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

2026/02/23 by Sam Frengley, Dylan Laird
Mathematics · #math.NT

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Abstract

We exhibit examples of geometrically simple abelian surfaces A/ℚ with conductor bounded by (10 000)² whose Tate--Shafarevich groups contain a subgroup isomorphic to (ℤ/pℤ)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 ∈ S₂^new(Γ₀(N)) and g ∈ S₂^new(Γ₀(M)) contained in the LMFDB (i.e., with N, M ≤ 10 000) and with coefficient fields of degree ≤ 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 (ℤ/7ℤ)² ⊂ Sha(A/ℚ) which is (conjecturally) not visible in any abelian threefold.

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