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.
Categories: math.NT math.AG