Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Modular abelian surfaces of small conductor with nontrivial Tate--Shafarevich groups

Sam Frengley, Dylan Laird

Abstract

We exhibit examples of geometrically simple abelian surfaces $A/\mathbb{Q}$ with conductor bounded by $(10\,000)^2$ whose Tate--Shafarevich groups contain a subgroup isomorphic to $(\mathbb{Z}/p\mathbb{Z})^2$ for each $p = 5, 7, 11, 13$. To find these examples we generalise work of Cremona--Freitas to enumerate all congruences of a certain type between pairs of weight $2$ newforms $f \in S_2^{\mathrm{new}}(Γ_0(N))$ and $g \in S_2^{\mathrm{new}}(Γ_0(M))$ contained in the LMFDB (i.e., with $N, M < 10\,000$) and with coefficient fields of degree $\leq 4$. Passing from the modular forms to the corresponding abelian varieties we use visibility to (unconditionally) prove the existence of non-trivial elements of the Tate--Shafarevich group. Finally we construct an example of an abelian surface with $(\mathbb{Z}/7\mathbb{Z})^2 \subset \mathrm{Sha}(A/\mathbb{Q})$ which is (conjecturally) not visible in any abelian threefold.

Modular abelian surfaces of small conductor with nontrivial Tate--Shafarevich groups

Abstract

We exhibit examples of geometrically simple abelian surfaces with conductor bounded by whose Tate--Shafarevich groups contain a subgroup isomorphic to for each . To find these examples we generalise work of Cremona--Freitas to enumerate all congruences of a certain type between pairs of weight newforms and contained in the LMFDB (i.e., with ) and with coefficient fields of degree . Passing from the modular forms to the corresponding abelian varieties we use visibility to (unconditionally) prove the existence of non-trivial elements of the Tate--Shafarevich group. Finally we construct an example of an abelian surface with which is (conjecturally) not visible in any abelian threefold.
Paper Structure (13 sections, 10 theorems, 14 equations, 5 tables)

This paper contains 13 sections, 10 theorems, 14 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Congruences of weight 2 newforms
  3. Outline of the paper
  4. Acknowledgements
  5. Proving congruences of modular forms
  6. Promoting congruences to Galois representations
  7. Sieving the LMFDB for congruences between modular forms
  8. Filtering for visible III and Theorems 1.1 and 1.4
  9. Sieving for possible non-trivial III(A/Q)[p]
  10. On Conjecture 1.5
  11. On the theorem of Agashe--Stein and Fisher
  12. Tables
  13. How to read the tables

Key Result

Theorem 1.1

Let $(A, p)$ be any of the pairs in Table table:examples where $A/\mathbb{Q}$ is an abelian surface, $p$ is a prime number. Then $A/\mathbb{Q}$ is geometrically simple and the Tate--Shafarevich group $\Sha(A/\mathbb{Q})$ contains a subgroup isomorphic to $(\mathbb{Z}/p\mathbb{Z})^2$.

Theorems & Definitions (25)

  • Theorem 1.1
  • Remark 1.2
  • Definition 1.3
  • Theorem 1.4
  • Conjecture 1.5
  • Definition 1.7
  • Theorem 1.8
  • Remark 1.9
  • Lemma 2.1
  • proof
  • ...and 15 more