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

Abstract

We exhibit examples of geometrically simple abelian surfaces $A/\mathbb{Q}$ with conductor bounded by $(10000{)}^2$ 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 give a candidate list of all congruences of a certain type between pairs of weight $2$ newforms $f\in S_2^{\mathrm{new}}({\Gamma }_0(N))$ and $g\in S_2^{\mathrm{new}}({\Gamma }_0(M))$ contained in the LMFDB (i.e., with $N,M\le 10000$) and with coefficient fields of degree $\le 4$. 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.

21 pages, comments welcome