Bielliptic-Shimura-Curves

Magma code for the paper Bielliptic Shimura curves $X_0^D(N)$ with nontrivial level by Oana Padurariu and Frederick Saia. (arXiv link)

Total computation time ($< 3$ minutes with a 2020 Apple M1 chip) and memory usage ($< 81$ MB) for running all main files is not significant. The contents of this repository, sorted roughly in order of use in the paper, are as follows:

Main Files

• narrow_to_candidates.m: Here, we narrow our list of candidate bielliptic pairs $(D,N)$ with $D,N > 1$ and $D,N$ relatively prime using the genus bound: if $X_0^D(N)$ is bielliptic, then $g(X_0^D(N)) \leq 39$, and the fact that if $X_0^D(N)$ is bielliptic then $X_0^D(1)$ is bielliptic, hyperelliptic, or of genus at most $1$. We then prove that certain candidate pairs have $\text{Aut}(X_0^D(N)) = W_0(D,N)$.

• quot_genus.m: The main function, quot_genus, computes for a given indefinite rational quaternion discriminant $D$, positive integer $N$ coprime to $D$, and $W \subseteq W_0(D,N)$ a subgroup of the full group of Atkin--Lehner involutions on $X_0^D(N)$, the genus of the quotient Shimura curve $X_0^D(N)/W$.

• genus_1_quotients_and_ranks.m: Using the list of candidate pairs with $N$ squarefree computed in narrow_to_candidates.m and the quot_genus function in quot_genus.m for the genus of an Atkin--Lehner quotient $X_0^D(N)/\langle w_m \rangle$, we determine the genus $1$ quotients of this form. We then determine the rank of the elliptic curve jacobian of each such quotient, using Ribet's isogeny. We also compute the genus $1$ quotients of this form for the candidates with $N$ not squarefree.

• rationality_by_CM.m: Here, we aim to prove that certain bielliptic quotients are elliptic over $\mathbb{Q}$ by showing that the genus one quotient has a rational CM point.

• narrow_sporadics.m: The aim of this code is to use our results on Bielliptic Shimura curves $X_0^D(N)$ to decrease the number of pairs $(D,N)$ for which we remain unsure of whether $X_0^D(N)$ has a sporadic point, following initial work of Saia 2024.

• generating_trigonal_candidates.m: Here, we narrow our list of candidate geometrically trigonal pairs $(D,N)$ with $D,N > 1$ and $D,N$ relatively prime using the genus bound given by Abramovich's gonality bound: if $X_0^D(N)$ is geometrically trigonal, then $g(X_0^D(N)) \leq 29$.

• trigonal_checks.m: In this code, we narrow down the pairs in trigonal_candidate_pairs.m using the two results of Schweizer on geometrically trigonal curves.

Required Files

• cond_disc_list_allO.m: list of all (not just maximal) imaginary quadratic orders of class number up to $100$. The $i^\text{th}$ element is the complete list of sequences $[f,d_K] = [\text{conductor}, \text{fundamental disc}]$ for imaginary quadratic orders of class number $i$. Generated using list of maximal orders by M. Watkins. This list is used in least degree computations in shimura_curve_CM_locus.m.

• shimura_curve_CM_locus.m: The aim of this code, from the paper CM points on Shimura curves via QM-equivariant isogeny volcanoes (Saia 2024) is to compute the $\Delta$-CM locus on $X_0^D(N)$ for any imaginary quadratic discriminant $\Delta$ and positive integer $N$ coprime to a given quaternion discriminant $D$. This is done via the QM-equivariant isogeny volcano approach of the referenced paper of Saia in the $D>1$ case, and of work of Clark--Saia in the $D=1$ case. In particular, this file contains code to enumerate all CM points of a specified discriminant $\Delta$ with all possible residue fields up to isomorphism for one of these Shimura curves. This is used in quadratic CM point computations in rationality_by_CM and least degrees of CM points computations in narrow_sporadics.m.

• unknown_sporadics.m: List of all $329$ triples $[D,N,d_\text{CM}(X_0^D(N))]$ consisting of pairs $[D,N]$ for which we are unsure whether $X_0^D(N)$ has a sporadic CM point following work of Saia 2024. This list is used in narrow_sporadics.m, where we try to improve on the results of Saia 2024 by decreasing the number of unknowns.

Computed Lists

• candidate_pairs.m: list of $357$ pairs $(D,N)$ with $D,N>1$ and $(D,N) = 1$ so that $X_0^D(N)$ is possible bielliptic, given that $X_0^D(1)$ is of genus at most $2$ or is hyperelliptic or bielliptic and that $g(X_0^D(N))$ is at most $39$. This is computed in the file narrow_to_candidates.m.

• all_atkin_lehner.m: list of $275$ candidate pairs $(D,N)$ for which we know that full automorphism group of $X_0^D(N)$ is the Atkin--Lehner group $W_0(D,N)$. This is computed in the file narrow_to_candidates.m.

• sqfree_candidate_pairs.m : list of $301$ pairs $(D,N)$ from candidate_pairs.m with $N$ squarefree.

• not_sqfree_candidate_pairs.m : list of $56$ pairs $(D,N)$ from candidate_pairs.m with $N$ not squarefree.

• genus_1_quotients_N_sqfree.m: list of all $77$ triples $(D,N,m)$ with $D>1$, $D$ relatively prime to $N$, $N$ squarefree and $m$ a Hall Divisor of $DN$ such that $X_0^D(N)/\langle w_m \rangle$ has genus $1$. This is computed in genus_1_quotients_and_ranks.m.

• genus_1_quotients_N_not_sqfree.m: list of all $5$ triples $(D,N,m)$ with $D>1$, $D$ relatively prime to $N$, $N$ not squarefree and $m$ a Hall Divisor of $DN$ such that $X_0^D(N)/\langle w_m \rangle$ has genus $1$. This is computed in genus_1_quotients_and_ranks.m.

• rank0.m: list of all $68$ triples $(D,N,m)$ in genus_1_quotients_N_sqfree such that $J(X_0^D(N)/\langle w_m \rangle)$ has rank $0$. This is computed in genus_1_quotients_and_ranks.m.

• rank1.m: list of all $9$ triples $(D,N,m)$ in genus_1_quotients_N_sqfree such that $J(X_0^D(N)/\langle w_m \rangle)$ has rank $1$. This is computed in genus_1_quotients_and_ranks.m.

• to_check_quad_pts.m: for all the pairs $(D,N)$ for which we are unsure whether $X_0^D(N)$ is bielliptic over $\mathbb{Q}$ (via an Atkin--Lehner involution), we list all of the quadratic CM points on $X_0^D(N)$. This is computed in rationality_by_CM.m.

• rational_CM_pts_info.m: info on rational CM points on genus $1$ Atkin--Lehner quotients $X_0^D(N)/\langle w_m \rangle$, computed in rationality_by_CM.m.

• has_sporadic_CM_X0.m: list of $264$ pairs $[D,N]$ for which we know $X_0^D(N)$ has a sporadic CM point, by virtue of having a degree $2$ CM point and not having infinitely many deg $2$ points. This is computed in narrow_sporadics.m.

• has_sporadic_CM_X1.m: list of $73$ pairs $[D,N]$ for which we know $X_1^D(N)$ has a sporadic CM point, by virtue of having a degree $2$ CM point and not having infinitely many deg $2$ points. This is computed in narrow_sporadics.m.

• new_no_sporadics_XD0.m: list of all $73$ pairs $[D,N]$ for which we know that $X_0^D(N)$ has no sporadic points, by virtue of having infinitely many degree $2$ points. This list includes (all but at most $3$) cases where $X_0^D(N)$ is bielliptic with a bielliptic quotient of positive rank over $\mathbb{Q}$. This is computed in narrow_sporadics.m.

• no_sporadic_CM_X1.m: list of $4$ pairs $[D,N]$ for which we determine that $X_1^D(N)$ has no sporadic CM points, by virtue of $X_0^D(N)$ having infinitely many degree $2$ points and having $\text{a.irr}(X_1^D(N)) \leq \text{max}(2,\phi(N)) \leq d_{\text{CM}}(X_1^D(N))$. This is computed in narrow_sporadics.m.

• new_unknowns_X0.m: narrowed list of $56$ triples $[D,N,d_\text{CM}(X_0^D(N))]$, where $[D,N]$ is such that we remain unsure whether $X_0^D(N)$ has a sporadic point. This is computed in narrow_sporadics.m.

• new_unknowns_X1.m: narrowed list of $263$ triples $[D,N,d_\text{CM}(X_1^D(N))]$, where $[D,N]$ is such that we remain unsure whether $X_1^D(N)$ has a sporadic point. This is computed in narrow_sporadics.m.

• trigonal_candidate_pairs.m: All $455$ geometrically trigonal candidate pairs $(D,N)$, i.e., pairs $(D,N)$ with $\text{gcd}(D,N) = 1$ and $D>1$ such that $g(D,N) \leq 29$.

• narrowed_trigonal_candidates.m: List of all 6 relatively prime pairs $(D,N)$ with $D>1$ with $2 \leq g(X_0^D(N)) \leq 29$ (Abramovich bound required to be trigonal), with $g(X_0^D(N)) \not\equiv 1 \pmod{4}$ (Schweizer Corollary 3.5) and with $X_0^D(N)$ having the required fixed point counts for all Atkin--Lehner involutions (via Schweizer Lemma 3.4).