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 \(J(C_f)\).


14H40 Jacobians, Prym varieties
14K22 Complex multiplication and abelian varieties
11G10 Abelian varieties of dimension \(> 1\)
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