Integral points on coarse Hilbert moduli schemes

TL;DR

This paper develops explicit bounds on the height and number of integral points on coarse Hilbert moduli schemes by combining Faltings' method with modularity and isogeny estimates, advancing effective Diophantine geometry.

Contribution

It introduces explicit height and point bounds on coarse Hilbert moduli schemes, extending previous theoretical results with concrete applications to classical surfaces.

Findings

Bound the height of integral points on Hilbert moduli schemes.

Explicitly bound the number of integral points outside the branch locus.

Apply results to classical algebraic surfaces by Klein and Clebsch.

Abstract

We continue our study of integral points on moduli schemes by combining the method of Faltings (Arakelov, Parsin, Szpiro) with modularity results and Masser-W\"ustholz isogeny estimates. In this work we explicitly bound the height and the number of integral points on coarse Hilbert moduli schemes outside the branch locus. In the first part we define and study coarse Hilbert moduli schemes with their heights and branch loci. In the second part we establish the effective Shafarevich conjecture for abelian varieties $A$ over a number field $K$ such that $A_{\bar{K}}$ has CM or $A_{\bar{K}}$ is of GL2-type and isogenous to all its $G_\mathbb Q$-conjugates. In the third part we continue our explicit study of the Parsin construction given by the forgetful morphism of Hilbert moduli schemes. We now work out our strategy for arbitrary number fields $K$ and we explicitly bound the number of polarizations and module structures on abelian varieties over $K$ with real multiplications. In the last part we illustrate our results by applying them to two classical surfaces first studied by Clebsch (1871) and Klein (1873): We explicitly bound the Weil height and the number of their integral points.