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

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.