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