Un peu d'effectivit\'e pour les vari\'et\'es modulaires de Hilbert-Blumenthal

TL;DR

This paper establishes an effective bound on the degree of isogenies between certain abelian varieties of $ ext{GL}_2$-type over number fields, leading to bounds on $S$-integral points on Hilbert modular varieties.

Contribution

It provides a height-free, effective isogeny estimate for $ ext{GL}_2$-type abelian varieties, which was not previously known.

Findings

Effective bound on isogeny degree depending only on $g$, $K$, and $S$.

Bound on the number of $S$-integral points on Hilbert modular varieties.

Application to finiteness results in arithmetic geometry.

Abstract

We prove a "height-free" effective isogeny estimate for abelian varieties of $\mathrm{GL}_2$-type. More precisely, let $g\in \mathbb{Z}^+$, $K$ a number field, $S$ a finite set of places of $K$, and $A,B/K$ $g$-dimensional abelian varieties with good reduction outside $S$ which are $K$-isogenous and of $\mathrm{GL}_2$-type over $\overline{\mathbb{Q}}$. We show that there is a $K$-isogeny $A\to B$ of degree effectively bounded in terms of $g$, $K$, and $S$ only. We deduce among other things an effective upper bound on the number of $S$-integral $K$-points on a Hilbert modular variety.