Un peu d'effectivité pour les variétés modulaires de Hilbert-Blumenthal

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.

39 pages, title's in French (because I like Szpiro's modest phrase) but the paper's in English

References in corpus (1)

• On heights of motives with semistable reduction