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Modularity theorems for abelian surfaces
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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.
Forward citations
Cited by 2 Pith papers
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The residual monodromy for the Dwork family in even characteristic and its applications to Galois representations
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Higher Hida theory for Drinfeld modular curves
Higher Hida theory, including an interpolated Serre duality pairing, is constructed for Drinfeld modular curves.
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