euniv}: constants, quasi-logarithms
$\zeta^*(u)+\Qb$, $u\in\Lg_E$, and $\eta^*(\Lambda_E)$). The following lemma describes the mixed subgroups of two fibre factors.

\begin{lemma}[Fibre images are rational]\label{lem:F}
Let $K\supseteq k$ have degree $n$, $\theta$ a $\Q$-basis of $K$, $\phi(m)=\sum\theta_j\bar m_j$ on $k^n$, and
$Q=(E^\natural)^n/(B^\natural_{\ker\phi}+\{\theta^{\mathsf T}w=0\})$, with fibre coordinate $\varsigma=\theta^{\mathsf T}w$ (the
quotient $Q''$ of the proof of \cite[Thm.~4.3]{Note}). Put $Q_1=Q_3=Q$. If $H\subset Q_1\times Q_3$ is a connected algebraic
subgroup which is not a vector group and $(\varsigma_1,\varsigma_3)(T_H)$ is a line $\ell$, then $\ell$ is spanned by a
vector of $K^2$, and the abelian image of $H$ has dimension at most $n/2$.
\end{lemma}

\begin{proof}
Let $\tilde H\subset(E^\natural)^{2n}$ be the connected preimage, with abelian image $B_S$, $S\subset k^{2n}$, and let $M$
be a matrix over $\mathcal O_k$ whose columns span $S$. Since $\mathcal O_k$ acts on $\mathrm{Lie}(E^\natural)$ by
$\mathrm{diag}(a,\bar a)$, $\mathrm{Lie}(B_S^\natural)=\{(Mu,\bar Mu')\}$, and $\tilde H\supseteq B_S^\natural$ by the
universal property of vector extensions ((F3) for $g=1$). The fibre vectors $(0,\bar Mu')$ have fibre coordinates
$(\phi(M_1)u',\phi(M_3)u')$, which span the $K$-rational space $R_S=\mathrm{span}\{(\phi(s_1),\phi(s_3)):s\in S\}$;
$R_S=0$ exactly when $S\subset\ker\phi\oplus\ker\phi$, i.e.\ when $H$ is a vector group. So $0\ne R_S\subseteq\ell$ and
$\ell=R_S$. The map $S/(\ker\phi)^2\to K^2$, $s\mapsto(\phi(s_1),\phi(s_3))$, is injective with image in the $K$-line
$\ell\cap K^2$, whence the dimension bound.
\end{proof}

\begin{theorem}[Theorem QL]\label{thm:QL}
Let $p_2$ be one of: $2\pi i$; a period $\Omega'$ of a CM curve $E'$ ($E'=E$ allowed); the quasi-period
$\eta'=\eta^*(\Omega')$ of a CM curve $E'$ with CM field $k'\neq k$; a first-kind eigen-period $P$ of a simple CM abelian
variety $A$. Let $X_2$ be the corresponding set of values $\Lg$, $\Lg_{E'}$, $\Fib_{E'}$, $z_\sigma(\Lambda(A))$. Let
$c\in\Qb^\times$ and $t=cp_2/\eta$. If $f_1,f_2\in\Qb\Fib_E$ and $\eta,f_1,f_2$ are linearly independent over $\Qb$, then
$tf_1$ and $tf_2$ are not both in $\Qb\Fib_E+\Qb X_2$.
\end{theorem}


\begin{corollary}\label{cor:QL}
Let $t\in\Qb^\times\{\Omega,\ \Omega\Omega'/\pi,\ \Omega/\Omega',\ \Omega P/\pi\}^{\pm1}$, with $\Omega'$, $P$ as in
Theorem~\ref{thm:QL} (in the class $\Omega/\Omega'$ the curve $E'$ has CM field $k'\neq k$). If $f_1,f_2\in\Qb\Fib_E$ are
linearly independent over $\Qb$, then $tf_1,tf_2$ are not both in $\Qb\Fib_E$. For example (class $\Omega/\Omega'$ with
$E=E_i$, $E'=E_\rho$), if $u_1,u_2\in\Lg_{E_i}$ and $\vp$ are linearly independent over $\Q(i)$, then
$(\vp/\Omega_\rho)\zeta_i(u_1)$ and $(\vp/\Omega_\rho)\zeta_i(u_2)$ are not both $\Qb$-linear combinations of $1$ and
quasi-logarithms $\zeta_i(u)$, $u\in\Lg_{E_i}\setminus\vp\Z[i]$.
\end{corollary}



\noindent In the class $\Qb^\times\Omega^{\pm1}$ ($p_2=2\pi i$) the statement is of degree one: by Laurent's identity
(\ref{eq:laurent}), $\Omega\zeta^*(u)=\lambda(u,\Omega)+\eta u$ with $\eta u\in\Qb^\times(\pi/\Omega)u$ a $\Qb$-combination of
third-kind periods, so the hypothesis $tf_1,tf_2\in\Qb\Fib_E$ is a pair of $\Qb$-linear relations among periods of
$1$-motives, and that case is within reach of the analytic subgroup theorem (we keep it in Theorem~\ref{thm:QL} as a
calibration case). The bilinear content is in the other three classes, where a period of another slot multiplies a
fibre value, as in the example.

\begin{proof}[Proof of Theorem~\ref{thm:QL}]
Let $K$ be a number field of degree $n\ge6$ containing $k$, $c$, the CM field of the second factor (the Galois closure of
$F$ if $X_2$ comes from $A$) and all coefficients needed to write $f_j\in\Qb\Fib_E$ and $tf_j\in\Qb\Fib_E+\Qb X_2$
($j=1,2$), so that $v_j\in\Psi(\Lambda_K(G))$. Take $G=E^\natural\times X\times E^\natural$ with
$X=\Gm,E',E'^\natural,A$ respectively, $\Psi=(w_1,\,x_2+w_3)$ ($w_1,w_3$ the fibre coordinates, $x_2$ the coordinate of
$X$ read by $X_2$) and $V=\C(1,t)$. If $X=E'^\natural$, the two fibre coordinates read in the second target are first
merged (the vector subgroup $\{\theta^{\mathsf T}w'+\theta^{\mathsf T}w_3=0\}$ lies in the kernel). Then
$G''=Q_1\times X''\times Q_3$ with $Q_1,Q_3$ as in Lemma~\ref{lem:F} (for $X=E'^\natural$: $Q_1\times(X''\times Q_3)/D$,
$D=\{\varsigma'+\varsigma_3=0\}$, with one fibre $\Ga$ over the abelian part of $X''\times Q_3$). The data are
$w=(\eta,cp_2)$ and $v_j=(f_j,tf_j)$; $m=3n$, $\kappa=n$ (a period in $Y''$ has first coordinate in $K\eta$), and
$\Psi(\Omega_K(G))=K\eta\times(K\eta+Kp_2)$. Here $t\notin K$: $t\in\Qb$ would put $p_2$ in $\Qb\eta\subset\Qb\Fib_E$, which
W\"ustholz's theorem \cite{Wu89} excludes. For $r'=1$, (\ref{eq:41}) reads $\mathrm{rk}\,Y_{G'}+l+2\dim(G'/T')\ge4n$,
where $T'$ is the torus part of $G'$ and $l=\dim T'$. Subgroups with image $\C(0,1)$ have $\mathrm{rk}\,Y_{G'}\le n$ and
$l+2\dim(G'/T')\le2n+2$. Subgroups of $Q_1$ have image $\C(1,0)$, dimension $\le n/2+1$ and $\mathrm{rk}\,Y_{G'}\le2n$.
Mixed subgroups of $Q_1\times Q_3$ that are not vector groups have, by Lemma~\ref{lem:F}, a $K$-rational image line, dimension
$\le n/2+1$, and $\mathrm{rk}\,Y_{G'}\le2n$: if $v_1,v_2\in\ell+\Psi(\Omega_K(G))$ with $\ell=\C(p,1)$, $p\in K$, then
$f_j(1-pt)\in K\eta+Kp_2=\eta(K+Kt)$ (period datum $p_2=t\eta/c$), and if $\ell=\C(1,0)$ then $tf_j\in\eta(K+Kt)$; either
way $\eta,f_1,f_2$ lie in a $2$-dimensional $K$-space, against their $K$-independence. Vector lines have dimension $1$ and
$\mathrm{rk}\,Y_{G'}\le3n$. These cases are complete. If $X''$ has an abelian part isogenous to the base of $E$ ($E'=E$, or
$A$ isogenous to a power of $E$), abelian subvarieties may mix $Q_1$, $X''$ and $Q_3$ (and, for $X=E'^\natural$, $Q_1$ and
the merged factor); if the abelian image of $G'$ projects nontrivially to $X''$ (resp.\ to the $E'$-part of the merged
factor), its image contains $(0,1)$, so $r'=1$ forces image $\C(0,1)$, hence $\phi(S_1)=0$ and $G'\subset X''\times Q_3$,
which is the case of image $\C(0,1)$ above; otherwise Lemma~\ref{lem:F} applies verbatim (to $Q_1\times Q_3$, resp.\ to
$Q_1$ and the merged factor). So every subgroup with $r'=1$ has total at most $3n+2<4n$. For $G'=0$ the inequality fails
with margin $n-4>0$, since $n\ge6$ (it obstructs exactly for $n\le4$); this is the minimum margin. No subgroup satisfies
(\ref{eq:41}). Condition (\ref{eq:NR}) holds
($d_0=0$, $2N=2d-2\ge3n+2\ge d_1+\kappa$).
\end{proof}

\begin{proof}[Proof of Corollary~\ref{cor:QL}]
The statement is invariant under $t\mapsto\gamma t$ ($\gamma\in\Qb^\times$) and under $t\mapsto1/t$ (replace $f_j$ by
$tf_j$). By $\Omega\eta\in\Qb^\times\pi$ and $\Omega'\eta'\in\Qb^\times\pi$, the classes $\Qb^\times p_2/\eta$ of
Theorem~\ref{thm:QL} are $\Qb^\times\Omega$, $\Qb^\times\Omega\Omega'/\pi$, $\Qb^\times\Omega/\Omega'$ and
$\Qb^\times\Omega P/\pi$. If $\eta\in\Qb f_1+\Qb f_2$, then $tf_1,tf_2\in\Qb\Fib_E$ gives $cp_2=t\eta\in\Qb\Fib_E$, which
W\"ustholz's theorem \cite{Wu89} excludes; otherwise Theorem~\ref{thm:QL} applies. For the example:
$t=\vp/\Omega_\rho\in\Qb^\times\eta_\rho/\eta_i$ is the class $\Qb^\times p_2/\eta$ with $p_2=\eta'=\eta(\Omega_\rho)$ of
$E'=E_\rho$, whose CM field $\Q(\rho)$ differs from $\Q(i)$; and $f_j=\zeta_i(u_j)\in\Fib_{E_i}$ ($\kappa_1=0$ on $E_i$).
By W\"ustholz's theorem on $(E_i^\natural)^3$, whose smallest algebraic subgroup through
$((\vp,\eta_i),(u_1,\zeta_i(u_1)),(u_2,\zeta_i(u_2)))$ is $(E_i^\natural)^3$ when $\vp,u_1,u_2$ are $\Q(i)$-independent,
the numbers $\eta_i,\zeta_i(u_1),\zeta_i(u_2)$ are $\Qb$-independent, so in particular $f_1,f_2$ are. Finally
$\Qb\Fib_{E_i}$ is the $\Qb$-span of $1$ and the $\zeta_i(u)$, $u\in\Lg_{E_i}$ off the lattice, since
$\eta_i=2\zeta_i(\vp/2)$ and $\eta(\Lambda)\subset\Q(i)\eta_i$.
\end{proof}

\begin{remark}
Without Lemma~\ref{lem:F} a mixed line $\C(\mu,1)$ with $\mu\notin K$ would only give $\mathrm{rk}\,Y_{G'}\le3n$, which is
an obstruction. The same lemma, with $\psi_{\tilde\sigma}$ in place of
$\phi$ and $K\supseteq\tilde\sigma(F)$, gives, by (F1)--(F4), the analogue for second-kind eigen-coordinates of CM
abelian varieties (not written out).
\end{remark}

\section{Conjecture 42 at CM ratios}\label{sec:C42}
In this section
$E/\Qb$ has CM by $k$, $\Omega\ne0$ is a period, $\eta^*$ the eigen quasi-period (\S\ref{sec:Euniv}), so
$\eta^*(\Omega)\in\Qb^\times\pi/\Omega$; $\Fib_E$ is the set of fibre values; $\Lt=\Qb+\Qb\Lg$. A CM curve $E'$ ``not isogenous
to $E$'' is defined over $\Qb$ with CM field $k'\ne k$, and $\Omega'$ is a nonzero period of $E'$.

Waldschmidt's Conjecture~42 \cite[p.~173]{W06} asks that a rank-one matrix $\left(\begin{smallmatrix}u_1&\log\alpha_1\\
u_2&\log\alpha_2\end{smallmatrix}\right)$ with $u_j\in\Lg_E$, $\log\alpha_j\in\Lg$ have $\Q$-proportional rows; equivalently,
the middle inequality $\delta(e^z,\wp(tz))\le1$ of Ramachandra's $\delta$-conjecture in the form of \cite[\S2(a)]{W13}.
Theorem~M of \cite[Thm.~6.1]{Note} is its CM case with a period row.

\begin{theorem}[Theorem C42]\label{thm:C42}
Let $\rho\in R_E:=\Qb^\times\cdot\{\Omega,\ \Omega',\ \pi/\Omega',\ \Omega'/\Omega,\ \pi/\Omega^2,\ \pi/(\Omega\Omega')\}$. If
$u_1,u_2\in\Lg_E\setminus\{0\}$ satisfy $\rho u_1,\rho u_2\in\Lt$, then $u_2/u_1\in k$. If moreover $\rho u_j\in\Lg$ ($j=1,2$),
then $u_2/u_1\in\Q$.
\end{theorem}


\begin{corollary}\label{cor:C42}
\begin{itemize}
\item[(a)] Conjecture~42 of \cite{W06} holds for every rank-one matrix whose column ratio $\log\alpha_j/u_j$ lies in $R_E$.
\item[(b)] If $\log\alpha,\log\beta$ are linearly independent over $\Q$ and
\[
t\in\Qb^\times\cdot\{1/\Omega,\ 1/\Omega',\ \Omega'/\pi,\ \Omega/\Omega',\ \Omega^2/\pi,\ \Omega\Omega'/\pi\},
\]
then $\wp_E(t\log\alpha)$ and $\wp_E(t\log\beta)$ are not both algebraic (a pole counts as algebraic). For instance
$\wp_i(\log2/\vp)$ and $\wp_i(\log3/\vp)$ are not both algebraic, nor are $\wp_i(\Omega_\rho\log2/\pi)$ and $\wp_i(\Omega_\rho\log3/\pi)$.
\item[(c)] For $\rho\in R_E$ and $k$-independent $u_1,u_2\in\Lg_E$, $\rho u_1$ and $\rho u_2$ are not both in $\Lt$; e.g.\
$e^{\vp u_1}$ and $e^{\vp u_2}$ are not both algebraic for $\Q(i)$-independent $u_1,u_2\in\Lg_{E_i}$, nor are
$e^{\pi u_j/\vp^2}$, nor $e^{\Omega_\rho u_j/\vp}$.
\end{itemize}
\end{corollary}



\begin{proof}[Proof of Corollary~\ref{cor:C42}]
(a) and (c) are restatements. For (b) put $\rho=1/t$ and $u_j=t\log\alpha_j$; if both values are algebraic then
$u_j\in\Lg_E$ and $\rho u_j\in\Lg$, so $\log\beta/\log\alpha=u_2/u_1\in\Q$.
\end{proof}

\noindent\emph{The line form.} Each class of $\rho$ is a cell $\sigma_1\,|\,\sigma_2+\sigma_3$: a factor $\sigma_1$ read into
the first coordinate of $\C^2$, two factors $\sigma_2,\sigma_3$ read into the second, a slope $r=c\,p_2/p_1$ with $p_i$ the
period of $\sigma_i$, and the set
\[
\mathcal S_K=\{\lambda\in\Lambda_K(\sigma_1):\ r\lambda\in\Lambda_K(\sigma_2)+\Lambda_K(\sigma_3)\},
\]
where $\Lambda_K(\Gm)=K\Lg$, $\Lambda_K(E)=K\Lg_E$ and $\Lambda_K(\text{fibre})=K\cdot\Fib$.
\begin{center}
{\small
\begin{tabular}{llllll}
row & $\sigma_1$ & $\sigma_2$ & $\sigma_3$ & slope $r$ & $p_1\mapsto rp_1$\\\hline
(i) & $\Gm$ & fibre of $E$ & $E$ & $c/\Omega$ & $2\pi i\mapsto\Qb^\times\eta^*(\Omega)$\\
(ii) & $\Gm$ & fibre of $E'$ & $E$ & $c/\Omega'$ & $2\pi i\mapsto\Qb^\times\eta^{*\prime}(\Omega')$\\
(iii) & $\Gm$ & $E'$ & $E$ & $c\Omega'/\pi$ & $2\pi i\mapsto2ic\Omega'$\\
(iv) & $E$ & $E'$ & $\Gm$ & $c\Omega'/\Omega$ & $\Omega\mapsto c\Omega'$\\
(v) & $E$ & fibre of $E$ & $\Gm$ & $c\pi/\Omega^2$ & $\Omega\mapsto\Qb^\times\eta^*(\Omega)$\\
(vi) & $E$ & fibre of $E'$ & $\Gm$ & $c\pi/(\Omega\Omega')$ & $\Omega\mapsto\Qb^\times\eta^{*\prime}(\Omega')$
\end{tabular}
}
\end{center}
with $\rho=1/r$ in rows (i)--(iii) and $\rho=r$ in rows (iv)--(vi).

\begin{proposition}[Line form]\label{prop:C42line}
In each row, for every number field $K\supseteq kk'(i,c)$ with $[K:\Q]\ge4$, $\dim_K\mathcal S_K\le2$, and $p_1\in\mathcal S_K$.
The same holds with a factor $\Ga$ added to the coordinate that carries the ordinary logarithms (target $1$ in rows
(i)--(iii), target $2$ in rows (iv)--(vi)), i.e.\ with $K\Lg$ replaced by $K+K\Lg$.
\end{proposition}


\begin{proof}
We apply Theorem~\ref{thm:RW} with the upper bound for the inner weight of \S\ref{sec:RW}. Fix a row, let $n=[K:\Q]$ and let
$\theta$ be a basis of $K/\Q$. Let $G$ be the product of three groups carrying the three factors: $\Gm$; $E$ or $E'$, read through
the base coordinate $z$, resp.\ $z'$; $E^\natural$ or $E'^\natural$, read through the conjugate-eigen fibre coordinate $w$, resp.\
$w'$, of \S\ref{sec:Euniv}. (In rows (i) and (v) the curve $E$ occurs twice, as $E$ and as $E^\natural$; these are two different
factors of $G$.) Let $\Psi\colon T_G\to\C^2$ read $\sigma_1$ into the first coordinate and the sum of $\sigma_2$ and $\sigma_3$ into
the second, $V=\C(1,r)$ and $W=0$. Since $\Lambda_K(G)$ is the product of the $K$-spans of the logarithms of the factors,
$\Psi(\Lambda_K(G))=\Lambda_K(\sigma_1)\times(\Lambda_K(\sigma_2)+\Lambda_K(\sigma_3))$, and so
\[
V\cap\Psi(\Lambda_K(G))=\{(\lambda,r\lambda):\ \lambda\in\mathcal S_K\},\qquad\dim_K\mathcal S_K=\dim_K\bigl(V\cap\Psi(\Lambda_K(G))\bigr).
\]

\emph{The condition on $V$.} Up to $\Qb^\times$, $r$ is $1/\Omega$, $1/\Omega'$, $\Omega'/\pi$, $\Omega'/\Omega$, $\eta^*(\Omega)/\Omega$
or $\eta^{*\prime}(\Omega')/\Omega$ (rows (i)--(vi); recall $\eta^*(\Omega)\in\Qb^\times\pi/\Omega$). None of these is algebraic, by
(W): $\Omega$, $\Omega'$, $\eta^*(\Omega)$, $\eta^{*\prime}(\Omega')$ are periods of four distinct eigen-slots, and periods of distinct
slots are linearly independent over $\Qb$ together with $1$ and $\pi$. So $V$ has no nonzero $\Qb$-rational point, and
Theorem~\ref{thm:RW} applies.

\emph{The period datum.} By Legendre's relation and $\eta^*(a\lambda)=\bar a\,\eta^*(\lambda)$, $\Omega\,\eta^*(\Omega)\in\pi i\,k^\times$:
if $\lambda_1,\lambda_2=\tau\lambda_1$ is a basis of $\Lambda_E$ with $\mathrm{Im}\,\tau>0$, then
$\eta^*(\lambda_1)\lambda_2-\eta^*(\lambda_2)\lambda_1=2\pi i$ gives $(\tau-\bar\tau)\lambda_1\eta^*(\lambda_1)=2\pi i$, and
$\Omega\eta^*(\Omega)=a\bar a\,\lambda_1\eta^*(\lambda_1)$ for $\Omega=a\lambda_1$, $a\in k^\times$. Likewise for $E'$. Since
$K\supseteq kk'(i,c)$, in each row $rp_1$ lies in the $K$-span of the periods of $\sigma_2$: $K\eta^*(\Omega)$, $K\eta^{*\prime}(\Omega')$,
$K\Omega'$, $K\Omega'$, $K\eta^*(\Omega)$, $K\eta^{*\prime}(\Omega')$ (here $\eta^*(\Omega)$ is the fibre coordinate of the period of
$E^\natural$ above $\Omega$, \S\ref{sec:Euniv}). And $p_1\in\Omega(\sigma_1)$. So $(p_1,rp_1)\in V\cap\Psi(\Omega_K(G))$, $p_1\in\mathcal S_K$
(periods are logarithms), and $\kappa\ge1$.

\emph{The inner weight.} Let $\psi,\phi$ (resp.\ $\psi',\phi'$) be the maps of Lemma~\ref{lem:K} for $k$ (resp.\ $k'$). In $E^n$ the
abelian subvariety $B_{\ker\psi}$ has dimension $n/2$ and $\theta^{\mathsf T}z$ vanishes on its tangent space. In $(E^\natural)^n$,
put $U^\natural=B^\natural_{\ker\phi}+\{\theta^{\mathsf T}w=0\}$: the tangent space of $B^\natural_S$ is $\{(Mu,\bar Mu')\}$ for a matrix
$M$ whose columns span $S$ ((F4) for $g=1$), so $\theta^{\mathsf T}w=\phi(M)u'$ vanishes on it for $S=\ker\phi$, and the quotient
$(E^\natural)^n/U^\natural$ is an extension of $E^n/B_{\ker\phi}$ (dimension $n/2$) by $\Ga$, of dimension $n/2+1$ (as in
\cite[proof of Thm.~4.3]{Note}). The same holds for $E'$ with $\psi',\phi'$. Let $U\subset G^n$ be the product of these subgroups in
the factors $E$, $E'$, $E^\natural$, $E'^\natural$, and of $0$ in the factor $\Gm^n$. Then $U$ is connected, $T_U\subset\ker\Psi_\theta$,
and
\[
G^n/U=\Gm^n\times A''\times A'''\quad\text{(rows (iii), (iv))},\qquad G^n/U=\Gm^n\times A''\times Q''\quad\text{(rows (i), (ii), (v), (vi))},
\]
with $\dim A''=\dim A'''=n/2$ and $\dim Q''=n/2+1$. It surjects onto $\Gm^n$, so $\alpha(G^n/U)\le2\cdot2n-n=3n$, resp.\
$\le2(2n+1)-n=3n+2$.

Hence $\dim_K\mathcal S_K\le\alpha_{K,\Psi}(G)-\kappa\le3-1=2$ in rows (iii), (iv), and $\le3+2/n-1<3$ in the other rows, since
$n\ge4$; so $\dim_K\mathcal S_K\le2$.

\emph{Affine variant.} Add a factor $\Ga$ to $G$, read into the coordinate that carries $\Gm$, and add $\{\theta\cdot a=0\}$ to
$U$. Then $\Lambda_K(\Ga)=\Qb\supseteq K$, the quotient $G^n/U$ acquires a factor $\Ga$, which raises $2\dim$ by $2$ and $2d_0$ by
$2$, so the bound for $\alpha(G^n/U)$ is unchanged; the condition on $V$ and the period datum are as before. So
$\{\lambda\in\Qb+K\Lg:\ r\lambda\in\Lambda_K(\sigma_2)+\Lambda_K(\sigma_3)\}$ in rows (i)--(iii), and $\mathcal S_K$ with $\Lambda_K(\Gm)$
replaced by $\Qb+K\Lg$ in rows (iv)--(vi), have $K$-dimension $\le2$, which contains the statement.
\end{proof}

\begin{remark}[A direct proof of row (iv)]\label{rem:C42direct}
Let $G=E\times E'\times\Gm$ with coordinates $(z,z',x)$, $\Psi(z,z',x)=(z,z'+x)$, $V=\C(1,r)$,
$n=[K:\Q]$, $\theta$, $p_\theta$, $\Psi_\theta$ as in \S\ref{sec:RW}. Connected algebraic subgroups of $E^n\times E'^n\times\Gm^n$
are products $B_S\times B'_{S'}\times T_L$ ($\mathrm{Hom}(E,E')=0$), and such a subgroup has tangent space in $\ker\Psi_\theta$
iff $\psi(S)=0$, $\psi'(S')=0$ and $L=0$. So $U=B_{\ker\psi}\times B'_{\ker\psi'}$, and $G''=G^n/U=A''\times A'''\times\Gm^n$
with $\dim A''=\dim A'''=n/2$: $d_0=0$, $d_1=n$, $d_2=n$, $d=2n$, weight $3n$. $\Psi''$ is onto $\C^2$, $V''$ is a hyperplane
and $W''=\ker\Psi''$. The period datum is $w=(\Omega,c\Omega')\in\Psi(\Omega_K(G))$. Suppose $\Omega,u_1,u_2\in\mathcal S_K$ are
$K$-independent, put $v_j=(u_j,ru_j)$, and let $Y''$ be generated by lifts of $D\theta_hw$, $\theta_hv_1$, $\theta_hv_2$: then
$m=3n$, and $\kappa=n$ (a period of $G''$ has first coordinate in $K\Omega$, which kills the $v$-parts). With $d-N=1$ and
$\delta-\tau=2-r'$, $r'=\dim\Psi''(T_{G'})$, an obstruction needs
$(m-\mathrm{rk}\,Y_{G'})+(3n-\mathrm{wt}\,G')\le(2-r')\cdot2n$, where $Y_{G'}=Y''\cap(T_{G'}+\Omega(G''))$. Since $U$ is
maximal, a nonzero $B''$ has image $\C(1,0)$ and a nonzero $B'''\times T_L$ has image containing $\C(0,1)$; so $r'\le1$ leaves:
$G'=0$ ($5n\le4n$ fails, margin $n$); $G'=B'''\times T_L\ne0$ with image $\C(0,1)$ ($\mathrm{rk}\,Y_{G'}\le n$, so the left
side is $\ge3n>2n$); $G'=B''\ne0$ with image $\C(1,0)$: an element of $Y_{G'}$ has $\Psi''$-image $(X,rX)$ with
$rX\in K\Omega'+K\cdot2\pi i$, i.e.\ $X\in K\Omega+K\cdot2\pi i\Omega/(c\Omega')$, a $2$-dimensional $K$-space, so
$\mathrm{rk}\,Y_{G'}\le2n$ and the left side is $\ge n+2n>2n$. No $G'$ satisfies (\ref{eq:41}). (\ref{eq:NR}) reads
$2(2n-1)\ge2n$.
The minimum margin is $n$, for every even $n$.
\end{remark}

\begin{proof}[Proof of Theorem~\ref{thm:C42}]
Take $K$ large (containing the coefficients needed below). \emph{Rows (i)--(iii).} $x_j:=\rho u_j\in\Lt\subset K+K\Lg$ and
$rx_j=u_j\in K\Lg_E$, so $2\pi i,x_1,x_2\in\mathcal S_K$ (affine variant) and $a\cdot2\pi i+bx_1+c'x_2=0$ with $(a,b,c')\ne0$.
If $a\ne0$, then $bu_1+c'u_2=-2\pi ia/\rho$ is a nonzero element of $\Qb^\times\eta^*(\Omega)$, $\Qb^\times\eta^{*\prime}(\Omega')$
or $\Qb^\times\Omega'$ lying in $\Qb\Lg_E$. This is impossible by W\"ustholz's theorem \cite{Wu89}: $\Qb\Lg_E\cap\Qb\eta^*(\Omega)=0$
(apply it to $(u_1,\dots,u_r,(\Omega,\eta^*(\Omega)))$ in $E^r\times E^\natural$: the smallest algebraic subgroup whose tangent space
contains this point contains the conjugate-eigen part of $E^\natural$ (F3), which carries the fibre coordinate $w$ independently of the
coordinates $z(u_i)$, so a form $\sum_i\beta_iz(u_i)-\gamma w$ with $\gamma\ne0$ does not vanish on it), $\Qb\Lg_E\cap\Qb\eta^{*\prime}=0$ (on $E\times E'^\natural$) and
$\Qb\Lg_E\cap\Qb\Omega'=0$ (on $E\times E'$). So $a=0$. \emph{Rows (iv)--(vi).} $\Omega,u_1,u_2\in\mathcal S_K$ (affine variant
in target $2$), so $a\Omega+bu_1+c'u_2=0$; if $a\ne0$ then $\rho(bu_1+c'u_2)=-a\rho\Omega\in\Lt$, but
$\rho\Omega\in\Qb^\times\{\Omega',\eta^*(\Omega),\eta^{*\prime}(\Omega')\}$, none of which lies in $\Lt$ (W\"ustholz on
$\Ga\times\Gm^N\times E'$, $\times E^\natural$, $\times E'^\natural$). So $a=0$. In all rows $u_1,u_2$ are $\Qb$-dependent, so
$u_2/u_1\in k$ by Masser's theorem \cite{Ma75}. If $\rho u_j=\log\alpha_j$ and $u_2=\gamma u_1$ with $\gamma\in k$, then
$\log\alpha_2=\gamma\log\alpha_1\ne0$, and the Gel'fond--Schneider theorem forces $\gamma\in\Q$.
\end{proof}

\noindent\emph{Provenance.} Rows (iii), (iv) of Proposition~\ref{prop:C42line} are direct instances of
\cite[Th\'eor\`eme (iii)]{RW93} as printed (Remark~\ref{rem:printed}); for the fibre rows the printed weight gives no conclusion on
the groups used here, and the inner weight is used. The linearisation filter: on a CM curve only $\eta^*\cdot u$ and
$\Omega\cdot\zeta^*(u)$ are periods of $1$-motives (Laurent's identity (\ref{eq:laurent})), so $\rho\in\Qb^\times\pi/\Omega$
(Theorem~M of \cite[Thm.~6.1]{Note}) is the unique linearisable CM ratio, and we do not expect Theorem~\ref{thm:C42} to follow from
the analytic subgroup theorem \lab{Heuristic}. On what is known: all six classes satisfy $\bar\rho/\rho\in\Qb^\times$ and $\bar E$ is
isogenous to $E$, so a datum $u$ gives a second datum $\bar u$. Write $V(\alpha,\beta)$ for the $\Q$-span of
$\log\alpha,\log\beta,\overline{\log\alpha},\overline{\log\beta}$. If $\dim V(\alpha,\beta)=4$, Corollary~\ref{cor:C42}(b) already
follows \lab{Proof sketch} from the Lang--Ramachandra mixed six exponentials theorem \cite[Thm.~24]{W06} plus conjugation (Ramachandra; Diaz
\cite{Di04}); if $\dim V(\alpha,\beta)=3$, it is a one-line consequence of Theorem~\ref{thm:RW} on $E\times\Gm$ ($\alpha_K=2$,
$\kappa=0$) plus conjugation \lab{Proof sketch}. As far as we found, the conjugation-stable case $\dim V(\alpha,\beta)=2$ (for instance $\alpha,\beta$
real), which contains all our examples, is not in print. Corollary~\ref{cor:C42}(b) is the inequality $\delta(e^z,\wp(tz))\le1$ of
Ramachandra's $\delta$-conjecture in the form of \cite[\S2(a)]{W13}, at six CM classes of $t$; the best bounds listed there are
$\delta\le3$ in general and $\delta\le2$ at the slope of Theorem~M, with no case of $\delta\le1$.

\section{Applications in codimension $\ge2$}\label{sec:H2}
The two results of this section live on subspaces $V''$ of codimension $\ge2$, where Theorem~\ref{thm:W88} applied subgroup by
subgroup is stronger than the uniform bound of Theorem~\ref{thm:RW}: at $G'=0$, (\ref{eq:41}) divides the threshold by $d-N$
\cite[(4.2)]{W88}. In the configurations we use, Theorem~\ref{thm:RW} gives no conclusion or a weaker one (bound $3$ against $2$
data for Theorem~\ref{thm:CL4}, $k\le g-2$ for (U1), and its condition on $V$ fails when $K$ contains the imaginary quadratic field
of Theorem~\ref{thm:U}). We apply Theorem~\ref{thm:W88} as printed, subgroup by subgroup; both applications satisfy
(\ref{eq:NR}). The other application of this kind we know is Theorem~P of \cite[\S7]{Note}. Notation: $\Lt=\Qb+\Qb\Lg$; a CM
curve is an elliptic curve over $\Qb$ with complex multiplication, and $\eta_j^*$ is the conjugate-eigen quasi-period of a period
$\Omega_j$ (\S\ref{sec:Euniv}), so $\eta_j^*\in\Qb^\times\pi/\Omega_j$.

\subsection{Collinearity of period points}\label{sec:CL4}

\begin{theorem}[Theorem CL4]\label{thm:CL4}
Let $E_1,E_2,E_3$ be CM elliptic curves over $\Qb$ with pairwise distinct CM fields $k_1,k_2,k_3$, and $\Omega_j$ a nonzero
period of $E_j$. If $s,t\in\C$ satisfy
\[
s+2\pi i\,t\in\Qb\Lg\quad\text{and}\quad s+\Omega_jt\in\Qb\Lg_{E_j}\ (j=1,2,3),
\]
then $s=0$. Equivalently, if the four points $(2\pi i,x)$, $(\Omega_j,y_j)$ of $\C^2$ with $x\in\Qb\Lg$, $y_j\in\Qb\Lg_{E_j}$ lie on
one line, the line passes through the origin. The same holds with $\Gm$ replaced by a fourth CM curve $E_0$ with a CM field
distinct from the others.
\end{theorem}


\begin{corollary}\label{cor:CL4}
Put $X=\Omega_1(2\pi i-\Omega_2)/(\Omega_1-\Omega_2)$ and $Y=\Omega_1(\Omega_3-\Omega_2)/(\Omega_1-\Omega_2)$. Then $e^X$ and
$\wp_{E_3}(Y)$ are not both algebraic.
\end{corollary}

\begin{proof}
Take $y_1=\Omega_1$, $y_2=0$: then $t=\Omega_1/(\Omega_1-\Omega_2)$ and $s=-\Omega_2t\ne0$, and the theorem forbids $x=X\in\Lg$
together with $y_3=Y\in\Lg_{E_3}$. (Some such proviso is needed: $y_j=\Omega_j/2$ gives $s=0$, $x=\pi i$.)
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:CL4}]
Let $K\supseteq k_1k_2k_3(i)$, $n=[K:\Q]\ge8$, $G=\Gm\times E_1\times E_2\times E_3$, $\Psi=\mathrm{id}$ (one slot per
coordinate). Subgroups of $G^n$ are products, $U=0\times B_{\ker\psi_1}\times B_{\ker\psi_2}\times B_{\ker\psi_3}$, and
$G''=\Gm^n\times A_1''\times A_2''\times A_3''$ with $\dim A_j''=n/2$: $d=5n/2$, $d_0=0$, $d_1=n$, $d_2=3n/2$, weight $4n$. Put
$w=(1,1,1,1)$, $p=(2\pi i,\Omega_1,\Omega_2,\Omega_3)$, $V=\C w\oplus\C p$, $W=\C w$ ($\Qb$-rational). So $V''=\Psi''^{-1}(V)$
has codimension $2$ and $W''=\Psi''^{-1}(W)$ is $\Qb$-rational of dimension $d-3$. A datum $v=sw+tp$ with $s\ne0$ lies in
$\Psi(\Lambda_K(G))$; $Y''$ is generated by lifts of $D\theta_hp$ and $\theta_hv$, so $m=2n$.

\emph{$\kappa=n$.} A $K$-period vector on $V$ is $s'w+t'p$ with $s'+2\pi it'\in K\cdot2\pi i$ and $s'+\Omega_jt'\in K\Omega_j$.
Writing $s'=2\pi i(a-t')=\Omega_j(b_j-t')$ and eliminating $t'$ between $j=1,2$ gives
$(b_1-b_2)/(2\pi i)-(b_1-a)/\Omega_2+(b_2-a)/\Omega_1=0$; as $1/\Omega_j\in\Qb^\times\eta_j^*/\pi$, this is a $\Qb$-linear relation
among $1,\eta_1^*,\eta_2^*$, trivial by W\"ustholz's theorem on $\Ga\times E_1^\natural\times E_2^\natural$. So $b_1=b_2=a$,
$s'=0$, and $V\cap\Psi(\Omega_K(G))=Kp$.

\emph{Lemma S: if three of the four coordinates of a datum $v=sw+tp$ are $K$-periods, then $s=0$.} If they are the
$E_j$-coordinates, $s=\Omega_j(b_j-t)$ for $j=1,2,3$; eliminating $t$ pairwise gives
$(b_2-b_3)/\Omega_1+(b_1-b_2)/\Omega_3+(b_3-b_1)/\Omega_2=0$, a $\Qb$-relation among $\eta_1^*,\eta_2^*,\eta_3^*$ (after
multiplying by $\pi$), trivial by W\"ustholz on $E_1^\natural\times E_2^\natural\times E_3^\natural$; so $b_1=b_2=b_3=t$ and $s=0$.
If they include the $\Gm$-coordinate, the computation of $\kappa$ applies. (The eliminations use $2\pi i\notin\Qb\Omega_j$ and
$\Omega_1/\Omega_2\notin\Qb$.)

\emph{Obstructions.} Connected subgroups are $G'=T_L\times B_1\times B_2\times B_3$; every nonzero factor maps onto its
coordinate line ($U$ is maximal), so $\Psi''(T_{G'})=\C^S$ for the set $S$ of nonzero factors, and
$\delta-\tau=3-|S|+[S=\{0,1,2,3\}]$. (\ref{eq:41}) reads $(\lambda+4n-\mathrm{wt}\,G')\cdot2\le(\delta-\tau)\cdot3n$:
for $S=\emptyset$, $\lambda=n$ and $10n\le9n$ fails; for one slot, $\mathrm{wt}\le n$ and $\lambda=n$ (by Lemma S, the datum's
other three coordinates are not all periods), so $8n\le6n$ fails (without Lemma S, $\lambda=0$ would give $6n\le6n$); for two
slots the left side is $\ge4n>3n$; for three or four slots $\delta>\tau$ fails. Condition (\ref{eq:NR}):
$2(5n/2-2)\ge2n$. So no datum with $s\ne0$ exists. The four-curve version has the same table (weight $4n$, Lemma S with any
three of the four $\eta^*$'s, (\ref{eq:NR}) $4n-4\ge n$).
\end{proof}

\noindent\emph{Controls.} With $W=0$, or without $\kappa$, one datum is allowed; a non-CM $E_3$ gives $G'=0$ at
equality. Theorem~\ref{thm:RW} on this plane ($\alpha_K=4$, $\kappa=1$, $\dim V/W=1$) gives the bound $3$ against $2$ vectors,
hence nothing.

\subsection{A unitary source of the period term}\label{sec:U}
Let $\mathbf K\subset\C$ be imaginary quadratic, $\iota$ the inclusion, and $A/\Qb$ an abelian variety of dimension $g$,
simple over $\Qb$, with $\mathcal O_{\mathbf K}\subset\mathrm{End}(A)$. Then $\mathrm{Lie}\,A=L_\iota\oplus L_{\bar\iota}$,
where $[a]$ acts by $\iota(a)$ on $L_\iota$ and by $\bar\iota(a)$ on $L_{\bar\iota}$; both are $\Qb$-rational. The signature is
$(r,s)=(\dim L_\iota,\dim L_{\bar\iota})$; for $A$ simple and $r\ge2$ we have $s\ge1$. Examples: Picard curves $y^3=f_4(x)$
(signature $(2,1)$; $dx/y^2$, $x\,dx/y^2$ span one eigenspace) and $y^3=f_6(x)$ (signature $(3,1)$), whose Jacobians are simple
for generic $f$ (simplicity must be checked for each curve). These are generically not of CM type. For $x\in\mathrm{Lie}\,A$
write $x=x_\iota+x_{\bar\iota}$.

\begin{lemma}\label{lem:Uper}
Assume $r\ge2$. For a nonzero period $\omega$ of $A$, $\omega_\iota\ne0$ and $\omega_{\bar\iota}\ne0$. If $r>s$ (for instance
if $s=1$) and $\omega,\omega'$ are periods with $\omega'_\iota\in\C\omega_\iota$, then $\omega'\in\mathbf K\omega$ in
$\Omega(A)\otimes\Q$.
\end{lemma}

\begin{proof}
If $\omega\in L_{\bar\iota}$, then $(\mathbf K\otimes\R)\omega$ is a complex line spanned by the rank-$2$ lattice
$\mathcal O_{\mathbf K}\omega$, an elliptic curve in $A$, impossible for $A$ simple of dimension $\ge2$. For the second claim
with $s=1$: if $\omega,\omega'$ are $\mathbf K$-independent with $\omega'_\iota=t\omega_\iota$, take $\theta_t\in\mathbf K\otimes\R$
with $\iota(\theta_t)=t$; then $0\ne\omega'-\theta_t\omega\in U_\R\cap L_{\bar\iota}$ for $U=\mathbf K\omega+\mathbf K\omega'$, and
$U_\R\cap L_{\bar\iota}$ is a $\mathbf K\otimes\R$-submodule, on which $\mathbf K\otimes\R$ acts through $\bar\iota$ by all complex
scalars; so it is a nonzero complex subspace of the line $L_{\bar\iota}$, and $L_{\bar\iota}\subset U_\R$. As $g\ge3$, $U$ lies
in the kernel of a nonzero $\mathbf K$-linear $\varphi\colon\Omega(A)_\Q\to\mathbf K$, whose extension kills $L_{\bar\iota}$ and is
complex-linear on $L_\iota$, hence is a holomorphic $1$-form with periods in $\iota(\mathbf K)$; the $\mathbf K$-span of
$\mathrm{Tr}\circ\varphi$ is a $2$-dimensional rational sub-Hodge structure of $H^1$, i.e.\ an elliptic quotient of $A$, a
contradiction. For $s\ge2$, $r>s$: Theorem~\ref{thm:W88} with $Y=\mathcal O_{\mathbf K}\omega+\mathcal O_{\mathbf K}\omega'$ as in the
proof below ($\kappa=4$, $\lambda=0$, and $d-N=r-1\ge1$ since $r\ge2$) gives $2g(r-1)\le r(2g-4)$, i.e.\ $r\le s$, a
contradiction; here (\ref{eq:NR}) holds strictly.
\end{proof}

\noindent The first claim needs only $g\ge2$. In the second, $r\ge2$ is what makes ``$s=1$'' imply $r>s$: the formulation ``if $s=1$
or $r>s$'' would fail at $(r,s)=(1,1)$, $g=2$, where $L_\iota$ is a line, so $\omega'_\iota\in\C\omega_\iota$ holds for
all periods $\omega,\omega'$, while $\Omega(A)\otimes\Q$ has $\mathbf K$-dimension $2$. (The proof uses $g\ge3$ for $s=1$, and
$d-N\ge1$ for $s\ge2$.) Theorem~\ref{thm:U} assumes $r\ge2$.

\begin{theorem}[Theorem U]\label{thm:U}
Let $A$ be as above, of signature $(r,s)$ with $r\ge2$, and $\omega$ a nonzero period.
\begin{itemize}
\item[(U1)] If $r\ge s+2$ and $u\in\Lambda(A)$ has $u_\iota\in\C\omega_\iota$, then $\exp_A(u)$ is torsion.
\item[(U2)] The $u\in\Lambda(A)$ with $u_\iota\in\C\omega_\iota$ form a group whose image in $A(\Qb)/\mathrm{tors}$ is an
$\mathcal O_{\mathbf K}$-module of rank $\le\lfloor s/(r-1)\rfloor$ ($\le1$ for Picard curves).
\item[(U3)] If $r\ge s+1$ and $u\in\Lambda(A)$ has $u_\iota=t\,\omega_\iota$ with $2\pi it\in\Lg+\iota(\tau)\Lg$
($\mathcal O_{\mathbf K}=\Z[\tau]$), then $t\in\mathbf K$ and $\exp_A(u)$ is torsion; in particular
$u_\iota=(\log\alpha/2\pi i)\,\omega_\iota$ with $\alpha\in\Qb^\times$ forces $\alpha$ to be a root of unity.
\item[(U4)] If $r\ge s+1$, $E$ is a CM curve with CM by $\mathcal O_{\mathbf K}$ (acting on $\mathrm{Lie}\,E$ through $\iota$) with
period $\Omega$, and $u\in\Lambda(A)$, $v\in\Lg_E$ satisfy $u_\iota=(v/\Omega)\,\omega_\iota$, then $\exp_E(v)$ and $\exp_A(u)$ are
torsion.
\end{itemize}
\end{theorem}

\noindent Concretely, on a simple Picard Jacobian,
\[
\Bigl(\int^P\frac{dx}{y^2},\int^P\frac{x\,dx}{y^2}\Bigr)=\frac{\log\alpha}{2\pi i}\Bigl(\int_\gamma\frac{dx}{y^2},\int_\gamma\frac{x\,dx}{y^2}\Bigr)
\]
only for $P$ torsion and $\alpha$ a root of unity: the unitary analogue of $\wp(\vp\log\alpha/\pi)\notin\Qb$ (\cite[Cor.~6.2, (M2)]{Note}).

\begin{proof}
Let $G=A\times E^a\times T^b$ ($a,b\in\{0,1\}$; $T=\Gm\otimes\mathcal O_{\mathbf K}$, a split torus of dimension $2$), so
$d=g+a+2b$, $d_0=0$, $d_1=2b$, $d_2=g+a$, weight $2g+2a+2b$. $L_\iota(G)=L_\iota(A)\oplus\mathrm{Lie}\,E^a\oplus\C_\iota^b$ has
dimension $R=r+a+b$, and $L_{\bar\iota}(G)=L_{\bar\iota}(A)\oplus\C_{\bar\iota}^b$ has dimension $S=s+b$; both are $\Qb$-rational. Let
$w=(\omega,\Omega^a,(2\pi i\otimes1)^b)$ be the period vector, and
\[
V=\C w_\iota\oplus L_{\bar\iota}(G),\quad W=L_{\bar\iota}(G),\quad N=1+S,\quad t=S,\quad d-N=R-1 .
\]
Since $[a]$ acts on $L_\iota(G)$ by the scalar $\iota(a)$, $\mathcal O_{\mathbf K}w\subset V$, and every datum below lies in $V$.

\emph{Data.} A datum is $y\in\Lambda(G)$ with $y_\iota=t'w_\iota$; put $Y=\mathcal O_{\mathbf K}w+\sum_j\mathcal O_{\mathbf K}y_j$.
If a component of $y$ is torsion, the datum is trivial: e.g.\ if its torus component is torsion then $t'\in\iota(\mathbf K)$ and
$y-[c]w\in W$ for some $c\in\mathbf K$, a $\Qb$-rational subspace containing a logarithm of an algebraic point, so by
W\"ustholz's theorem it lies in the Lie algebra of an algebraic subgroup inside $L_{\bar\iota}(G)$, which is $0$; if the
$A$-component is torsion and $r>s$, Lemma~\ref{lem:Uper} gives $t'\in\mathbf K$. So for nontrivial data each orbit
$\mathcal O_{\mathbf K}y_j$ has rank $2$ modulo $\mathcal O_{\mathbf K}w$, $\kappa=2$, and for $G'\subset G$,
$\lambda(G')\ge2$ per independent datum unless $G'$ contains all factors of the datum (a rational line $l\subset T$ meets an
orbit $\mathcal O_{\mathbf K}z$ of a non-torsion $z$ in rank $\le1$).

\emph{Subgroups.} $A$ and $E$ are simple, $\mathrm{Hom}(A,E)=0$, and there are no homomorphisms between abelian varieties and
tori; so $G'=A'\times E'\times T_L$ with $A'\in\{0,A\}$, $E'\in\{0,E\}$, $T_L\in\{0,T,\text{a rational line }l\}$, and
$\tau=S-\dim(W\cap T_{G'})$ with $W\cap T_{G'}=L_{\bar\iota}(A')+(\C_{\bar\iota}$ if $T_L=T)$.

\emph{The inequality.} $G'$ obstructs iff $\delta>\tau$ and $(\lambda+\delta_1+2\delta_2)(R-1)\le(\delta-\tau)(2b+2g+2a-2)$.
For $G=A$ ($a=b=0$), $G'=0$ and $k$ data: $\lambda=2k$, $\delta=g$, $\tau=s$, and the inequality is
$(2k+2g)(r-1)\le r(2g-2)$, i.e.\ $k(r-1)\le s$. So $k$ data are excluded iff $k>s/(r-1)$; with $G'=A$ one has
$\delta=0$. This gives (U1) ($k=1$, $r\ge s+2$) and (U2). For (U3), (U4) take $G=A\times T$, resp.\ $A\times E$, and one nontrivial datum $y$. Then $d-N=R-1=r$ and
$d_1+2d_2-\kappa=2g$ in both cases, and $\lambda=2$ for every $G'$ in the table below except $A\times l$, by the remark on orbits above (for $G'=l$ a point of
$\Gamma\cap G'$ has torsion $A$-component). The tangent line of
a one-dimensional subtorus $l\subset T$ is not an eigenline of $\mathcal O_{\mathbf K}$ (it comes from a rank-one subgroup of the
cocharacters, which is not an $\mathcal O_{\mathbf K}$-module), so $W\cap T_l=0$. The proper connected subgroups are:
\begin{center}
{\small
\begin{tabular}{llllllll}
$G$ & $G'$ & $\delta$ & $\delta_1$ & $\delta_2$ & $\tau$ & $(\lambda+\delta_1+2\delta_2)\,r$ & $(\delta-\tau)\cdot2g$\\\hline
$A\times T$ & $0$ & $g+2$ & $2$ & $g$ & $s+1$ & $(2g+4)r$ & $2g(r+1)$\\
 & $A$ & $2$ & $2$ & $0$ & $1$ & $4r$ & $2g$\\
 & $T$ & $g$ & $0$ & $g$ & $s$ & $(2g+2)r$ & $2gr$\\
 & $l$ & $g+1$ & $1$ & $g$ & $s+1$ & $(2g+3)r$ & $2gr$\\
 & $A\times l$ & $1$ & $1$ & $0$ & $1$ & \multicolumn{2}{l}{$\delta=\tau$}\\
$A\times E$ & $0$ & $g+1$ & $0$ & $g+1$ & $s$ & $(2g+4)r$ & $2g(r+1)$\\
 & $A$ & $1$ & $0$ & $1$ & $0$ & $4r$ & $2g$\\
 & $E$ & $g$ & $0$ & $g$ & $s$ & $(2g+2)r$ & $2gr$
\end{tabular}
}
\end{center}
A subgroup obstructs iff the left column is at most the right one. Since $g=r+s$, both $(2g+4)r\le2g(r+1)$ and $4r\le2g$ are
equivalent to $r\le s$, and the rows $T$, $l$, $E$ never obstruct. So for $r\ge s+1$ no subgroup satisfies (\ref{eq:41}). For
Picard $(2,1)$ ($g=3$) the rows read $20>18$, $8>6$, $16>12$, $18>12$ for $A\times T$ and $20>18$, $8>6$, $16>12$ for $A\times E$.
Condition (\ref{eq:NR}): $2+2S\ge2b+2$, strict as $s\ge1$. For (U3), $t\in\mathbf K$ and $\log\alpha=2\pi i
(c_1+c_2\sqrt{-D})$ give $|\alpha|=e^{-2\pi c_2\sqrt D}$, and Gel'fond--Schneider forces $c_2=0$.
\end{proof}

\noindent\emph{Controls.} With the period term deleted, or with $W=0$, the count obstructs; for Picard $(2,1)$ one datum is
allowed ($8\le8$ at $G'=0$); for a non-simple $A=B\times E_\rho$ of signature $(3,1)$, (U1) is false and the count finds the
obstruction $G'=B$. \emph{Provenance.} (U2) for $r=2$ (Picard curves included) is also the case $K=\Q$, $\Psi=\mathrm{id}$, $G=A$, $V$ a
hyperplane, of Theorem~\ref{thm:RW}, i.e.\ of Theorem~\ref{thm:W88k} ($\alpha=2g$, $\kappa=2$, so $2+2k\le2g-2$, i.e.\ $k\le s$). For $r\ge3$, and for (U1), (U3), (U4), $V$ has codimension
$r-1\ge2$ in $T_G$ (resp.\ $R-1$), and the subgroup-by-subgroup count gives more than Theorem~\ref{thm:RW}; for $K\supseteq\mathbf K$
the condition on $V$ in Theorem~\ref{thm:RW} fails, since $L_{\bar\iota}\in S_K(A)$. The relation $u_\iota\parallel\omega_\iota$ is bilinear in a logarithm and a
period of the same abelian variety; it is $\Qb$-linear only for algebraic $t$ \lab{Heuristic: not linearisable}.

\section{Formal verification and literature}\label{sec:formal}

\emph{Formal verification.} Two statements of this paper have been checked in Lean~4 with Mathlib, both relative to explicit
hypotheses; everything else is not formalised.
\begin{itemize}
\item Theorem~\ref{thm:X7}, with its torsion clause, and Corollary~\ref{cor:X7}(a), (b) (for Abel--Jacobi integrals) and (c) are
checked in the certificate \cite{LM} (declarations \texttt{CM4E.\allowbreak AbVar.\allowbreak thmX7}, \texttt{thmX7\_\allowbreak torsion}, \texttt{x7a\_\allowbreak AJ},
\texttt{x7b\_\allowbreak AJ}, \texttt{x7c}) \emph{conditionally} on a named instance of Waldschmidt's theorem for the genus-three group
(\texttt{Thm11KappaX7}), which neither \cite{LM} nor \cite{LSG} proves, and on named inputs (Chowla--Selberg at level seven,
and forms of W\"ustholz's theorem; for the torsion clause and the Abel--Jacobi forms also a named torsion step). Lean has no
Jacobians: the logarithms of $A$ enter as a parameter set, and Abel--Jacobi integrals with endpoints at the branch points are not
covered. The generic Theorem~\ref{thm:X}, without the torsion
clause, is checked conditionally in the same way (\texttt{AbVar.\allowbreak XSetup.\allowbreak thmXA}). In the terminology of \cite[\S12]{KrL} these are of
status W (verified conditionally on a named form of \cite{W88} that is not proved).
\item Theorem~\ref{thm:SE}(1) for $(E,E')=(E_i,E_\rho)$, at $t=c\,\Omega_\rho/\vp$ ($c\in\Qb^\times$), is checked
(\texttt{CM4E.\allowbreak Unconditional.\allowbreak thmSE\_\allowbreak varpi\_\allowbreak omegaRho}) with no form of \cite{W88} assumed: the instance of Waldschmidt's theorem it
uses is proved in the Lean project \cite{LSG} described in \cite{KrL}, and the one hypothesis left is that $g_3$ of the lattice
$\Omega_\rho\Z[\rho]$ is algebraic ($g_3=4$ is classical but not formalised). This is status I of \cite[\S12]{KrL}.
\end{itemize}
In both cases the Lean proofs go through the hyperplane theorem (Theorem~\ref{thm:W88k}), not through our case checks, and the group
models are hand-written and reviewed, not derived in Lean. Theorems~\ref{thm:X924}, \ref{thm:Xk}, \ref{thm:SE}(2),
\ref{thm:EA}, \ref{thm:QL}, \ref{thm:C42}, \ref{thm:CL4}, \ref{thm:U} and their corollaries are not formalised.

\subsection{What we read}\label{sec:lit}
\emph{Quadratic relations.} Gao, Ullmo and Yafaev \cite{GUY} and Gao and Ullmo \cite{GU} describe algebraic relations among
holomorphic CM periods beyond the linear ones as largely open, apart from cases reducible to the linear case such as
$\theta\theta'/\pi\notin\Qb$ \cite[Cor.~3.1]{GUY}, and ask whether all quadratic relations are elementary \cite[Question~1.3]{GUY};
the relations used in Theorems~\ref{thm:X} and~\ref{thm:Xk} are elementary in their sense. Wolfart and W\"ustholz \cite{WW85}
determine the $\Qb$-linear relations among Beta values; Corollary~\ref{cor:X7intro}(b) involves the quotient
$B(\frac37,\frac37)/B(\frac17,\frac17)$, whose Koblitz--Ogus function takes three values, so it is not covered. Huber and
W\"ustholz \cite{HW} prove the period conjecture for $1$-periods; they exclude products of periods explicitly
(\cite[Remark~13.2]{HW}, and the Prologue), and their Beta material is linear. By Remark~\ref{rem:multiplier}, no Gamma monomial
reduces our lists to theirs within the Koblitz--Ogus dictionary.

\emph{The other results.} For Theorem~\ref{thm:C42}: Waldschmidt's survey \cite{W06} (Conjecture~42 and Thm.~24), the account of
Ramachandra's $\delta$-conjecture in \cite{W13}, and \S3.5 of \cite{W73} (one-parameter subgroups; mixed $E\times\Gm$ is not treated
there, and there is no CM refinement). For Theorems~\ref{thm:SE}, \ref{thm:EA}: \cite[Cor.~1]{RW93} (elliptic and abelian strong six
exponentials) and the conjugation arguments of Diaz \cite{Di04} and Waldschmidt \cite[Cor.~2.7, Thm.~4.1]{W07}, which give the
non-conjugation-stable cases as said in the introduction; \cite[Thms.~4.2, 4.3]{W05}, which need many more data and involve no
period slope. For Theorem~\ref{thm:QL}: Laurent \cite{La82}, whose results are of degree one. For Theorem~\ref{thm:CL4}: the count
\cite[(4.2), Cor.~4.3]{W88} and \cite[Thm.~4.2]{W05} (mixed $\Ga\times\Gm\times E$, seven columns, no CM); the numbers in
Corollary~\ref{cor:CL4} are artificial, and the content is the mechanism. For Theorem~\ref{thm:U}: \cite{W88,W83,RW93} (for
signature $(r,s)$ with $r,s\ge1$ the weight of \cite{RW93} is $\alpha_K(A)=2g$, with no saving) and \cite{HW}, which gives only
the case $t\in\Qb$. Roy's Luminy paper \cite{R1} (read in full) has only general bounds, not statements at particular periods. We
also looked in Waldschmidt's survey \cite{W22} (read in full) and in \cite{W05,W07}, and searched arXiv for 2024--26 papers stating
four exponentials or a subgroup-theorem count at CM, Beta or abelian periods, without finding any.

\emph{Not consulted.} We could not consult Bertrand \cite{Be83} (Schneider--Lang theorems on abelian varieties with endomorphisms;
the most likely earlier source for Theorem~\ref{thm:U}), Waldschmidt \cite{W90}, Baker--W\"ustholz \cite[\S6.2]{BW07}, Ramachandra's
papers in Acta Arith.\ 14 (1968), or the MathSciNet citation lists of \cite{RW93} and \cite{W88} (the zbMATH lists we saw are nearly
empty). Any of these might contain some of our statements. Every result here is a specialisation of the same published theorems,
so this residual risk is shared by all of them.

\emph{Review.} All review so far is by AI referees; no human expert has reviewed the definitions or the proofs. The results and
proofs were found and written with substantial use of AI tools. Recent literature in this area includes unreviewed, partly
AI-generated preprints; we cite none of them.

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