  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 $n=2^{3m}+1,
\Gal(f)=\U_3(2^m):=\PSU_3(\F_{2^m})$.
