Modularity theorems for abelian surfaces

TL;DR

The paper proves modularity for a positive fraction of abelian surfaces over ${f Q}$ under precise residuosity and ordinarity hypotheses at $3$, via a 2-3 switch and a new classicality theorem for ordinary $p$-adic Siegel modular forms. The authors develop a modern framework combining condensed mathematics, Lie algebra (co)homology, and equivariant sheaves on flag varieties to relate $p$-adic and classical automorphic data, culminating in a p-adic Eichler–Shimura theory for Hodge-type Shimura varieties. Central to the argument is proving that certain ordinary $p$-adic Siegel eigenforms with de Rham Galois representations are classical, enabling modular lifts from residual data to full modularity of abelian surfaces. The work relies on Arthur’s endoscopic classification, recent modularity lifting results at small primes, and a geometric localization theory (Higher Coleman theory) that interprets representations as sheaves on Bruhat strata with explicit Ext- and cohomology calculations. The results yield concrete modularity statements for Jacobians of genus two curves under computable congruence conditions, and they pave the way toward a broader modularity program for abelian surfaces and related Shimura varieties.

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.

Theorems & Definitions (658)

• Theorem A
• Theorem B
• Definition 1.8.10
• Remark 1.8.11
• Theorem 1.8.13
• Remark 1.8.15
• Definition 1.8.16
• Theorem 1.8.17
• proof
• Remark 1.8.20