Finiteness theorems for torsion of abelian varieties over large algebraic fields. (English) Zbl 1124.14304

Jacobson, Marcel; Jarden, Moshe

Acta Arith. 98, No. 1, 15-31 (2001).

From the introduction: Let \(K\) be a field finitely generated over its prime field. We denote the separable (resp. algebraic) closure of \(K\) by \(K_s\) (resp. \(\widetilde K\)) and let \(G(K)={\mathcal G}(K_s/K)\) be the absolute Galois group of \(K\). Each \(\sigma\in G(K)\) uniquely extends to an automorphism of \(\widetilde K\) having the same notation \(\sigma\). We consider the Cartesian product \(G(K)^e\) of \(e\) copies of \(G(K)\). For an abelian variety \(A\) over \(K\) and for each positive integer \(n\) let \(A_n\) be the kernel of multiplying \(A\) by \(n\). If \(M\) is an extension of \(K\), then \(A(n)=\{p\in A(M)\mid np= 0\}\). We use \(l\) to denote prime numbers and let \(A_{l^\infty}(M)= \bigcup^\infty_{i=1} A_{l^i}(M)\). We also let \(A_{\text{tor}}(M)\) be the group of all points of \(A(M)\) of finite order. Finally, we equip \(G(K)^e\) with the unique Haar measure \(\mu_K\) which is normalized with \(\mu_K(G(K))= 1\).
The main goal of our work is to prove the following result:
Main theorem. For almost all \(\sigma\in G(K)^e\), the following holds:
(a) \(A_{l^\infty}(\widetilde K(\sigma))\) is a finite group for all prime numbers \(l\);
(b) If \(e\geq 2\) and \(\text{char}(K)= 0\), then \(A_{\text{tor}}(\widetilde K(\sigma))\) is a finite group.
The main theorem solves part C of the W.-D. Geyer and M. Jarden conjecture [Isr. J. Math. 31, 157–197 (1978; Zbl 0406.14025)] in all cases and part B for \(\text{char}(K)= 0\).

MSC:

14K15	Arithmetic ground fields for abelian varieties
11G10	Abelian varieties of dimension \(> 1\)
14G27	Other nonalgebraically closed ground fields in algebraic geometry