The Shafarevich conjecture for hypersurfaces in abelian varieties

TL;DR

The paper proves a Shafarevich-type finiteness result for smooth hypersurfaces in a fixed abelian variety A with good reduction outside a finite set S, provided dim A ≥ 4 (and a conditional refinement when dim A = 3). It develops a novel framework combining p-adic variation of Hodge structure with the Tannakian theory of sheaf convolution to establish a uniform big monodromy result for families of hypersurfaces in A. Central to the argument are Hodge–Deligne systems with H0-algebras, a careful treatment of disconnected reductive groups, and period-map techniques (both complex and p-adic) that translate monodromy bounds into non-density statements for integral points, hence finiteness. The approach is designed to be broadly applicable to families with locally injective period maps and highlights a deep connection between arithmetic finiteness and monodromy in the setting of abelian varieties. The work also situates its results within the broader program of Shafarevich-type finiteness by leveraging big monodromy and Tannakian convolution to control the arithmetic of cohomology in high degrees.”

Abstract

Faltings proved that there are finitely many abelian varieties of genus $g$ over a number field $K$, with good reduction outside a finite set of primes $S$. Fixing one of these abelian varieties $A$, we prove that there are finitely many smooth hypersurfaces in $A$, with good reduction outside $S$, representing a given ample class in the Néron-Severi group of $A$, up to translation, as long as the dimension of $A$ is at least $4$. Our approach builds on the approach of arXiv:1807.02721 which studies $p$-adic variations of Hodge structure to turn finiteness results for $p$-adic Galois representations into geometric finiteness statements. A key new ingredient is an approach to proving big monodromy for the variations of Hodge structure arising from the middle cohomology of these hypersurfaces using the Tannakian theory of sheaf convolution on abelian varieties.

Theorems & Definitions (274)

• Theorem 1.1: Theorem \ref{['intro_1']}
• Theorem 1.2: Theorem \ref{['intro_2']}
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Definition 2.3
• Definition 2.4
• Lemma 2.5
• proof