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Hyperelliptic Jacobians without complex multiplication. (English) Zbl 0959.14013
Let $$K$$ be a field of characteristic different from 2 and $$K_a$$ its algebraic closure. The main result proved in the paper is the following:
Let $$f(x)\in K[x]$$ be an irreducible and separable polynomial of degree $$n\geq 5$$ whose Galois group is $$S_n$$ or $$A_n$$ and let $$J(C_f)$$ be the Jacobian of the hyperelliptic curve of equation $$y^2=f(x)$$ whose $$K_a$$-endomorphism ring is denoted by $$\text{End} (J(C_f))$$. Then either $$\text{End} (J(C_f))= \mathbb{Z}$$ or $$\text{char}(K)>0$$ and $$J(C_f)$$ is a supersingular abelian variety.
The proof is based on the study of representation of $$\text{End} (J(C_f))$$ in the $$\mathbb{F}_2$$-vector space of 2-torsion points of $$J(C_f)$$.

##### MSC:
 14H40 Jacobians, Prym varieties 14K22 Complex multiplication and abelian varieties 11G10 Abelian varieties of dimension $$> 1$$
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