Hyperelliptic jacobians and projective linear Galois groups

Abstract

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 $m>2$.