Special Cases of the Shafarevich Conjecture for Complete Intersections in Abelian Varieties

TL;DR

This paper proves the Shafarevich conjecture for specific complete intersections in abelian varieties over number fields, utilizing the Lawrence-Venkatesh method and new monodromy and Euler characteristic computations.

Contribution

It introduces novel computations of Euler characteristics and a big monodromy result for the Hodge structures of these complete intersections, advancing the understanding of the conjecture.

Findings

Proved the Shafarevich conjecture for certain complete intersections in abelian varieties.

Established a big monodromy statement for the variation of Hodge structures.

Computed key Euler characteristics of the complete intersections.

Abstract

In this paper, we prove the Shafarevich conjecture for certain complete intersections of hypersurfaces in abelian varieties defined over a number field $K$ using the Lawrence-Venkatesh method. The main new inputs we need are computation of certain Euler characteristics of these complete intersections and a big monodromy statement for the variation of Hodge structure arising from the middle cohomology of a family of such complete intersections. Following \cite{ls25}, we prove the latter by relating this monodromy statement to a statement about Tannaka groups, which we then convert into a combinatorial statement.