Hyperelliptic jacobians and simple groups $\mathbf {U}_3(2^m)$

Abstract:

In a previous paper, 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 $\operatorname {Gal}(f)$ of the irreducible polynomial $f(x) \in K[x]$ is either the symmetric group $\mathbf {S}_n$ or the alternating group $\mathbf {A}_n$. Here $n>4$ is the degree of $f$. In another paper by the author this result was extended to the case of certain “smaller” Galois groups. In particular, the infinite series $n=2^r+1, \operatorname {Gal}(f)=\mathbf {L}_2(2^r):=\operatorname {PSL}_2 (\mathbf {F}_{2^r})$ and $n=2^{4r+2}+1, \operatorname {Gal}(f)=\mathbf {Sz}(2^{2r+1})$ were treated. In this paper the case of $\operatorname {Gal}(f)=\mathbf {U}_3(2^m):=\operatorname {PSU}_3 (\mathbf {F}_{2^m})$ and $n=2^{3m}+1$ is treated.