A "height-free" effective isogeny estimate for abelian varieties of $\GL_2$-type.

Abstract: 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 $\GL_2$-type over $\overline{\mathbb{Q}}$. We show that there is a $K$-isogeny $A\rightarrow B$ 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.

algebraic geometrynumber theory

Audience: researchers in the topic

( paper )

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