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

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.