The rank of abelian varieties over large algebraic fields (Q2490659)

Let \(K\) be an infinite finitely generated field over its prime field and \(A\) an abelian variety over \(K\). Let \(K_s\) be the separable closure of \(K\). Let \(\text{Gal}(K)= \text{Gal}(K_s/K)\) be the absolute Galois group of \(K\). let \(e\) be a positive integer, then for each \(\sigma= (\sigma_1,\sigma_2 ,\dots, \sigma_e)\in \text{Gal} (K)^e\) we write \(K_s(\sigma)\) for the fixed field of \(\sigma_1,\sigma_2,\dots,\sigma_e\). The following theorem was proved by Frey-Jarden in 1974. Theorem. Let \(K\) be an infinite finitely generated field, \(A\) an abelian variety over \(K\) of positive dimension, and \(e\) a positive integer. Then \(\text{rank} (A(K_s(\sigma))= \infty\) for almost all \(\sigma\in\text{Gal} (K)^e\). The authors of the paper under review use the stability of fields [\textit{K. Neumann}, Isr. J. Math. 104, 221--260 (1998; Zbl 0923.12006)] and \textit{B.-H. Im}'s method [Can. J. Math. 58, No. 4, 796--819 (2006; Zbl 1160.11031)] to sharpen the above theorem.