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Modularity theorems for abelian surfaces

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arxiv 2502.20645 v1 pith:IDDJIPLP submitted 2025-02-28 math.NT

classification math.NT
keywords abelianmodularitysurfaceshypothesisordinaryproveactionadic
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abstract

We prove the modularity of a positive proportion of abelian surfaces over $\mathbf{Q}$. More precisely, we prove the modularity of abelian surfaces which are ordinary at $3$ and are $3$-distinguished, subject to some assumptions on the $3$-torsion representation (a "big image" hypothesis, and a technical hypothesis on the action of a decomposition group at $2$). We employ a 2-3 switch and a new classicality theorem (in the style of Lue Pan) for ordinary $p$-adic Siegel modular forms.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. The residual monodromy for the Dwork family in even characteristic and its applications to Galois representations

    math.NT 2025-06 conditional novelty 7.0 of 10

    The mod 2 monodromy of the Dwork family is classified, with symmetric group images when the dimension is a power of two, yielding automorphy of rank-4 symplectic Galois representations over totally real fields.

  2. Higher Hida theory for Drinfeld modular curves

    math.NT 2025-07 conditional novelty 6.0 of 10

    Higher Hida theory, including an interpolated Serre duality pairing, is constructed for Drinfeld modular curves.

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