Hyperelliptic jacobians without complex multiplication and Steinberg representations in positive characteristic

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 $\mathrm{\ne }2$ 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 $\text{Gal}(f)$ of the irreducible polynomial $f(x)\in K[x]$ is either the symmetric group $\text{Sn}$ or the alternating group ${\text{A}}_n$. 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 $n=2^m+1,\text{Gal}(f)={\text{L}}_2(2^m):={\text{PSL}}_2({\text{F}}_{2^m})$, $n=2^{4m+2}+1, \Gal(f)=\Sz(2^{2m+1})= {^2\B_2}(2^{2m+1})$and$n=2^{3m}+1, \Gal(f)=\U_3(2^m):=\PSU_3(\F_{2^m})$.