Hyperelliptic jacobians and $\U_3(2^m)$

Yuri G. Zarhin

paper

Hyperelliptic jacobians and $\U_3(2^m)$

arXiv:math/0103082

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 $K_{a}$ of the ground field $K$ if the Galois group $G a l (f)$ 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 math.AG/0003002 we extended this result to the case of certain ``smaller'' Galois groups. In particular, we treated the infinite series $n = 2^{r} + 1, G a l (f) = L_{2} (2^{r})$ and $n = 2^{4 r + 2} + 1, G a l (f) = S z (2^{2 r + 1})$ . In this paper we do the case of $Gal(f)=\U_3(2^m)$ and $n = 2^{3 m} + 1$ .