Modular abelian surfaces of small conductor with nontrivial Tate--Shafarevich groups
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| Format: | Preprint |
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2026
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| author | Frengley, Sam Laird, Dylan |
| author_facet | Frengley, Sam Laird, Dylan |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_19813 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Modular abelian surfaces of small conductor with nontrivial Tate--Shafarevich groups Frengley, Sam Laird, Dylan Number Theory 11G30, 11G10, 14H40 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. |
| title | Modular abelian surfaces of small conductor with nontrivial Tate--Shafarevich groups |
| topic | Number Theory 11G30, 11G10, 14H40 |
| url | https://arxiv.org/abs/2602.19813 |