Hyperelliptic jacobians and modular representations
Abstract
Description
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.
The paper will appear in Texel volume "Moduli of abelian varieties" (Texel Island 1999), Birkhäuser
The paper will appear in Texel volume "Moduli of abelian varieties" (Texel Island 1999), Birkhäuser