Hyperelliptic jacobians without complex multiplication and Steinberg representations in positive characteristic

Details Hyperelliptic jacobians without complex multiplication and Steinberg representations in positive characteristic Hyperelliptic jacobians without complex multiplication and Steinberg representations in positive characteristic

2003-01-16

arxiv.org/abs/math/0301177v3

Mathematics

Number Theory

LaTeX2e, 11 pages

Scientific paper

In his previous papers (Math. Res. Letters 7 (2000), 123--13; Math. Res. Letters 8 (2001), 429--435; Moscow Math. J. 2 (2002), issue 2, 403-431) the author proved that in characteristic $\ne 2$ the jacobian $J(C)$ of a hyperelliptic curve $C: y^2=f(x)$ has only trivial endomorphisms over an algebraic closure $K_a$ 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 $\Sn$ or the alternating group $\A_n$. Here $n\ge 9$ is the degree of $f$. The goal of this paper is to extend this result to the case of certain ``smaller'' doubly transitive simple Galois groups. Namely, we treat the infinite series $n=2^m+1, \Gal(f)=\L_2(2^m):=\PSL_2(\F_{2^m})$, $n=2^{4m+2}+1, \Gal(f)=\Sz(2^{2m+1})= {^2\B_2}(2^{2m+1})$ and $n=2^{3m}+1, \Gal(f)=\U_3(2^m):=\PSU_3(\F_{2^m})$.

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