TORSION OF ABELIAN VARIETIES OVER LARGE ALGEBRAIC EXTENSIONS OF

Abstract

Let$K$be a finitely generated extension of$\mathbb{Q}$, and let$A$be a nonzero abelian variety over$K$. Let$\tilde{K}$be the algebraic closure of$K$, and let$\text{Gal}(K)=\text{Gal}(\tilde{K}/K)$be the absolute Galois group of$K$equipped with its Haar measure. For each$\unicode[STIX]{x1D70E}\in \text{Gal}(K)$, let$\tilde{K}(\unicode[STIX]{x1D70E})$be the fixed field of$\unicode[STIX]{x1D70E}$in$\tilde{K}$. We prove that for almost all$\unicode[STIX]{x1D70E}\in \text{Gal}(K)$, there exist infinitely many prime numbers$l$such that$A$has a nonzero$\tilde{K}(\unicode[STIX]{x1D70E})$-rational point of order$l$. This completes the proof of a conjecture of Geyer–Jarden from 1978 in characteristic 0.