DOI: 10.1017/s0010437x26103091 ISSN: 0010-437X

The Northcott property in the double struck upper Q overbar

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Abstract

Let A be an abelian variety over a number field

sans serif upper K comma K , $\mathsf{K},$
with algebraic closure
sans serif upper K overbar K ¯ $\bar{\mathsf{K}}$
. Assuming the Mumford–Tate conjecture for A , we show that the isogeny class of A over
sans serif upper K overbar K ¯ $\bar{\mathsf{K}}$
contains only finitely many isomorphism classes of bounded Faltings height. As the Mumford–Tate conjecture is known for many abelian varieties, our theorem is unconditional in those cases.

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