%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{The Hecke sieve and push-forwards to smaller levels}\label{sec:sieve} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% In this section we exclude the groups $\torz n$ for $$n\in\{44,46,48,54,56,58,60,62,66,70,74\}.$$ In most of the cases we reduce the trace divisor $D$ of a point of order $n$ over a decic field (\Cref{subsec:trace}) modulo a small prime $p$, and we exclude all the possible reductions $D'+C'$ with the Hecke sieve and the push-forward class test of \Cref{subsec:sieve}, applied on $X_1(n)$ itself or after pushing $D$ forward to curves of smaller level, such as $X_1(n/2)$ and $X_1(2,n/2)$. For some of the groups, such as $\torz{54}$, $\torz{56}$ and $\torz{58}$, we also use the reduction modulo $2$, Riemann-Roch computations on explicit models, or known points of degree $10$ on curves of smaller level. Over nonic fields the groups $\torz{44}$, $\torz{46}$, $\torz{48}$, $\torz{54}$, $\torz{58}$ and $\torz{62}$ were excluded in \cite[Propositions 6.7, 6.8, 6.11, 10.4 and 11.15]{N69} by the same methods, and for $n=44$ and $n=46$ we reuse the computations made there for the reductions whose non-cuspidal part has degree at most $9$. The numbers of non-cuspidal closed points of $X_1(n)_{\F_p}$ that we use for $n\in\{44,46,48,56\}$ are listed in \Cref{table:sv-counts}, together with the numbers of their orbits under the diamond operators. For $n\in\{44,46,48\}$ the numbers of points of degree at most $9$ are those of \cite[Table 3 and the proof of Proposition 11.15]{N69}, and for $X_1(44)$ the numbers of orbits of degree $6,\dots,9$ are those of \cite[proof of Proposition 6.8]{N69}. We \gitlink{continuation_20261031a/C44/new_finite_jets.m}{compute} the other numbers by enumerating the closed points of degree $10$ on the models of $X_1(44)_{\F_3}$ and $X_1(46)_{\F_3}$ (see also \githubbare{continuation_20261031a/C46/new_support_filter.m}), and, for $X_1(48)_{\F_5}$ and $X_1(56)_{\F_3}$, by running through all the elliptic curves over $\F_{p^k}$, $k\leq10$, with a point of order $n$ (see \githubbare{computations/d31a_even_p5_torsion_common.m} and \githubbare{computations/d31a_even_p3_census_common.m}). \begin{table}[H] \centering \small \begin{tabular}{llll} \toprule $X$ & $p$ & & degree $1,2,\dots,10$\\ \midrule $X_1(44)$ & $3$ & points & $0,0,0,0,60,160,240,920,2260,5854$\\ & & orbits & $0,0,0,0,6,16,24,92,226,590$\\ $X_1(46)$ & $3$ & points & $0,0,0,55,44,55,264,781,1958,6292$\\ & & orbits & $0,0,0,5,4,5,24,71,178,572$\\ $X_1(48)$ & $5$ & points & $0,0,16,64,656,2712,10784,48992,217520,977528$\\ & & orbits & $0,0,2,8,82,342,1348,6172,27190,122318$\\ $X_1(56)$ & $3$ & points & $0,0,0,0,72,48,324,588,2112,6300$\\ & & orbits & $0,0,0,0,6,4,27,54,176,530$\\ \bottomrule \end{tabular} \medskip \caption{Non-cuspidal closed points of $X_{\F_p}$ of degree at most $10$, and the numbers of their orbits under the diamond operators.} \label{table:sv-counts} \end{table} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{The curve \texorpdfstring{$X_1(44)$}{X1(44)}}\label{subsec:sv-44} Over nonic fields, $\torz{44}$ is excluded in \cite[Proposition 6.8]{N69}, and we reuse the computations made there. The new reductions in degree $10$ are treated with the canonical ring of $X_1(44)_{\F_3}$. \begin{lemma}\label{lem:sv-44} Let $D'$ be an effective divisor on $X_1(44)_{\F_3}$. \begin{itemize} \item[a)] If $D'$ is a closed point of degree $5$, or the sum of two (possibly equal) closed points of degree $5$, then $A_5D'$ is not principal. \item[b)] If $D'$ is a closed point of degree $10$, then $A_5D'$ or $A_7D'$ is not principal. \end{itemize} \end{lemma} \begin{proof} As $A_q$ commutes with the diamond operators, we may take a closed point $D'$ of degree $5$ or $10$ among the $6$ and $590$ representatives of the diamond orbits of \Cref{table:sv-counts}, and for a sum of two closed points of degree $5$ we may take the first one among the $6$ representatives and the second one among all the $60$ closed points of degree $5$. This gives $6+590+360=956$ divisors $D'$. Write $A_qD'=T-U$ with $T=T_qD'$ and $U=D'+q\dm qD'$. We \gitlink{continuation_20261031a/C44/new_finite_jets.m}{compute} $T$ from the $q+1$ isogenies of degree $q$ of the elliptic curves of the points in the support of $D'$, including the residue degrees, and $U$ by multiplying the point of order $44$ by $q$, both on the moduli model. Both are effective, of degree $30$ or $60$ for $q=5$ and of degree $80$ for $q=7$. Let $K_X$ be a canonical divisor of $X=X_1(44)_{\F_3}$. We use the canonical model of $X$, of genus $36$ and degree $70$ in $\PP^{35}$, and the multiplication map $R_2\times R_2\to R_4$, where $R_i=H^0(X,iK_X)$, $\dim R_2=105$ and $\dim R_4=245$, which were computed in the proof of \cite[Proposition 6.8]{N69} as the canonical embedding of a plane model of degree $19$. From the plane model we also obtain the local parameters at the points in the support of $U$. We \gitlink{continuation_20261031a/C44/new_matrix_sieve.m}{check} with \Cref{lem:mult}, with $\mathcal L=\mathcal M=\mathcal O(2K_X)$ and non-zero sections $F,H\in H^0(X,2K_X-T)$, that the linear map of \Cref{lem:mult} has rank $105=\dim R_2$, and hence $\ell(T-U)=0$, for $q=5$ for $954$ of the $956$ divisors. The remaining $2$ divisors are closed points of degree $10$, and for them we \gitlink{continuation_20261031a/C44/q7_matrix_sieve.m}{check} in the same way that $\ell(T-U)=0$ for $q=7$. \end{proof} \begin{proposition}\label{prop:44} The group $\torz{44}$ is not a subgroup of the torsion of an elliptic curve over a number field of degree $10$. \end{proposition} \begin{proof} Let $X=X_1(44)$, and let $D$ be the trace divisor of a point of order $44$ over a decic field. We have $\rk J_1(44)(\Q)=0$ \cite[Theorem 3.1(2)]{DEHMZ} and $\gon_\Q X=15$ \cite[Theorem D]{Varivoda26}, so by \Cref{prop:main}~a) with $p=3$ the reduction of $D$ modulo $3$ is $D'+C'$ with $D'\neq0$. By \Cref{table:sv-counts}, $D'$ is a closed point of degree $5\leq\deg D'\leq10$ or the sum of two closed points of degree $5$. By \Cref{prop:tools}~d) we have $[A_q(D)]=0$, and hence $[A_q(D')]=-[A_q(C')]$, for every prime $q\nmid 66$. We \gitlink{continuation_20261031a/C44/cusp_audit.py}{compute} that the Galois orbits of cusps of $X$ over $\Q$ have degrees $1^{15},2^5,5,10^2$, and that the closed cusps of $X_{\F_3}$ have degrees $1^{15},2^5,5^3,10$. So every closed cusp of $X_{\F_3}$ of degree at most $4$ is the reduction of a rational cuspidal divisor of the same degree, and if $\deg C'\leq4$, then $A_qC'=0$ for every prime $q\nmid 66$. As $5\equiv3^8\pmod{44}$, \Cref{lem:cuspHecke} also gives $A_5C'=0$ for every $C'$. Suppose first that $\deg D'\in\{6,7,8\}$. Then $\deg C'\leq4$, and in the proof of \cite[Proposition 6.8]{N69} it is computed that for every closed point $D'$ of $X_{\F_3}$ of degree $6$, $7$ or $8$ one of the classes $[A_q(D')]$, $q\in\{5,7,13,17,19\}$, is non-zero, which is a contradiction. Suppose now that $\deg D'=9$. Then $C'$ is a closed cusp of degree $1$, so $A_5D'$ is principal, and this is excluded in the proof of \cite[Proposition 6.8]{N69}, where it is shown that $\ell(T_5P-P-5\dm5P)=0$ for all the closed points $P$ of degree $9$ of $X_{\F_3}$. Finally, if $D'$ is a closed point of degree $5$, then $A_5D'$ is principal as $A_5C'=0$, and if $\deg D'=10$, then $C'=0$ and $A_qD'$ is principal for every $q\nmid 66$. Both are impossible by \Cref{lem:sv-44}. \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{The curve \texorpdfstring{$X_1(46)$}{X1(46)}}\label{subsec:sv-46} Over nonic fields, $\torz{46}$ is excluded in \cite[Proposition 6.7]{N69}, and we reuse the computations made there. Let $\pi:X_1(46)\to X_1(23)$ be the map $(E,P)\mapsto(E,2P)$, which has degree $3$ and commutes with the diamond operators. The group $J_1(23)(\Q)\simeq\torz{408991}$ is generated by rational cuspidal divisor classes \cite[Theorem 3.1(2), Corollary 4.14 and Table 2]{DEHMZ}, and its reduction $G$ has index $23$ in $J_1(23)(\F_3)\simeq\torz{9406793}$ \cite[proof of Proposition 6.7]{N69}. Let $y_0$ be a rational cusp of $X_1(23)$. \begin{lemma}\label{lem:sv-46} Let $D'$ be an effective divisor of degree $10$ on $X_1(46)_{\F_3}$ with non-cuspidal support, of type $[10]$, $[6,4]$ or $[5,5]$. Then $[\pi_*D'-10\overline{y_0}]\notin G$, or $A_5D'$ is not principal. \end{lemma} \begin{proof} Both conditions are invariant under the diamond operators, as these commute with $\pi$ and $A_5$, and the diamond operators of $X_1(23)$ preserve $G$ and map $\overline{y_0}$ to reductions of rational cusps, whose differences have classes in $G$. Taking a representative of each diamond orbit, and for the types $[6,4]$ and $[5,5]$ the first point up to the diamond operators and the second one arbitrary, we obtain by \Cref{table:sv-counts} $572$ divisors of type $[10]$, $5\cdot55=275$ of type $[6,4]$ and $4\cdot44=176$ of type $[5,5]$, $1023$ in total. We \gitlink{continuation_20261031a/C46/new_support_filter.m}{compute} that $[\pi_*D'-10\overline{y_0}]\notin G$ for $552$, $264$ and $170$ of them, respectively. For the remaining $20$, $11$ and $6$ divisors we \gitlink{computations/d31a_46_parent_exceptional_hecke_common.m}{compute} that $\ell(A_5D')=0$ on the function field of $X_1(46)_{\F_3}$, which has genus $45$, where $T_5D'$ is computed from the isogenies of degree $5$ of the elliptic curves of the points in the support of $D'$ on the moduli model, including the residue degrees. \end{proof} \begin{proposition}\label{prop:46} The group $\torz{46}$ is not a subgroup of the torsion of an elliptic curve over a number field of degree $10$. \end{proposition} \begin{proof} Let $D$ be the trace divisor on $X_1(46)$ of a point of order $46$ over a decic field. We have $\rk J_1(46)(\Q)=0$ \cite[Theorem 3.1(2)]{DEHMZ} and $\gon_\Q X_1(46)=19$ \cite[Theorem D]{Varivoda26}, so by \Cref{prop:main}~a) with $p=3$ the reduction of $D$ modulo $3$ is $D'+C'$ with $D'\neq0$ of degree $e\geq4$ (\Cref{table:sv-counts}). We \gitlink{continuation_20261031a/C46/cusp_audit.py}{compute} that the Galois orbits of cusps of $X_1(46)$ over $\Q$ and the closed cusps of $X_1(46)_{\F_3}$ both have degrees $1^{22},11^2$. So $C'$, which has degree at most $6$, is the reduction of a rational cuspidal divisor, and $A_qC'=0$ for every prime $q\nmid138$ by \Cref{prop:tools}~c). By \Cref{prop:tools}~d) it follows that $[A_q(D')]=0$ for all these $q$. The closed cusps of $X_1(23)_{\F_3}$ have degrees $1^{11},11$, so $\pi_*C'$ is the reduction of a rational cuspidal divisor, and \eqref{eq:pushclass} gives $[\pi_*D'-e\overline{y_0}]\in G$. Note that neither of the two conditions depends on the degree of $C'$. By \Cref{table:sv-counts} the type of $D'$ is one of $[4]$, $[5]$, $[6]$, $[7]$, $[8]$, $[9]$, $[4,4]$, $[5,4]$, and $[10]$, $[6,4]$, $[5,5]$. In the proof of \cite[Proposition 6.7]{N69} it is computed that every divisor $D'$ of one of the first $8$ types, of which there are $782$ up to the diamond operators, satisfies $[\pi_*D'-e\overline{y_0}]\notin G$ or $[A_q(D')]\neq0$ for some $q\in\{5,7,11,13,17,19\}$. The last $3$ types are excluded by \Cref{lem:sv-46}. \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{The curve \texorpdfstring{$X_1(48)$}{X1(48)}}\label{subsec:sv-48} Over nonic fields, $\torz{48}$ is excluded in \cite[Proposition 11.15]{N69}. In degree $10$ we push the trace divisor forward to $X_1(2,24)$, and use $A_{13}$ together with a description of its action on the cusps. Let $\pi:X_1(48)\to X_1(2,24)$ be the map $(E,P)\mapsto(E/\dm{24P},T,\overline P)$ of \cite[proof of Proposition 11.15]{N69}, where $T$ is the non-zero point of the kernel of the dual isogeny and $\overline P$ is the image of $P$, which has order $24$. It has degree $2$ and commutes with the diamond operators. The group $J_1(2,24)(\Q)\simeq\torz2\times(\torz4)^2\times\torz{120}\times\torz{240}$ is generated by rational cuspidal divisor classes \cite[Proposition 4.16 and Table 3]{DEHMZ}, and its reduction $H$ has order $921600$ and index $4608$ in $J_1(2,24)(\F_5)\simeq(\torz2)^2\times(\torz4)^2\times\torz{12}\times\torz{24}\times(\torz{480})^2$ \cite[proofs of Propositions 11.2 and 11.15]{N69}. We fix a rational cusp $c_\ast$ of $X_1(2,24)$, and we write $\mathcal S_j$ for the set of the classes $[j\overline{c_\ast}-B]$ for the effective cuspidal divisors $B$ of degree $j$ on $X_1(2,24)_{\F_5}$, as in \eqref{eq:pushclass}. We \gitlink{continuation_20261031b/C48/d31b_C48_low_class_common.m}{compute} that the images of $\mathcal S_0,\dots,\mathcal S_7$ in $J_1(2,24)(\F_5)/H$ have $1$, $5$, $23$, $75$, $178$, $378$, $661$ and $1065$ elements. \begin{lemma}\label{lem:sv-48-cusps} For every effective cuspidal divisor $C$ on $X_1(48)_{\F_5}$, the divisor $A_{13}C$ is principal. \end{lemma} \begin{proof} It is enough to prove this for a closed cusp $c$. By \Cref{lem:cuspHecke}, $A_{13}c=(\sigma_{13}c-c)+13\dm{13}(\sigma_{13}^{-1}c-c)$. The $56$ geometric cusps of $X_1(48)$ are given by the pairs $(a,b)\in(\torz{48})^2$ of order $48$ modulo $(a,b)\sim\pm(a+kb,b)$, on which $\sigma_u(a,b)=(ua,b)$ and $\dm u(a,b)=(ua,u^{-1}b)$. We \gitlink{continuation_20261031b/C48/d31b_C48_independent_cusp_labels.py}{compute} that the closed cusps of $X_1(48)_{\F_5}$ have degrees $1^{16},2^{12},4^4$, and that $\sigma_{13}$ fixes $28$ of these $32$ closed cusps, and hence $A_{13}c=0$ for them. The remaining $4$ are the $2$ closed cusps $C_2,C_2'$ of degree $2$ at which $j$ has a pole of order $2$, and the $2$ closed cusps $C_4,C_4'$ of degree $4$ at which $j$ has a pole of order $1$. Both $\sigma_{13}$ and $\dm{13}$ interchange $C_i$ and $C_i'$ for $i=2,4$, and as $\sigma_{13}^2=\sigma_{25}=\sigma_5^2$ fixes every closed cusp, we obtain $$A_{13}C_i=(C_i'-C_i)+13(C_i-C_i')=12(C_i-C_i')=-A_{13}C_i'.$$ Note that this does not depend on the conventions for the diamond operators of \Cref{subsec:hecke}. We \gitlink{continuation_20261031b/C48/d31b_C48_cusp_corrections_v2.m}{compute} functions $f_2$ and $f_4$ on $X_1(48)_{\F_5}$ with $\ddiv(f_i)=12(C_i-C_i')$, which proves the claim. \end{proof} \begin{lemma}\label{lem:sv-48-sieve} Let $N\neq0$ be an effective divisor of degree $r\leq10$ on $X_1(48)_{\F_5}$ with non-cuspidal support, such that $[\pi_*N-r\overline{c_\ast}]\in H+\mathcal S_{10-r}$. If $r=10$, then $A_7N$ is not principal, and if $r<10$, then $A_{13}N$ is not principal. \end{lemma} \begin{proof} The diamond operators commute with $\pi$, $A_7$ and $A_{13}$, and they act on $X_1(2,24)$ by level automorphisms defined over $\Q$, which preserve $H$ and the sets $\mathcal S_j$ and map $\overline{c_\ast}$ to reductions of rational cusps. So both the condition and the conclusion are invariant under the diamond operators. By \Cref{table:sv-counts} the possible types of $N$ are $[k]$ for $3\leq k\leq10$, $[3,3]$, $[3,4]$, $[3,5]$, $[3,6]$, $[3,7]$, $[4,4]$, $[4,5]$, $[4,6]$, $[5,5]$, $[3,3,3]$ and $[3,3,4]$. Suppose first that $N$ is a closed point of degree $9$ or $10$. There are $27190+122318=149508$ representatives of the diamond orbits of these points. Before computing the classes we use $4$ characters $\chi_f$ as in \Cref{lem:quadchar}, for functions $f$ on $X_1(2,24)_{\F_5}$ with $\ddiv(f)=2A_f$, which we check to be trivial on $H$ (the functions are given in \githubbare{continuation_20261031b/C48/inputs/four_characters.m}). If $N$ satisfies the condition, then the values of the $4$ characters at $[\pi_*N-r\overline{c_\ast}]$ are those at an element of $\mathcal S_{10-r}$. We \gitlink{continuation_20261031b/C48/d31b_C48_character_high_common_v3.m}{compute} that this holds for $12603$ of the representatives, and we keep $1$ further representative, at which one of the functions $f$ has a zero or a pole on the support of $\pi_*N$. For these $12604$ representatives we \gitlink{continuation_20261031b/C48/d31b_C48_high_class_common.m}{compute} the classes, and find that the condition holds for only $107$ of them, $57$ of degree $10$ and $50$ of degree $9$. For the other types we take the first point of $N$ up to the diamond operators and the others arbitrary. This gives $2$, $8$, $82$, $342$, $1348$ and $6172$ divisors of the types $[3],\dots,[8]$, and \[ \begin{array}{c|ccccccccccc} \text{type}&[3,3]&[3,4]&[3,5]&[3,6]&[3,7]&[4,4]&[4,5]&[4,6]&[5,5]&[3,3,3]&[3,3,4]\\\hline \text{divisors}&32&128&1312&5424&21568&512&5248&21696&53792&272&2048 \end{array} \] of the other types, $119986$ in total. We \gitlink{continuation_20261031b/C48/d31b_C48_low_class_common.m}{compute} that the condition holds for only $243$ of them, which form $201$ orbits under the diamond operators: \[ \begin{array}{c|ccccccccccccc} \text{type}&[4]&[5]&[6]&[7]&[8]&[3,3]&[3,5]&[3,6]&[3,7]&[4,4]&[4,5]&[4,6]&[5,5]\\\hline \text{orbits}&4&6&20&24&55&4&10&4&8&16&8&12&30 \end{array} \] It remains to treat $107+201=308$ divisors $N$. For the $57+8+12+30=107$ of them of degree $10$ we \gitlink{continuation_20261031b/C48/d31b_C48_high_hecke_common.m}{compute} that $\ell(A_7N)=0$, and for the other $201$ we \gitlink{continuation_20261031b/C48/d31b_C48_low_hecke_common.m}{compute} that $\ell(A_{13}N)=0$, on the function field of $X_1(48)_{\F_5}$, which has genus $37$. Here $T_qN$ is computed from the cyclic isogenies of degree $q$ of the elliptic curves of the points in the support of $N$ on the moduli model, including the lengths of their Frobenius orbits and the residue degrees. \end{proof} \begin{proposition}\label{prop:48} The group $\torz{48}$ is not a subgroup of the torsion of an elliptic curve over a number field of degree $10$. \end{proposition} \begin{proof} Let $D$ be the trace divisor on $X_1(48)$ of a point of order $48$ over a decic field. We have $\rk J_1(48)(\Q)=0$ \cite[Theorem 3.1(2)]{DEHMZ} and $\gon_\Q X_1(48)=16$ \cite[Theorem D]{Varivoda26}, so by \Cref{prop:main}~a) with $p=5$ the reduction of $D$ modulo $5$ is $N+C'$ with $N\neq0$ effective of degree $r$ with non-cuspidal support, and $C'$ effective and cuspidal of degree $10-r$. By \eqref{eq:pushclass}, with $y_0=c_\ast$ and $G=H$, we have $[\pi_*N-r\overline{c_\ast}]\in H+\mathcal S_{10-r}$. By \Cref{prop:tools}~d) the classes $[A_7(D)]$ and $[A_{13}(D)]$ are $0$. If $r=10$, then $C'=0$ and $A_7N$ is principal. If $r<10$, then $A_{13}N+A_{13}C'$ is principal, and as $A_{13}C'$ is principal by \Cref{lem:sv-48-cusps}, so is $A_{13}N$. Both contradict \Cref{lem:sv-48-sieve}. \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{The curve \texorpdfstring{$X_1(54)$}{X1(54)}}\label{subsec:sv-54} Over nonic fields, $\torz{54}$ is excluded in \cite[Section 10]{N69}. We write the points of $X_1(54)$ as triples $(E,P_2,Q)$ with $P_2$ of order $2$ and $Q$ of order $27$, and use the two degeneracy maps \[ \alpha,\beta:X_1(54)\to Z:=X_1(27),\qquad \alpha(E,P_2,Q)=(E,Q),\qquad \beta(E,P_2,Q)=(E/\dm{P_2},\phi(Q)), \] where $\phi:E\to E/\dm{P_2}$ is the quotient isogeny. Both are defined over $\Q$, have degree $3$, map cusps to cusps and non-cusps to non-cusps, and commute with the diamond operators. We write $\delta=\dm2$ for the diamond operator of $Z$. At the unique prime above $2$ at which the elliptic curve has good reduction, the reductions of the push-forwards of the trace divisor under $\alpha$ and $\beta$ have the same non-cuspidal part, and their cuspidal parts are related by $\delta$. We combine this with Riemann-Roch computations on $Z_{\F_2}$ and with class computations on $Z_{\F_5}$. \begin{lemma}\label{lem:sv-54-gon} We have $\gon_\Q X_1(54)\geq12$. \end{lemma} \begin{proof} The curve $X_1(54)$ has genus $52$ and $i(54)=972$, so Abramovich's bound gives $\gon_\Q X_1(54)\geq\gon_\C X_1(54)\geq\tfrac{325}{2^{15}}\cdot972>9.6$. Let $W_{54}=X_1(54)/\dm{17}$, where $\dm{17}$ has order $3$ in $\Delta(54)$, as $17^3\equiv-1\pmod{54}$. We \gitlink{computations/d31a_54_parent_gonality_controls.py}{compute} from the action of $\PSL_2(\Z)$ on the cosets that the group of $W_{54}$ has index $324$ in $\PSL_2(\Z)$, no elliptic points and $36$ cusps, so $W_{54}$ has genus $1+324/12-36/2=10$. Suppose that there is a function $f$ on $X_1(54)$ over $\Q$ of degree $n\in\{10,11\}$. As $[\Q(X_1(54)):\Q(W_{54})]=3$ is prime, either $f\in\Q(W_{54})$, in which case $3\mid n$, or $\Q(X_1(54))=\Q(W_{54})(f)$, in which case the Castelnuovo-Severi inequality \cite[Theorem 3.11.3]{stichtenoth} gives $52\leq3\cdot10+(3-1)(n-1)\leq50$. Both are impossible. \end{proof} \begin{lemma}\label{lem:sv-54-good2} Let $E$ be an elliptic curve over a number field $K$ of degree $10$ with a point of order $54$. Then $E$ has good reduction at some prime of $K$ above $2$. Every such prime is unramified and has residue degree between $6$ and $10$, and the reduction of $E$ at it is ordinary. \end{lemma} \begin{proof} This is \cite[Lemma 10.2]{N69} for number fields of degree $d\in\{7,8,9\}$, and its proof applies verbatim to $d=10$. It uses only that $J_1(54)(\Q)$ is finite of odd order \cite[Theorem 3.1(2), Corollary 4.14 and Table 2]{DEHMZ}, that $\gon_\Q X_1(54)>d$, which holds for $d=10$ by \Cref{lem:sv-54-gon}, that $Z_{\F_2}$ has no non-cuspidal closed points of degree at most $5$, and that the ramification index $e$ and the residue degree $f\geq6$ of a prime above $2$ of good reduction satisfy $ef\leq d$, which forces $e=1$ also for $d=10$. The last step of that proof, which shows that the reduction is ordinary, is the case of \Cref{lem:sw-twoisog}~a) in which the isogeny of degree $2$ is the quotient by the subgroup generated by the $K$-rational point of order $2$, as the prime is unramified over $2$. \end{proof} \begin{lemma}\label{lem:sv-54-Z} \begin{itemize} \item[a)] The group $J_Z(\Q)\simeq(\torz3)^2\times\torz{52497}$ is generated by rational cuspidal divisor classes. Reduction modulo $2$ is an isomorphism $J_Z(\Q)\to J_Z(\F_2)$, and reduction modulo $5$ is injective on $J_Z(\Q)$. \item[b)] The Galois orbits of cusps of $Z$ over $\Q$ have degrees $1^9,2^3,6,9$, and each of them reduces modulo $2$ and modulo $5$ to a single closed cusp of the same degree. The valuations of the functions $b$ and $b^4/\Delta$, where $\Delta$ is the discriminant of the Tate normal form $E(b,c)$ of \eqref{eq:tate}, are different at the $14$ orbits, and they are the same at the corresponding closed cusps of $Z_{\F_2}$ and $Z_{\F_5}$. \item[c)] The curve $Z_{\F_2}$ has no non-cuspidal closed points of degree at most $5$. The non-cuspidal closed points of $Z_{\F_2}$ of degree $6$, $7$, $8$, $9$ and $10$ at which the elliptic curve is ordinary number $9$, $9$, $18$, $90$ and $99$, and they form $1$, $1$, $2$, $10$ and $11$ orbits under the diamond operators. \end{itemize} \end{lemma} \begin{proof} a) The first statement is \cite[Theorem 3.1(2), Theorem 4.13, Corollary 4.14 and Table 2]{DEHMZ}, the statement on the reduction modulo $2$ is shown in \cite[Section 10]{N69} (the paragraph after Lemma 10.2), and reduction modulo $5$ is injective by Katz's result. b) The cusps are defined over subfields of $\Q(\zeta_{27})$, and both $2$ and $5$ generate $(\torz{27})^\times$, so each Galois orbit stays irreducible modulo $2$ and modulo $5$. The functions $b$ and $b^4/\Delta$ are units on $Y_1(27)$ over $\Z[\tfrac1{27}]$, so their valuations at the cusps do not change under reduction. They are used in \cite[Section 10]{N69} (the paragraph after Lemma 10.2) to match the closed cusps modulo $2$ and modulo $5$, and we \gitlink{continuation_20261031b/C54/d31b_C54_dyadic_rigidity_v2.m}{compute} the degrees of the orbits and that these valuations distinguish them. c) The first statement is in the proof of \cite[Lemma 10.2]{N69}, and we \gitlink{continuation_20261031b/C54/d31b_C54_dyadic_rigidity_v2.m}{compute} the second one. Note that an elliptic curve over a finite field of characteristic $2$ is ordinary if and only if its $j$-invariant is non-zero. \end{proof} The following lemma describes the reductions modulo $2$. By the reduction of a divisor on $Z$ modulo $2$ we mean its reduction on the smooth model of $Z$ over $\Z_{(2)}$. \begin{lemma}\label{lem:sv-54-shape} Let $K$ be a number field of degree $10$, and let $E/K$ be an elliptic curve with points $P_2,Q\in E(K)$ of orders $2$ and $27$. Suppose that $E$ has good ordinary reduction at a prime $\mathfrak p$ of $K$ above $2$, that $P_2$ reduces to $0$ modulo $\mathfrak p$, and that $E$ has bad reduction at all the other primes of $K$ above $2$. Let $D$ be the trace divisor of $x=(E,P_2,Q)$ on $X_1(54)$, and put $D_0=\alpha_*D$ and $D_1=\beta_*D$. Then the residue degree $f$ of $\mathfrak p$ satisfies $6\leq f\leq10$, and there are a non-cuspidal closed point $N$ of $Z_{\F_2}$ of degree $f$ and effective rational cuspidal divisors $C_a$ and $C_e$ on $Z$ with $\deg C_a+\deg C_e=10-f$, such that the reductions of $D_0$ and $D_1$ modulo $2$ are \[ \overline{D_0}=N+\overline{C_a}+\overline{C_e},\qquad \overline{D_1}=N+\overline{C_a}+\delta\overline{C_e}, \] and such that $D_1+C_e\sim D_0+\delta C_e$. \end{lemma} \begin{proof} The divisors $D_0$ and $D_1$ are the trace divisors of the points $\alpha(x)$ and $\beta(x)$ of $Y_1(27)(K)$. For a prime $\mathfrak q$ of $K$ above $2$ let $y_{\mathfrak q}$ and $y'_{\mathfrak q}$ be the closed points of $Z_{\F_2}$ given by the reductions of $\alpha(x)$ and $\beta(x)$ modulo $\mathfrak q$, and let $e_{\mathfrak q}$ and $f_{\mathfrak q}$ be the ramification index and the residue degree of $\mathfrak q$. By \Cref{subsec:trace}, $\overline{D_0}$ is the sum of $e_{\mathfrak q}f_{\mathfrak q}/\deg(y_{\mathfrak q})\cdot y_{\mathfrak q}$ over these primes, and similarly for $\overline{D_1}$, with $\deg(y'_{\mathfrak q})=\deg(y_{\mathfrak q})$ in all the cases below. At $\mathfrak p$ the point $Q$, of odd order, reduces to a point of order $27$, so $y_{\mathfrak p}$ is non-cuspidal. By \Cref{lem:sv-54-Z}~c) we have $\deg y_{\mathfrak p}\geq6$, and as $\deg y_{\mathfrak p}$ divides $f$ and $e_{\mathfrak p}f\leq10$, it follows that $\deg y_{\mathfrak p}=f$ and $e_{\mathfrak p}=1$. As the reduction is ordinary and $P_2$ reduces to $0$, the subgroup generated by $P_2$ is the generic fibre of the connected subgroup scheme of order $2$ of the $2$-torsion of the N\'eron model of $E$ at $\mathfrak p$. Hence $\phi$ reduces to an isogeny of degree $2$ with connected kernel, which is the Frobenius isogeny of the reduction of $E$, and the reduction of $\beta(x)$ is the image of the reduction of $\alpha(x)$ under the $2$-power Frobenius. It follows that $y'_{\mathfrak p}=y_{\mathfrak p}=:N$, with coefficient $1$ in both $\overline{D_0}$ and $\overline{D_1}$. Let $\mathfrak q\neq\mathfrak p$ be a prime above $2$. By \Cref{lem:additive}, applied to the point $Q$ of odd order, $E$ has multiplicative reduction at $\mathfrak q$, so $y_{\mathfrak q}$ and $y'_{\mathfrak q}$ are cusps. Over the completion of the maximal unramified extension of $K_{\mathfrak q}$, the curve $E$ is a Tate curve $\mathbb G_m/t^\Z$. Its subgroups of order $2$ are $\mu_2$ and the $2$ subgroups generated by $\pm t^{1/2}$, and the computation in the proof of \Cref{lem:cuspHecke}, with $q=2$, which is prime to $27$, shows the following. If $P_2=-1$, then the cusp to which $\beta(x)$ reduces is $\sigma_2(c)$, where $c$ is the cusp to which $\alpha(x)$ reduces, and otherwise it is $\dm2\sigma_2^{-1}(c)$. As $\sigma_2$ acts on the reductions of the cusps as the Frobenius, we get $y'_{\mathfrak q}=y_{\mathfrak q}$ in the first case and $y'_{\mathfrak q}=\delta y_{\mathfrak q}$ in the second one. Let $\overline{C_a}$ and $\overline{C_e}$ be the sums of the contributions of the primes of the first and of the second kind to $\overline{D_0}$. By \Cref{lem:sv-54-Z}~b) every closed cusp of $Z_{\F_2}$ is the reduction of a rational cuspidal divisor of the same degree, so $\overline{C_a}$ and $\overline{C_e}$ are the reductions of effective rational cuspidal divisors $C_a$ and $C_e$, and the reductions of $D_0$ and $D_1$ are as claimed. Finally, the classes $[D_1-D_0]$ and $[\delta C_e-C_e]$ lie in $J_Z(\Q)$ and have the same reduction modulo $2$, so they are equal by \Cref{lem:sv-54-Z}~a). \end{proof} \begin{lemma}\label{lem:sv-54-rigid} Let $K$ be a number field of degree $10$, let $E/K$ be an elliptic curve with points $P_2,Q\in E(K)$ of orders $2$ and $27$, let $D$ be the trace divisor of $(E,P_2,Q)$ on $X_1(54)$, and put $D_0=\alpha_*D$ and $D_1=\beta_*D$. Then $D_0\neq D_1$. \end{lemma} \begin{proof} As $E(K)$ has a point of order $54$, \Cref{lem:CM} shows that $E$ has no complex multiplication. Suppose that $D_0=D_1$, and let $S$ be the support of $D_0$, a set of at most $10$ geometric points of $Z$. Consider the directed graph with vertex set $S$ and an edge from $\alpha(x^\sigma)$ to $\beta(x^\sigma)$ for every embedding $\sigma$ of $K$ into $\Qbar$, where $x=(E,P_2,Q)$. As $D_0=D_1$, every vertex has as many incoming as outgoing edges, so the graph contains a directed cycle $s_1\to s_2\to\dots\to s_r\to s_1$ with $r\leq10$ distinct vertices. Every edge $s_i\to s_{i+1}$ comes with an isogeny of degree $2$ between elliptic curves representing $s_i$ and $s_{i+1}$ which maps the point of order $27$ of the first to that of the second, up to sign. Composing these isogenies and an isomorphism, we obtain an endomorphism $\psi$ of degree $2^r$ of an elliptic curve $E_1$ isogenous to $E^\sigma$ for some $\sigma$, and a point $Q_1\in E_1$ of order $27$ with $\psi(Q_1)=\pm Q_1$. As $E_1$ has no complex multiplication, $\psi=\pm2^{r/2}$, so $r$ is even and $2^{r/2}\equiv\pm1\pmod{27}$. But $2$ has order $18$ modulo $27$ and $2^9\equiv-1\pmod{27}$, so $r\geq18$, which is a contradiction. \end{proof} \begin{lemma}\label{lem:sv-54-dyadic} Let $c_\ast$ be a rational cusp of $Z$, and let $\mathcal N$ be a set of representatives of the $25$ diamond orbits of the ordinary non-cuspidal closed points of $Z_{\F_2}$ of degree between $6$ and $10$ (\Cref{lem:sv-54-Z}~c)). Let $\mathcal T$ be the set of the triples $(N,C_a',C_e')$ with $N\in\mathcal N$ and $C_a'$, $C_e'$ effective cuspidal divisors on $Z_{\F_2}$ with $\deg N+\deg C_a'+\deg C_e'=10$, such that \[ \ell(N+C_a'+C_e'+\delta C_e')\geq2, \] and let $\mathcal P\subseteq J_Z(\Q)^2$ be the set of the pairs whose reductions modulo $2$ are $\bigl([N+C_a'+C_e'-10\overline{c_\ast}],[N+C_a'+\delta C_e'-10\overline{c_\ast}]\bigr)$ for $(N,C_a',C_e')\in\mathcal T$. Then $\mathcal P$ has $4103$ elements, and their first coordinates take $800$ values. \end{lemma} \begin{proof} As $\deg C_a'+\deg C_e'\leq4$, the divisors $C_a'$ and $C_e'$ are supported on the $12$ closed cusps of degree $1$ and $2$ (\Cref{lem:sv-54-Z}~b)). This gives $8825$ triples $(N,C_a',C_e')$ without the condition on $\ell$, and we \gitlink{continuation_20261031b/C54/d31b_C54_dyadic_rigidity_v2.m}{compute} that $\ell(N+C_a'+C_e'+\delta C_e')=1$ for $4721$ of them. The remaining $4104$ triples form $\mathcal T$, and we compute that they give $4103$ distinct pairs of classes in $J_Z(\F_2)$, with $800$ distinct first coordinates. By \Cref{lem:sv-54-Z}~a) these pairs are the reductions of a unique set of pairs in $J_Z(\Q)^2$. \end{proof} \begin{lemma}\label{lem:sv-54-five} Let $c_\ast$ and $\mathcal P$ be as in \Cref{lem:sv-54-dyadic}, and write $\overline\xi$ for the reduction modulo $5$ of $\xi\in J_Z(\Q)$. There are no $(\xi,\eta)\in\mathcal P$, no non-zero effective divisor $L$ on $X_1(54)_{\F_5}$ with non-cuspidal support and no effective cuspidal divisors $B_0,B_1$ on $Z_{\F_5}$ of degree $10-\deg L$ such that \[ [\alpha_*L+B_0-10\overline{c_\ast}]=\overline\xi\qquad\text{and}\qquad[\beta_*L+B_1-10\overline{c_\ast}]=\overline\eta. \] \end{lemma} \begin{proof} For each of the $800$ first coordinates $\xi$ we construct a divisor on $Z_{\F_5}$ in the class $10\overline{c_\ast}+\overline\xi$ from the expression of $\xi$ in terms of the classes of rational cuspidal divisors (\Cref{lem:sv-54-Z}~a)), using the identification of the cusps of \Cref{lem:sv-54-Z}~b), and we \gitlink{continuation_20261031b/C54/d31b_C54_transport_800_resume.m}{compute} its Riemann-Roch space. Only $724$ of these $800$ spaces are non-zero. We \gitlink{continuation_20261031b/C54/d31b_C54_enumerate_all_class_members.m}{list} all the effective divisors in these $724$ classes: $217$ of them are cuspidal, and $1067$ have a non-cuspidal part. If $(\xi,\eta)$, $L$, $B_0$ and $B_1$ are as in the statement, then $B=\alpha_*L+B_0$ is an effective divisor in the class of $10\overline{c_\ast}+\overline\xi$. As $L\neq0$ and $\alpha$ maps cusps to cusps and non-cusps to non-cusps, $B$ is one of the $1067$ divisors and $\alpha_*L$ is its non-cuspidal part. These $1067$ divisors involve $192$ distinct non-cuspidal closed points of $Z_{\F_5}$. Above a non-cuspidal closed point $y$ of $Z_{\F_5}$ of degree $k$, the closed points of $X_1(54)_{\F_5}$ correspond to the Frobenius orbits of the subgroups of order $2$ of the elliptic curve of $y$, and hence to the irreducible factors of its $2$-division polynomial over $\F_{5^k}$. A factor of degree $e$ gives a closed point of degree $ke$ whose push-forward under $\alpha$ is $ey$. So if $y$ occurs with multiplicity $m$ in $\alpha_*L$, then the points of $L$ above $y$ correspond to factors of degrees $e_i$ with multiplicities $n_i$ such that $\sum_ie_in_i=m$. We \gitlink{continuation_20261031b/C54/d31b_C54_member_two_isogenies_all.py}{compute} that there are $361$ closed points of $X_1(54)_{\F_5}$ above the $192$ points, that their images under $\beta$ are $334$ distinct closed points, and that there are only $147$ pairs $(B,L)$ of one of the $1067$ divisors $B$ and an effective divisor $L$ on $X_1(54)_{\F_5}$ such that $\alpha_*L$ is the non-cuspidal part of $B$: for $59$ of them $L$ has degree $10$, and for $88$ of them degree $9$. For each of these pairs we \gitlink{continuation_20261031b/C54/d31b_C54_member_all_classes.m}{compute} the class $[\beta_*L+B_1-10\overline{c_\ast}]$ for every effective cuspidal divisor $B_1$ of degree $10-\deg L$ on $Z_{\F_5}$, and find that it is never equal to $\overline\eta$ for a pair $(\xi,\eta)\in\mathcal P$ with $\overline\xi=[B-10\overline{c_\ast}]$. \end{proof} \begin{proposition}\label{prop:54} The group $\torz{54}$ is not a subgroup of the torsion of an elliptic curve over a number field of degree $10$. \end{proposition} \begin{proof} Suppose the opposite, and let $E/K$ be an elliptic curve over a decic field with a point of order $54$. Then $E(K)$ contains points $P_2$ and $Q$ of orders $2$ and $27$. By \Cref{lem:sv-54-good2}, $E$ has good ordinary reduction at a prime $\mathfrak p$ of $K$ above $2$, of residue degree at least $6$, and as two such primes would have residue degrees adding up to more than $10$, $\mathfrak p$ is the only prime above $2$ at which $E$ has good reduction. Suppose that $P_2$ does not reduce to $0$ modulo $\mathfrak p$. Then $\phi:E\to E'=E/\dm{P_2}$ reduces to a separable isogeny, and as the multiplication by $2$ on the ordinary reduction of $E$ has inseparable degree $2$, the dual isogeny $\widehat\phi$ reduces to an inseparable one. The kernel of $\widehat\phi$ is $\phi(E[2])$, whose non-zero point $R$ is $K$-rational, so $R$ reduces to $0$ modulo $\mathfrak p$. As $E'$ is isogenous to $E$, it has the same reduction types, and $(E',R,\phi(Q))$ is again a point of $X_1(54)$ over $K$. So, replacing $E$ by $E'$ if necessary, we may assume that $P_2$ reduces to $0$ modulo $\mathfrak p$. Let $D$ be the trace divisor of $x=(E,P_2,Q)$ on $X_1(54)$, and put $D_0=\alpha_*D$ and $D_1=\beta_*D$. By \Cref{lem:sv-54-shape} the reductions of $D_0$ and $D_1$ modulo $2$ are $N+\overline{C_a}+\overline{C_e}$ and $N+\overline{C_a}+\delta\overline{C_e}$, and $D_1+C_e\sim D_0+\delta C_e$. Replacing $Q$ by $aQ$ for a suitable $a\in(\torz{27})^\times$, which replaces $x$, $D$, $D_0$, $D_1$, $N$, $C_a$ and $C_e$ by their images under $\dm a$, we may assume that $N\in\mathcal N$. Suppose first that $\ell(N+\overline{C_a}+\overline{C_e}+\delta\overline{C_e})=1$ on $Z_{\F_2}$. The divisors $D_1+C_e$ and $D_0+\delta C_e$ are effective, linearly equivalent, and both reduce modulo $2$ to $N+\overline{C_a}+\overline{C_e}+\delta\overline{C_e}$. As $Z$ has good reduction at $2$, semicontinuity gives $\ell(D_0+\delta C_e)\leq1$ over $\Q$, so $D_1+C_e=D_0+\delta C_e$. Comparing the non-cuspidal parts we get $D_1=D_0$, which contradicts \Cref{lem:sv-54-rigid}. Hence $(N,\overline{C_a},\overline{C_e})$ lies in the set $\mathcal T$ of \Cref{lem:sv-54-dyadic}, and the pair $(\xi,\eta)=([D_0-10c_\ast],[D_1-10c_\ast])\in J_Z(\Q)^2$ lies in $\mathcal P$, as its reduction modulo $2$ is the pair of classes of this triple. Now we reduce modulo $5$. As $J_1(54)(\Q)$ is finite \cite[Theorem 3.1(2)]{DEHMZ}, by \Cref{lem:sv-54-gon} and \Cref{prop:main}~a) with $p=5$, the reduction of $D$ modulo $5$ is $L+\overline C$, where $L\neq0$ is effective with non-cuspidal support and $\overline C$ is effective and cuspidal. As $\alpha$ and $\beta$ map cusps to cusps and non-cusps to non-cusps, the reductions of $D_0$ and $D_1$ modulo $5$ are $\alpha_*L+\alpha_*\overline C$ and $\beta_*L+\beta_*\overline C$, whose classes relative to $10\overline{c_\ast}$ are $\overline\xi$ and $\overline\eta$. This contradicts \Cref{lem:sv-54-five}. \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{The curve \texorpdfstring{$X_1(56)$}{X1(56)}}\label{subsec:sv-56} The group $\tg{2}{28}$ occurs over a decic field (\Cref{prop:2x28-2x30}), so we cannot exclude $\torz{56}$ by \Cref{lem:isogeny}. We first sieve the reductions modulo $3$ with push-forwards to $X_1(2,28)$ and $X_1(28)$, and treat the few remaining cases with Riemann-Roch computations on an explicit model of $X_1(56)$ as a double cover of $X_1(2,28)$. As in \cite[Section 11]{N69}, the curve $X_1(2,28)$ parametrizes triples $(E,P,Q)$ with $P$ of order $2$ and $Q$ of order $28$ such that $P\notin\dm Q$, and its universal elliptic curve $\mathcal E$ has the model \begin{equation}\label{eq:sv-56-model} y^2=x^3+cx^2+b(1-b-c)x,\qquad P=(0,0),\qquad Q=(b,b). \end{equation} The functions $b$, $b(1-b-c)$ and $c^2-4b(1-b-c)$ are units on $Y_1(2,28)$ over $\Z[\tfrac1{56}]$ \cite[Section 11]{N69}. The maps \begin{equation}\label{eq:sv-56-aut} (E,P,Q)\longmapsto(E,P+14\beta Q,\gamma P+aQ),\qquad \beta,\gamma\in\{0,1\},\ a\in(\torz{28})^\times, \end{equation} taken modulo $-1$, form a group of $24$ level automorphisms of $X_1(2,28)$, which commute with $T_q$ for odd primes $q\nmid 28$ and with the diamond operators \cite[Section 11]{N69}. Let \[ \pi:X_1(56)\to X_1(2,28),\qquad (E,R)\longmapsto(E/\dm{28R},P_2,\phi(R)), \] where $\phi:E\to E/\dm{28R}$ is the quotient isogeny and $P_2$ is the non-zero point of the kernel of its dual $\widehat\phi$. The point $\phi(R)$ has order $28$, and $P_2\neq14\phi(R)$, as $\widehat\phi(14\phi(R))=28R\neq0$. The map $\pi$ has degree $2$, and it maps the diamond operator $\dm a$ of $X_1(56)$ to the automorphism \eqref{eq:sv-56-aut} with $\beta=\gamma=0$ and the same $a$. Let $\rho_1,\dots,\rho_4:X_1(2,28)\to X_1(28)$ be the $4$ maps of \cite[Section 11]{N69}, which keep $Q$ or $P+Q$, or divide $E$ by $\dm P$ or by $\dm{P+14Q}$ and keep the image of $Q$. By \cite[Theorem 3.1(2) and (3)]{DEHMZ} the groups $J_1(56)(\Q)$ and $J_1(2,28)(\Q)$ are finite. The curve $X_1(56)$ has genus $61$ and $i(56)=1152$, so $\gon_\Q X_1(56)\geq\gon_\C X_1(56)\geq \tfrac{325}{2^{15}}\cdot1152>11.4$ by Abramovich's bound, and $X_1(2,28)$ has genus $25$ and $\gon_\Q X_1(2,28)\geq12$ \cite[Table 1]{NV}. The group $J_1(28)(\Q)\simeq\torz2\times\torz4\times\torz{12}\times \torz{936}$ is generated by rational cuspidal divisor classes \cite[Theorem 3.1(2), Corollary 4.14 and Table 2]{DEHMZ}, and its reduction $G_{28}$ has index $4$ in $J_1(28)(\F_3)\simeq(\torz4)^2\times\torz{24}\times\torz{936}$ \cite[Section 11.2]{N69} (see also \githubbare{computations/d31a_C56_target28_setup.m}). We fix a rational cusp $z_\ast$ of $X_1(28)$, and we write $\mathcal S_j\subseteq J_1(28)(\F_3)$ for the sets of \eqref{eq:pushclass} with $y_0=z_\ast$. The closed cusps of $X_1(2,28)_{\F_3}$ have degrees $1^{12},2^6,3^4,6^2$, those of degree $1$ and $2$ are reductions of rational cuspidal divisors of the same degree, and for every closed cusp $c$ of degree $3$ the divisor $A_5c$ is one of $17$ explicit divisors \cite[Section 11.2]{N69}. We write $\mathcal A$ for the set of these $17$ divisors, which contains $0$. \begin{lemma}\label{lem:sv-56-sieve} Let $D'\neq0$ be an effective divisor of degree $e\leq10$ on $X_1(56)_{\F_3}$ with non-cuspidal support, and suppose that \begin{itemize} \item[a)] $[(\rho_i\circ\pi)_*D'-e\overline{z_\ast}]\in G_{28}+\mathcal S_{10-e}$ for $i=1,2,3,4$, and \item[b)] there is an effective cuspidal divisor $C_Y$ of degree $10-e$ on $X_1(2,28)_{\F_3}$ such that $A_5(\pi_*D'+C_Y)$ is principal. \end{itemize} Then, up to the diamond operators, $D'$ is one of the $11$ divisors of \Cref{table:sv-56}. \end{lemma} \begin{table}[H] \centering \begin{tabular}{c|c|c} \toprule divisors $D'$ & type & label of $\pi_*D'$\\ \midrule $65$ & $[8]$ & $8$\\ $271$, $327$ & $[10]$ & $14$\\ $491$, $578$ & $[10]$ & $26$\\ $492$, $737$ & $[10]$ & $27$\\ $818$, $837$ & $[5,5]$ & $36$\\ $978$, $1009$ & $[5,5]$ & $41$\\ \bottomrule \end{tabular} \medskip \caption{The divisors $D'$ of \Cref{lem:sv-56-sieve}, and the labels of their push-forwards to $X_1(2,28)_{\F_3}$ up to the automorphisms \eqref{eq:sv-56-aut}.} \label{table:sv-56} \end{table} \begin{proof} By \Cref{table:sv-counts} the type of $D'$ is one of $[5],\dots,[10]$ and $[5,5]$. Taking a representative of each diamond orbit, and for the type $[5,5]$ the first point among the $6$ representatives and the second one among all the $72$ closed points of degree $5$, we obtain $797+432=1229$ divisors $D'$, which we number as in \githubbare{continuation_20261031a/C56/fourmap_filter.m}. Condition a) is invariant under the diamond operators, as in the proof of \Cref{lem:sv-46}. We \gitlink{computations/d31a_C56_fourmap_filter_v4.m}{compute} that a) holds for only $135$ of the $1229$ divisors: $2$, $6$, $11$, $8$ and $48$ of the types $[6],\dots,[10]$, and $60$ of type $[5,5]$. Up to the $24$ automorphisms \eqref{eq:sv-56-aut}, the push-forwards $\pi_*D'$ of these $135$ divisors give $41$ divisors on $X_1(2,28)_{\F_3}$, which we number as in \githubbare{continuation_20261031b/Y228_shared/union_cases.m}. Condition b) is invariant under these automorphisms, as they commute with $A_5$ and permute the effective cuspidal divisors of each degree. If $\deg C_Y\leq2$, then $C_Y$ is supported on the closed cusps of degree $1$ and $2$, and $A_5C_Y=0$. If $3\leq\deg C_Y\leq4$, then $C_Y$ contains at most one closed cusp of degree $3$, and $A_5C_Y\in\mathcal A$. As $\deg C_Y=10-e$, only $A=0$ occurs if $e\geq8$. We \gitlink{computations/d31b_Y228_A5_common_v5.m}{compute} on the function field of $X_1(2,28)_{\F_3}$ that $A_5(\pi_*D')+A$ is not principal for all $A\in\mathcal A$ if $e\leq7$, and for $A=0$ if $e\geq8$, except for the $6$ divisors with labels $8$, $14$, $26$, $27$, $36$ and $41$. These are the push-forwards of $15$ of the $135$ divisors $D'$, which form the $11$ diamond orbits of \Cref{table:sv-56} (see \githubbare{continuation_20261031b/C56/d31b_C56_final_source_subset.py}). \end{proof} Next we give a model of $X_1(56)$ as a double cover of $X_1(2,28)$. \begin{lemma}\label{lem:sv-56-cover} Let $k$ be $\Q$ or $\F_3$, let $b,c$ be the functions of \eqref{eq:sv-56-model} on $X_1(2,28)_k$, and put $s=c^2-4b(1-b-c)$. Then $k(X_1(56))=k(X_1(2,28))(z)$ with $z^2=b$, and $\pi$ corresponds to this extension. The point \[ R=\bigl(2b+c+2z,\ 2z(2b+c+2z)\bigr) \] on $E'':y^2=x^3-2cx^2+sx$ has order $56$, and the isogeny $E''\to\mathcal E$, $(x,y)\mapsto\bigl((x+s/x-2c)/4,\ y(1-s/x^2)/8\bigr)$, whose kernel is generated by $(0,0)$, maps $R$ to $Q$. Here $E''=\mathcal E/\dm P$. The map $\pi$ is unramified outside the cusps. \end{lemma} \begin{proof} Write $u=2b+c+2z$. Using $z^2=b$, a direct computation gives $u^2-2cu+s=4bu$, so $u(u^2-2cu+s)=(2zu)^2$ and $R$ lies on $E''$, and $u^2-s=4zu$. Hence the isogeny maps $R$ to the point with coordinates $(u^2-2cu+s)/(4u)=b$ and $2zu(u^2-s)/(8u^2)=b$, which is $Q$. So $R$ has order $28$ or $56$, and we \gitlink{computations/d31b_C56_cover_hecke_common_v4.m}{check} that $28R\neq0$ in characteristic $3$, so also in characteristic $0$. Hence $28R$ is a non-zero point which the isogeny maps to $28Q=0$, so $28R=(0,0)$, and $E''/\dm{28R}\simeq\mathcal E$, with the kernel of the dual isogeny generated by $P$. So $(E'',R)$ is a point of $X_1(56)$ over $k(X_1(2,28))(z)$ whose image under $\pi$ is the generic point $(\mathcal E,P,Q)$ of $X_1(2,28)$. As $\pi$ has degree $2$ and $b$ is not a square (its divisor has odd valuations at $24$ cusps, as we \gitlink{computations/d31b_C56_cover_hecke_common_v4.m}{compute}), this identifies $k(X_1(56))$ with $k(X_1(2,28))(z)$, and we also check that the genus of this function field over $\F_3$ is $61$. As $b$ is a unit on $Y_1(2,28)$, the divisor of $b$ is supported on the cusps, so $\pi$ is unramified outside the cusps. \end{proof} \begin{lemma}\label{lem:sv-56-rr} Let $D'$ be one of the $11$ divisors of \Cref{table:sv-56}, and let $q=11$ if $D'$ has degree $8$ and $q=5$ otherwise. Then $A_qD'$ is not principal on $X_1(56)_{\F_3}$. \end{lemma} \begin{proof} Let $F_Y$ be the function field of $X_1(2,28)_{\F_3}$, and $L_Y=F_Y(z)$ that of $X_1(56)_{\F_3}$, with $z^2=b$ as in \Cref{lem:sv-56-cover}. We \gitlink{computations/d31b_C56_cover_hecke_common_v4.m}{compute} $T_qD'$ from the isogenies of degree $q$ of the elliptic curves of the points in the support of $D'$ on the moduli model, including the residue degrees, and we locate the places of $L_Y$ in the support of $G=A_qD'$ using the value of $z$ at the corresponding point of order $56$. Write $G=\sum n_{\mathfrak Q}\mathfrak Q$. Its support contains no cusp, so $\pi$ is unramified at it by \Cref{lem:sv-56-cover}. For a place $\mathfrak q$ of $F_Y$ put $a_{\mathfrak q}=\max(\{0\}\cup\{n_{\mathfrak Q}: \mathfrak Q\mid\mathfrak q\})$ and $H=\sum a_{\mathfrak q}\mathfrak q$. Then $G\leq\pi^*H$. As the divisor $\pi^*H$ is invariant under $z\mapsto-z$ and the characteristic is not $2$, we have \[ L(\pi^*H)=L(H)\oplus z\,L(H+R_b),\qquad R_b=\bigl\lfloor\tfrac12\ddiv(b)\bigr\rfloor, \] where the spaces on the right are Riemann-Roch spaces on $X_1(2,28)_{\F_3}$. A function $g\in L(\pi^*H)$ lies in $L(G)$ if and only if at every place $\mathfrak Q\mid\mathfrak q$ with $n_{\mathfrak Q}10$, \Cref{prop:main}~a) with $p=3$ shows that the reduction of $D$ modulo $3$ is $D'+C'$ with $D'\neq0$ of degree $e$. Applying \eqref{eq:pushclass} to $(\rho_i\circ\pi)_*D$ shows that $D'$ satisfies condition a) of \Cref{lem:sv-56-sieve}. By \Cref{prop:tools}~d) on $X_1(2,28)$ the divisor $A_5(\pi_*D)$ is principal, so $A_5(\pi_*D'+\pi_*C')$ is principal, and $D'$ satisfies condition b) with $C_Y=\pi_*C'$. Replacing $x$ by its image under a diamond operator, we may assume by \Cref{lem:sv-56-sieve} that $D'$ is one of the divisors of \Cref{table:sv-56}. If $D'$ has degree $8$, then $C'$ has degree $2$, and we \gitlink{continuation_20261031b/C56/d31b_C56_cusp_A11_audit.py}{compute} that $\sigma_{11}$ fixes each of the $27$ closed cusps of degree at most $2$ of $X_1(56)_{\F_3}$, so $A_{11}C'=0$ by \Cref{lem:cuspHecke}. Otherwise $C'=0$. So by \Cref{prop:tools}~d) on $X_1(56)$ the divisor $A_qD'$ is principal, with $q$ as in \Cref{lem:sv-56-rr}, which contradicts this lemma. \end{proof}