Finite descent obstruction for Hilbert modular varieties

TL;DR

This paper demonstrates that a finite Galois descent obstruction fully explains the existence of integral points on Hilbert modular varieties, extending known results from modular curves and connecting to deep conjectures.

Contribution

It extends the finite descent obstruction result from modular curves to Hilbert modular varieties, under certain conjectural assumptions.

Findings

Finite Galois descent obstruction is the only obstacle for integral points.

Extension of results from modular curves to Hilbert modular varieties.

Conditional proof assuming special cases of the Hodge and Serre's conjectures.

Abstract

Let $S$ be a finite set of primes. We prove that a form of finite Galois descent obstruction is the only obstruction to the existence of $\mathbb{Z}_{S}$-points on integral models of Hilbert modular varieties, extending a result of D.Helm and F.Voloch about modular curves. Let $L$ be a totally real field. Under (a special case of) the absolute Hodge conjecture and a weak Serre's conjecture for mod $\ell$ representations of the absolute Galois group of $L$, we prove that the same holds also for the $\mathcal{O}_{L,S}$-points.