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\begin{document}

\title{On the Hodge, Tate, and standard conjectures for abelian varieties}
\date{\today}
\author{J.S. Milne}
\maketitle

\hfill\begin{minipage}{3.2in}
\textit{To the memory of my teacher, John Tate, with regret
that he did not see this support for his vision.}
\end{minipage}\bigskip

This is a report on two preprints released by 
OpenAI on October 6, 2026,

\begin{itemize}
\item Milne's rationality conjecture for abelian varieties,\ 

\item The rational Hodge conjecture for CM abelian varieties.
\end{itemize}

and some of their consquences. In particular, I place the preprints in
context.\footnote{Expanded version of my notes for a talk at the University of
Michigan, Ann Arbor, November 5, 2026.}

\section{The rationality conjecture\hfill}

\begin{notation}
\label{h1}Everything is $\otimes\mathbb{Q}{}$. For example, abelian varieties
are objects in the category in which $\Hom$s are $\Hom^{0}%
(A,B)=\Hom(A,B)\otimes\mathbb{Q}$. The Lefschetz classes are the classes in
the $\mathbb{Q}$-algebra generated by divisor classes. An abelian variety in
characteristic zero is CM if its endomorphism algebra is largest possible.
\end{notation}

\subsection{Statement of the rationality conjecture}

Let $\mathbb{Q}{}^{\mathrm{al}}$ be an algebraic closure of $\mathbb{Q}{}$ and
$w$ a place of $\mathbb{Q}{}^{\mathrm{al}}$ lying over $p$. The residue field
$\mathbb{F}{}$ at $w$ is an algebraic closure of $\mathbb{F}{}_{p}$.

\begin{plain}
\label{h2}Let $A$ be an abelian variety of dimension $n$ over $\mathbb{Q}%
{}^{\mathrm{al}}$ with good reduction to an abelian variety $A_{0}$ over
$\mathbb{F}{}_{p}$. Let $\gamma$ be an algebraic class on $A$ and $\delta$ a
Lefschetz class on $A_{0}$ of complementary codimension,
\[
\begin{tikzcd}
A\arrow[-]{d}&H^{2\ast}(A,\mathbb{Q}_{\ell}(\ast))\arrow{d}{\simeq}
&\gamma\arrow[mapsto]{r}&\gamma_{\ell}\\
A_{0}&H^{2\ast}(A_0,\mathbb{Q}_{\ell}(\ast))&\delta\arrow[mapsto]{r}
&\delta_{\ell},\,\,(\gamma_{\ell})_0.
\end{tikzcd}
\]
Then
\[
\langle(\gamma_{\ell})_{0}\cdot\delta_{\ell}\rangle\in H_{\mathrm{et}}%
^{2n}(A_{0},\mathbb{Q}{}_{\ell}(n))\simeq\mathbb{Q}{}_{\ell},
\]
and we can ask whether this $\ell$-adic number is a rational number
independent of $\ell$. In this case, the answer is obviously yes, because
$\gamma$ reduces to an algebraic class $\gamma_{0}$ on $A_{0}$ and
$\langle(\gamma_{\ell})_{0}\cdot\delta_{\ell}\rangle=\langle\gamma_{0}%
\cdot\delta\rangle\in\mathbb{Q}$.
\[
\begin{tikzcd}
\gamma\arrow[mapsto]{r}&\gamma_\ell\\
\delta,\gamma_0\arrow[mapsto]{r}&\delta_\ell,(\gamma_0)_\ell=(\gamma_\ell)_0
\end{tikzcd}
\]
The conjecture says that the same should be true if $\gamma$ is an absolute
Hodge class.
\end{plain}

\begin{conjecture}
\label{h3}Let $A$ be an abelian variety over $\mathbb{Q}{}^{\mathrm{al}}$ with
good reduction at w. If $\gamma$ is an absolute Hodge class on $A$ and
$\delta$ is a Lefschetz class on of complementary dimension, then
$\langle\gamma_{0}\cdot\delta\rangle$, which a priori lies
in\footnote{Implicitly, $\ell\neq p$. There is also a $p$-factor, which I'm
ignoring for the sake of simplicity.} $\prod\mathbb{Q}_{\ell}$, actually lies
in $\mathbb{Q}\subset\prod\mathbb{Q}_{\ell}$.
\end{conjecture}

\begin{plain}
\label{h4}What is an absolute Hodge class? Let $A$ be an abelian variety over
$\mathbb{Q}{}^{\mathrm{al}}$. Deligne (1982)\nocite{deligne1982} showed that
there exists a $\mathbb{Q}{}$-subspace $\mathcal{B}{}^{r}(A)$ of
$\prod\nolimits_{\ell}H_{\mathrm{et}}^{2r}(A,\mathbb{Q}{}_{\ell}(r))$ mapping
isomorphically onto $\mathcal{B}{}^{r}(A_{\mathbb{C}{}})\overset{\df}{=}%
H^{2r}(A_{\mathbb{C}{}},\mathbb{Q}{}(r))\cap H^{p,p}\subset\prod
\nolimits_{\ell}H_{\mathrm{et}}^{2r}(A,\mathbb{Q}{}_{\ell}(r))$ for every
homomorphism $\mathbb{Q}^{\mathrm{al}}\rightarrow\mathbb{C}$,
\[
\begin{tikzcd}
\dstyle\prod\nolimits_{\ell}H^{2r}(A,\mathbb{Q}_{\ell}(r))\arrow{r}{\simeq}
&\dstyle\prod\nolimits_{\ell}H^{2r}(A_{\mathbb{C}},\mathbb{Q}_{\ell}(r))\\
\mathcal{B}^{r}(A)\arrow{r}{\simeq}\arrow[hook]{u}
&H^{2r}(A_{\mathbb{C}},\mathbb{Q}(r))\cap H^{r,r}\arrow[hook]{u}.%
\end{tikzcd}
\]
The classes in $\mathcal{B}{}^{r}(A)$ are the absolute Hodge classes on $A$.

The condition for a class in $\prod\nolimits_{\ell}H_{\mathrm{et}}%
^{2r}(A,\mathbb{Q}{}_{\ell}(r))$ to be absolutely Hodge is transcendental, and
there is no rational $\gamma_{0}$. Whence the problem.
\end{plain}

\begin{plain}
\label{h5}What was my motivation for the conjecture? Deligne's theorem allows
you to define a category of abelian motives over a field of characteristic
zero with all the properties Grothendieck's category would have if the Hodge
conjecture were known. The conjecture allows you to extend this to mixed
characteristic. There are many heuristic arguments proving things assuming a
sufficiently good theory of motives. With the proof of the conjecture, and the
existence of the categories of motives, some of these heuristic arguments
become proofs.
\end{plain}

\subsection{Proof of the rationality conjecture.}

\begin{theorem}
[OpenAI]\label{h6}The rationality conjecture is true for all abelian varieties.
\end{theorem}

\begin{plain}
\label{h7}There is an obvious approach to proving the conjecture. Let
$A,\gamma,\delta$ be as in the statement of the conjecture. If we can lift
$(A_{0},\gamma_{0},\delta)$ to $(A^{\prime},\gamma^{\prime},\delta^{\prime})$
in characteristic zero with $\gamma^{\prime}$ an absolute Hodge class and
$\delta^{\prime}$ a Lefschetz class,
\[
\begin{tikzcd}
(A,\gamma)\arrow{rd}
&&(A^{\prime},\gamma^{\prime},\delta^{\prime})\arrow{ld},\\
&(A_0,\gamma_0,\delta)
\end{tikzcd}
\]
then%
\[
\langle\gamma_{0}\cdot\delta\rangle=\langle\gamma_{0}^{\prime}\cdot\delta
_{0}^{\prime}\rangle=\langle\gamma^{\prime}\cdot\delta^{\prime}\rangle
\in\mathbb{Q},
\]
because $\gamma^{\prime}$ and $\delta^{\prime}$ are both absolute Hodge
classes. Sometimes it is possible to lift $(A_{0},\gamma_{0},\delta)$ and
sometimes it isn't. The problem is to show that it is possible to lift it
often enough to force rationality.
\end{plain}

\begin{plain}
\label{h8}First, two observations:

\begin{itemize}
\item an induction argument shows that it suffices to lift enough divisors
(not Lefschetz classes);

\item if $A$ is a polarized abelian variety, then%
\[
\text{NS}(A)_{\mathbb{Q}{}}\simeq\{u\in\End^{0}(A)\mid u^{\dagger}=u\},\quad
D_{u}\leftrightarrow u,
\]
where $\dagger$ is the Rosati involution defined by the polarization, so
lifting a divisor class is much the same as lifting an endomorphism.
\end{itemize}
\end{plain}

\begin{plain}
\label{h9}Lifting theorems for abelian varieties have a long history. In the
1930s, Deuring showed that it is always possible to lift an elliptic curve and
an endomorphism from characteristic $p$ to characteristic $0$. In the proof of
his part of the Honda-Tate theorem, Honda showed that you can always lift an
abelian variety to a CM abelian variety in characteristic zero.\footnote{It is
not always possible to lift an abelian variety with an endomorphism (even up
to isogeny). Let $E$ be a field of degree $2g$ over $\mathbb{Q}{}$ acting on
an abelian variety $A$ of dimension $g$. In characteristic zero, $E$ must be a
CM-field, but in characteristic $p$ this is not necessarily so.} The
conjecture of Langlands and Rapoport predicts that the reduction of any point
on a Shimura variety lifts to a CM-point\ldots The most recent lifting result
is Theorem 2.2.7 of \cite{kisinZ2025}.
\end{plain}

\begin{plain}
\label{h10}It has long been known that it suffices prove the rationality
conjecture for Weil classes on CM abelian varieties. This means that we have
the following.

\begin{itemize}
\item $A$ an abelian variety of CM-type over $\mathbb{Q}{}^{\mathrm{al}}$ (of
dimension $g$).

\item An action $\nu\colon E\rightarrow\End^{0}(A)$ of a CM field $E$ such
that $\bigwedge_{E}^{2r}H^{1}(A_{\mathbb{C}{}},\mathbb{Q}{})\subset
H^{2r}(A_{\mathbb{C}{}},\mathbb{Q}{})$ consists of Hodge classes (i.e.,
classes of type $(r,r)$. Here $2r=\dim_{E}H^{1}(A_{\mathbb{C}{}},\mathbb{Q}%
{})$. Correspondingly, we get a $\mathbb{Q}{}$-vector space $W_{E}^{r}(A)$ of
absolute Hodge classes on $A$, which we call Weil classes.

\item A polarization $\lambda$ of $A$ whose Rosati involution involution acts
as complex conjugation on $E$.
\end{itemize}

\noindent Given such a triple, we have to prove that $\langle\gamma_{0}%
\cdot\delta\rangle\in\mathbb{Q}{}$ for all Weil classes $\gamma$ on $A$ and
Lefschetz classes of codimension $g-r$ on $A_{0}$.
\end{plain}

\begin{plain}
\label{h11}We seek a set $S$ of Lefschetz classes on $A_{0}$ such that

\begin{enumerate}
\item $S$ is big enough:
\begin{align*}
\langle\gamma_{0}\cdot\delta\rangle &  \in\mathbb{Q}{}\text{ for all Weil
classes }\gamma\text{ and }\delta\in S\\
&  \implies\langle\gamma_{0}\cdot\delta\rangle\in\mathbb{Q}{}~\text{for all
Weil classes }\gamma\text{ and \textit{all }}\delta;
\end{align*}


\item $S$ is not too big: it is possible to lift $(A_{0},\nu_{0},\lambda
_{0},\delta)$ to characteristic zero.
\end{enumerate}

\noindent OpenAI gives the following answer. Let%
\[
U=\{u\in\End(A)\mid u=u^{\dagger},\quad ue=\bar{e}u\text{ for all }e\in E\};
\]
then%
\[
S=\{D_{u}^{r}\cdot\xi^{g-2r}\mid u\text{ is an invertible element of }U\}
\]
works. Here $\xi$ is the divisor class of the polarization.

Given $S$, it only takes a page to prove that it satisfies (a) and a couple of
pages to deduce (b) from the lifting theorem of Kisin and Zhou. See
\cite{madapusi2026}.
\end{plain}

\begin{plain}
\label{h12}There are precedents for this argument. For example, there is an
old theorem (\cite{zink1983}, 2.7) stating that, under some conditions, it is
possible to lift all semisimple subalgebras of $\End_{E}^{0}(A_{0})$, which
allows you to prove rationality for some Weil classes.
\end{plain}

\bigskip I was not surprised that my conjecture was proved --- indeed, for the
last twenty years I've been trying to persuade everyone that it was
accessible. However, as were most mathematicians, I was surprised by the next result.

\section{The Hodge conjecture for CM abelian varieties}

\begin{theorem}
[OpenAI]\label{h13}The Hodge conjecture is true for abelian varieties of CM type.
\end{theorem}

\begin{plain}
\label{h14}Why is this so important? After all, we know the Hodge conjecture
for many abelian varieties. Indeed, we know the Hodge conjecture for any
variety for which the \textquotedblleft known\textquotedblright\ algebraic
classes exhaust the $\mathbb{Q}{}$-algebra of Hodge classes. For example,
since we know the Hodge conjecture for the $(1,1)$ Hodge classes, we know the
conjecture for any variety for which these classes generate the $\mathbb{Q}{}%
$-algebra of Hodge classes. Results of this nature, while interesting, tell us
nothing about the veracity of the Hodge conjecture. For that, we have to prove
the conjecture for a class of algebraic variety for which the known classes do
not exhaust the Hodge classes. The abelian varieties of CM type are such a class.
\end{plain}

\begin{plain}
\label{h15}The Hodge conjecture for CM is not known to imply the Hodge
conjecture for all abelian varieties, but it would be strange if this wasn't
true. The CM-points are dense in the moduli space of abelian varieties. If the
$\mathbb{Q}{}$-algebra of algebraic classes differed from the $\mathbb{Q}%
$-algebra of Hodge classes at some point, it would be very odd if this doesn't
persist over a CM point.
\end{plain}

\begin{plain}
\label{h16}Finally, as we shall see, once the Hodge conjecture is known for CM
abelian varieties, it is possible to prove other very interesting results.
\end{plain}

We now sketch the proof of the conjecture. Note that this all takes place over
$\mathbb{C}$, so much of it is complex analysis.

\begin{E}
[\cite{hazama2000,hazama2003}]\label{h17} It suffices to prove the conjecture
in codimension $2$.
\end{E}

Fix a (large) CM-field $K$ finite and Galois over $\mathbb{Q}{}$ with Galois
group $G$. To give a CM abelian variety over $\mathbb{C}{}$ split by $K$,
i.e., such that $\End^{0}(A)\otimes_{\mathbb{Q}{}}K$ is a product of matrix
algebras over $K$, amounts to giving a finite $G$-set $S$ and a subset $S^{+}$
of $S$ such that $S=S^{+}\sqcup\iota S^{+}$. Here $\iota$ is complex
conjugation. For example, if $A$ has complex multiplication by a CM-algebra
$E$, then $S=\Hom(E,K)$ and $S^{+}$ is the CM type. If $A$ corresponds to
$(S,S^{+})$, then it is possible to read off $H^{\ast}(A,\mathbb{Q}{})$ with
its Hodge structure from $(S,S^{+})$, and even $H^{\ast}(A^{n},\mathbb{Q}{})$
because $A^{m}$ corresponds to the disjoint union of $n$ copies of $(S,S^{+}%
)$. Let $[K\colon\mathbb{Q}]=2m$. Let $H_{1},\ldots,H_{m}$ be the coordinate
hyperplanes in $\mathbb{R}^{m}$, and take $S$ to be the set of connected
regions of $\mathbb{R}{}^{m}\smallsetminus\bigcup H_{i}$, and $S^{+}$ the
subset on which $x_{1}>0.$ For a suitable action of $G$ on $\mathbb{R}^{m}$,
hence on $(S,S^{+})$, there is a CM abelian variety $A(S,S^{+})$ over
$\mathbb{C}{}$ and Hazama proves:

\begin{enumerate}
\item for every $n\geq0$, the $\mathbb{Q}{}$-algebra $\mathcal{B}{}^{\ast
}(A^{n})$ of Hodge classes is generated by the classes of degree $\leq2$;

\item every CM abelian variety split by $K$ is isogenous to an isogeny factor
of $A^{n}$ for some $n$.
\end{enumerate}

\noindent Thus, if the Hodge conjecture holds in codimension $\leq2$ for
abelian varieties split by $K$, then it holds in all codimensions for for
$A(S,S^{+})^{n}$ by (a), and hence for every CM abelian variety split by $K$
by (b).

\begin{E}
\label{h18} It suffices to prove the conjecture in dimension $2.$
\end{E}

The standard conjecture of Lefschetz type is known for abelian varieties: the
map on algebraic cycles%
\[
L^{n-4}\colon\mathcal{A}^{2}(A)\overset{\lsimeq}{\longrightarrow}\mathcal{A}%
{}^{n-2}(A)
\]
is an isomorphism. Here $L$ is the Lefschetz operator (cup-product with the
class of an ample divisor).

\begin{E}
\label{h19} Proof of the conjecture in dimension $2$.
\end{E}

Thus, the problem comes down to embedding surfaces into $A$. Let $S$ be a
surface. Suppose that the Hodge structure $H^{1}(A,\mathbb{Q}{})$ occurs in
$H^{1}(S,\mathbb{Q)}$. Then $A$ is an isogeny factor of the albanese of $S$,
and so we get a map $f\colon S\rightarrow A$. For $S$ we take Picard surfaces.
These are quotients of the 2-dimensional complex ball%
\[
\mathbb{B}{}^{2}=\left\{  (z_{1},z_{2})\in\mathbb{C}{}^{2}\mid|z_{1}%
|^{2}+|z_{2}|^{2}<1\right\}
\]
by a discrete group attached to a three-dimensional hermitian space%
\[
\psi\colon V\times V\rightarrow E
\]
over a CM-field $E$ such that $\psi$ has signature $(2,1)$ at one embedding of
the totally real subfield of $E$ and signature $(3,0)$ at all other
embeddings. From the \cite{liu2014,liu2021}, we find that the Hodge structure
$H^{1}(A,\mathbb{Q}{})$ occurs in $H^{1}(S,\mathbb{Q}{})$ for $S$ a Picard
surface. Now comes the hard part. We have to show that $f(S)$ is not
numerically equivalent to zero. For this, we have to prove that $Z\cdot
f(S)\neq0$ for some Hodge class $Z$ on $A$ of codimension 2. Here, we refer
the reader to \cite{hoften2026}, which in addition to the original OpenAI
manuscript draws on the manuscript \cite{bergeron2016}, which proves the Hodge
conjecture in small degrees for some arithmetic locally symmetric varieties.

\section{The Hodge, Tate, and standard conjectures}

The Hodge, Tate, and Grothendieck standard conjectures were stated in the
1950s and 1960s, and have haunted mathematicians ever since. Except for
divisors on abelian varieties (Tate, Zarhin, Faltings), little progress has
been made in proving the conjectures. Thus, the recent proofs for abelian
fourfolds have attracted considerable attention.\bigskip

\noindent%
\begin{tabular}
[c]{lll}\hline
Conjecture & Abelian fourfolds & General abelian varieties\\\hline
Hodge 1950 & Markman 2025\footnotemark & CM abelian varieties 2026 (OpenAI)\\
Tate 1963 & Broe 2026 & Divisors on AVs over finite fields 1966 (Tate)\\
&  & All abelian varieties over finite fields 1999* (Milne)\\
Grothendieck 1968 & Ancona 2021 & All abelian varieties 2002* (Milne)
\end{tabular}
\footnotetext{In his 2026 ICM talk, Markman said he had a program for proving that
Weil classes on all abelian varieties are algebraic. In particular, this would
imply OpenAI's result.} \footnotetext{On the last occasion that I saw Tate, I
asked him if he believed the Tate conjecture. He said he did for divisors, but
otherwise he had no idea. He would have been intereseted in this
result.}\medskip

\noindent Now we have the statements at right. The asterisk means that the
results were conditional until 2026. Since the proofs of the asterisked
results have suddenly become much more interesting (because unconditional), I
shall explain them. Both are applications of the theory of tannakian
categories.\footnote{The lesson of tannakian categories is that, as Galois
taught us 200 years ago, when you have a complicated situation, you should try
to understand it in terms of groups. See my book, \textit{Tannakian
Categories}, a near-final copy of which is available on website.}

\subsection{The Tate conjecture for abelian varieties over finite fields}

\begin{theorem}
[\textnf{\cite{milne1999LC,milne1999LM}}]\label{h20}The full Tate conjecture
holds for all abelian varieties over finite fields.
\end{theorem}

\begin{plain}
\label{h21}For a smooth projective variety $X$ over $\mathbb{F}_{q}$, the full
Tate conjecture says that%
\[
Z(X,t)\sim c(X)\cdot(1-q^{r}t)^{-\rho_{r}}\quad\text{as }t\rightarrow q^{-r},
\]
where $c(X)\neq0$ and $\rho_{r}$ is the is the rank of the group of numerical
equivalence classes of algebraic cycles of codimension $r$. Let $H^{i}%
(X_{Wet},\mathbb{Z}{}(r))$ denote the Weil-\'{e}tale cohomology of complex of
sheaves defined by Bloch's higher Chow groups, let $e\colon H^{2r}%
(X_{Wet},\mathbb{Z}{}(r))\rightarrow H^{2r+1}(X_{Wet},\mathbb{Z}{}(r))$ denote
cup-product with the canonical element of $H^{1}(k_{Wet},\mathbb{Z}{})$, and
let%
\[
\chi(X_{Wet},\mathbb{Z}{}(r))=\prod_{i\neq2r,2r+1}[H^{i}(X_{Wet},\mathbb{Z}%
{}(r))]^{(-1)^{i}}\frac{[\Ker(e)]}{[\Coker(e)].}%
\]
The conjecture of Lichtenbaum and Milne asserts that%
\[
\lim_{t\rightarrow q^{-r}}Z(X,t)=\pm\chi(X_{Wet},\mathbb{Z}(r))\cdot
q^{\chi(X,\mathcal{O}_{X},r)}\cdot(1-q^{r}t)^{-\rho_{r}},
\]
where $\chi(X,\mathcal{O}{}_{X},r)=\sum_{i\leq r,j}(-1)^{i+j}(r-i)\dim
H^{j}(X,\Omega_{X/k}^{i})$. This is now a theorem when $X$ is an abelian
variety. See \cite{milne2015}, Theorem 11 and Remark 12.
\end{plain}

In addition to Theorem \ref{h13}, the proof of Theorem \ref{h20} relies on the
magical properties of Lefschetz classes on abelian varieties and a
serendipitous equality.

\begin{plain}
[Lefschetz classes]\label{h22}Let $k$ be an algebraically closed field. The
Lefschetz classes on abelian varieties have remarkable properties. For
example, the Lefschetz standard conjecture holds for them%
\[
L^{g-2r}\colon\mathcal{D}{}^{r}\overset{\lsimeq}{\longrightarrow}\mathcal{D}%
{}^{g-r}.
\]
For any morphism $f\colon A\rightarrow B$ of abelian varieties, $f_{\ast}$
maps Lefschetz classes to Lefschetz classes.\footnote{This should surprise the
reader, because it is (very) false for maps $f\colon X\rightarrow Y$ of
algebraic varieties. For example, let $Z$ be an algebraic cycle on $Y$ that it
not a Lefschetz class, and choose a subvariety $X$ of $Y$ in which $Z$ has
codimension $1$.} Numerical and homological equivalence agree on Lefschetz
classes for every Weil cohomology theory. Using Lefschetz classes and abelian
varieties, it is possible to define a category $\LMot(k)$ of \textquotedblleft
Lefschetz motives\textquotedblright. This is a canonically polarized tannakian
category over $\mathbb{Q}{}$ with the expected fibre functors, and it is
possible to describe its fundamental group.
\end{plain}

\begin{plain}
[A serendipitous equality]\label{h23}Let $\mathbb{Q}^{\mathrm{al}}$ denote the
algebraic closure of $\mathbb{Q}{}$ in $\mathbb{C}{}$, let $w$ be a prime of
$\mathbb{Q}^{\mathrm{al}}$ lying over $p$, and let $\mathbb{F}{}$ be the
residue field at $w$. We have a diagram of tannakian categories at left, and
the corresponding diagram of tannakian fundamental groups at right,%
\[
\begin{tikzcd} \CM(\mathbb{Q}^{\mathrm{al}})\arrow{d}
&\LCM(\mathbb{Q}^{\mathrm{al}})\arrow{l}\arrow{d}
&S\arrow[hook]{r}&T\\
\Mot(\mathbb{F})&\LMot(\mathbb{F})\arrow{l}
&G\arrow[hook]{r}\arrow[hook]{u}&L\arrow[hook]{u}
\end{tikzcd}
\]
where%
\[%
\begin{tabular}
[c]{lll}\hline
Category & Based on & Correspondences\\\hline
$\CM(\mathbb{Q}{}^{\mathrm{al}})$ & CM abelian varieties over $\mathbb{Q}%
{}^{\mathrm{al}}$ & algebraic classes\\
$\LCM(\mathbb{Q}{}^{\mathrm{al}})$ & CM abelian varieties over $\mathbb{Q}%
{}^{\mathrm{al}}$ & Lefschetz classes\\
$\Mot(\mathbb{F}{})$ & abelian varieties over $\mathbb{F}{}$ & algebraic
classes\\
$\LMot(\mathbb{F}{})$ & abelian varieties over $\mathbb{F}{}$ & Lefschetz
classes.
\end{tabular}
\ \
\]
The groups at right are affine group schemes of multiplicative type over
$\mathbb{Q}{}$.

We know nothing about $G$ except that it is contained in $S$ (because of
\ref{h13}) and in $L$ (because Lefschetz classes are algebraic).

Let $P$ be the affine group scheme attached to the category of motives over
$\mathbb{F}{}$ when we assume the Tate and standard conjectures
(cf.~\cite{milne1994a}). It is called the Weil-number pro-torus, and fits into
the above diagram as follows,%
\[
\begin{tikzcd}
&S\arrow[hook]{r}&T\\
&G\arrow[hook]{r}\arrow[hook]{u}&L\arrow[hook]{u}\\
P\arrow[hook]{ru}\arrow[hook]{ruu}\arrow[hook]{rru}
\end{tikzcd}
\]
For any abelian variety $A$, there are natural actions of both $G$ and $P$ on
the cohomology of $A$. The first fixes the algebraic classes and the second
the Tate classes, so%
\[
\text{the Tate conjecture holds for abelian varieties over }\mathbb{F}{}\iff
P=G.
\]
Looking at the diagram, we see that we could prove that $P=G$ if $S$ and $L$
intersect in $P$. A priori, there is no reason to believe this should be true,
but serendipitously\footnote{Without this equality, we would have no hope of
identifying the group $G$, or proving the Tate conjecture.} it is,%
\[
P=L\cap S\quad\text{(intersection inside }T).
\]
This completes the sketch of the proof of Theorem \ref{h20}.
\end{plain}

This section summarizes parts of \cite{milne1999LC,milne1999LM}

\subsection{The Hodge standard conjecture for abelian varieties}

\begin{theorem}
[\cite{milne2002}]\label{h24}The standard conjecture of Hodge type holds for
all abelian varieties.
\end{theorem}

\begin{plain}
\label{h25}Let $A$ be a polarized abelian variety of dimension $g$. As the
Lefschetz standard conjecture holds for abelian varieties, we have an
isomorphism%
\[
L^{g-2r}\colon\mathcal{A}^{r}(A)\rightarrow\mathcal{A}^{g-r}(A)
\]
of spaces of algebraic classes. Here $L$, the Lefschetz operator, is
cup-product with the divisor class $\xi$ of the polarization. Let%
\[
P^{r}(A)=\Ker\Big(\mathcal{A}^{r}(A)\overset{L^{g-2r+1}}{\longrightarrow
}\mathcal{A}{}^{g-r+1}(A))\Big).
\]
The Hodge standard conjecture says that the pairing%
\[
P^{r}(A)\times P^{r}(A)\rightarrow\mathbb{Q},\quad a,b\mapsto a\cdot b\cdot
\xi^{g-2r}%
\]
is positive definite.
\end{plain}

\begin{plain}
\label{h26}By a specialization argument, it suffices to prove the conjecture
over finite fields.
\end{plain}

\begin{plain}
\label{h27}The Hodge standard conjecture has a tannakian interpretation. It
holds for a collection of algebraic varieties over a field $k$ if and only if
the category of motives $\Mot(k)$ generated by the varieties admits a
polarization for which the geometric pairings are positive.
\end{plain}

\begin{plain}
\label{h28}Roughly speaking, a polarization on a tannakian category is a
compatible family of positive hermitian forms on the objects. There is an
elaborate, and deep, theory of polarizations on tannakian categories. For
example, if $G$ is an algebraic group over a field, then the (tannakian)
category of finite-dimensional representations of $G$ admits a polarization if
and only if $G$ is an inner form of a compact group. See Chapter VI of
\cite{milne2027}.
\end{plain}

\begin{plain}
\label{h29}Hodge theory shows that the Hodge standard conjecture holds in
characteristic zero. Therefore $\CM(\mathbb{Q}{}^{\mathrm{al}})$ admits a
polarization for which the geometric pairings are positive. Using the diagram%
\[
\begin{tikzcd}
\CM(\mathbb{Q}^{\mathrm{al}})\arrow{d}
&\LCM(\mathbb{Q}^{\mathrm{al}})\arrow{l}\arrow{d}
\\
\Mot(\mathbb{F})&\LMot(\mathbb{F})\arrow{l}
\end{tikzcd}
\]
from \ref{h23}, it is possible to show that this polarization passes to the
quotient tannakian category $\Mot(\mathbb{F}{})$. Therefore, the Hodge
standard conjecture holds for all abelian varieties over $\mathbb{F}{}%
$.\footnote{Alongside the problem of resolution of singularities, Grothendieck
regarded the proof of the standard conjectures to be the most urgent task in
algebraic geometry. He once told Spencer Bloch that he hoped to see the
standard conjectures proved in his lifetime. In this, he was disappointed. At
least, we now know the conjectures for all abelian varieties, and so we have
part or Grothendieck's motivic paradise.}
\end{plain}

This section summarizes parts of \cite{milne2002}.

\subsection{Applications}

Let $\Mot(\mathbb{Q}^{\mathrm{al}})$ be the category of abelian motives over
$\mathbb{Q}{}^{\mathrm{al}}$ for absolute Hodge classes, and let
$\Mot^{\prime}(\mathbb{Q}{}^{\mathrm{al}})$ be the subcategory of those with
good reduction (suitably defined). With the proof of the rationality
conjecture, it is possible to show that there exists a tannakian category of
motives $\Mot(\mathbb{F}{})$ and reduction functor $\Mot^{\prime}%
(\mathbb{Q}^{\mathrm{al}})\rightarrow\Mot(\mathbb{F})$ having the properties
that Grothendiecks categories would have if the Hodge, Tate, and standard
conjectures were known (\cite{milne2009}). With the proof of the Hodge
conjecture for CM abelian varieties, we even know that $\Mot(\mathbb{F})$ is
defined by algebraic cycles and $\Mot^{\prime}(\mathbb{Q}^{\mathrm{al}})$ by
almost-algebraic cycles. This allows us to turn some heuristic arguments into proofs.

\begin{plain}
\label{h30}Let $A$ be an abelian variety over a number field $k$, and let $G$
be the Mumford-Tate group of $A_{\mathbb{C}{}}$. It follows from Deligne's
theorem on absolute Hodge classes that, after a finite extension of $k$, there
are well-defined Frobenius conjugacy classes in $G(\mathbb{Q}{}_{\ell})$,
$\ell$ a prime. It is an immediate consequence of the above theory that these
are rational, i.e., arise from a single conjugacy class in $G(\mathbb{Q}{}).$
See \cite{serre1994S}, 12.6.
\end{plain}

\begin{plain}
\label{h31}We leave it as a challenge to the reader to prove the conjecture of
Langlands and Rapoport (1987)\nocite{langlands1987} following the heuristic
argument they use to derive it.
\end{plain}

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