Arithmetic finiteness of very irregular varieties

TL;DR

This work establishes arithmetic finiteness (Shafarevich-type results) for a broad class of very irregular varieties, under a natural dimension constraint $d < \frac{1}{2}\dim\mathrm{Alb}(X)$ and mild numerical hypotheses. The authors blend the Lawrence–Venkatesh framework with a big monodromy criterion to show that integral points on moduli spaces of such varieties are not Zariski-dense, yielding finite families up to isomorphism. A key novelty is a robust p-adic Hodge-theoretic realization framework that handles general reductive groups and includes both étale and de Rham data, together with a careful analysis of period mappings and adjoint-filtration combinatorics. The results extend known Shafarevich finiteness to new higher-dimensional settings (including many complete intersections in abelian varieties) and advance the Lang–Vojta program by verifying finiteness for moduli spaces of very irregular canonically polarized varieties. The techniques have potential to illuminate broader questions about integral points, period maps, and monodromy in arithmetic geometry.

Abstract

We prove the Shafarevich conjecture for very irregular varieties of dimension less than half the dimension of their Albanese variety, subject to some mild numerical conditions. Our proof relies on the Lawrence-Venkatesh method as in the work of Lawrence-Sawin, together with the big monodromy criterion from our work with Javanpeykar and Lehn.

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