Hyperelliptic jacobians and modular representations

Abstract

In his previous paper (Math. Res. Letters 7(2000), 123--132) the author proved that in characteristic zero the jacobian $J(C)$ of a hyperelliptic curve $C: y^2=f(x)$ has only trivial endomorphisms over an algebraic closure of the ground field $K$ if the Galois group $Gal(f)$ of the irreducible polynomial $f(x) \in K[x]$ is either the symmetric group $S_n$ or the alternating group $A_n$. Here $n>4$ is the degree of $f$. In the present paper we extend this result to the case of certain ``smaller'' Galois groups. In particular, we treat the case when $n=11$ or 12 and $Gal(f)$ is the Mathieu group $M_{11}$ or $M_{12}$ respectively. The infinite series $n=2^r+1, Gal(f)=L_2(2^r)$ and $n=2^{4r+2}+1, Gal(f)=Sz(2^{2r+1})$ are also treated.