  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 $Gal(f)$ 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 $n=2^r+1,
Gal(f)=L_2(2^r)$ and $n=2^{4r+2}+1, Gal(f)=Sz(2^{2r+1})$.
  In the present paper we prove that $J(C)$ has only trivial endomorphisms over
$K_a$ if the set of roots of $f$ could be identified with the
$(m-1)$-dimensional projective space $P^{m-1}(F_q)$ over a finite field $F_q$
of odd characteristic in such a way that $Gal(f)$, viewed as its permutation
group, becomes either the projective linear group $PGL(m,F_q)$ or the
projective special linear group $L_m(q):=PSL(m,F_q)$. Here we assume that
$m>2$.
