Homomorphisms of Hyperelliptic Jacobians

Yu. G. Zarhin

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Number theory, algebra, and algebraic geometry, Tome 241 (2003), pp. 90-104

Résumé

Let $K$ be a field of characteristic different from $2$ and $K_a$ be its algebraic closure. Let $n\ge 5$ and $m\ge 5$ be integers. Assume, in addition, that if $K$ has positive characteristic, then $n\ge 9$. Let $f(x),h(x)\in K[x]$ be irreducible separable polynomials of degree $n$ and $m$, respectively. Suppose that the Galois group of $f$ is either the full symmetric group $\mathbf S_n$ or the alternating group $\mathbf A_n$ and the Galois group of $h$ is either the full symmetric group $\mathbf S_m$ or the alternating group $\mathbf A_m$. Let us consider the hyperelliptic curves $C_f\colon y^2=f(x)$ and $C_h\colon y^2=h(x)$. Let $J(C_f)$ be the Jacobian of $C_f$ and $J(C_h)$ be the Jacobian of $C_h$. Earlier, the author proved that $J(C_f)$ is an absolutely simple abelian variety without nontrivial endomorphisms over $K_a$. In the present paper, we prove that $J(C_f)$ and $J(C_h)$ are not isogenous over $K_a$ if the splitting fields of $f$ and $h$ are linearly disjoint over $K$.