Hyperelliptic jacobians without complex multiplication, doubly transitive permutation groups and projective representations

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 the next paper (Progress in Math. 195(2001), 473--490) we extended this result to the case of certain``smaller'' Galois groups. In particular, we treated the infinite series $n=2^r+1, Gal(f)=L_2(2^r)$. The case of small Mathieu groups $M_n$ (with $n=11,12)$ was also treated. In this paper we do the case of large Mathieu groups $M_n$ (with $n=22,23,24$). We also treat the infinite series $Gal(f)=L_m(2^r)$ (with $m>2$ except the cases $(m,r)=(3,2)$ or $(4,1)$), assuming that the set $R$ of roots of $f$ can be identified with the corresponding projective space $P^{m-1)(F_{2^r})$ over the finite field $F_{2^r}$ of characteristic 2 in such a way that the Galois action on $R$ becomes the natural action of $L_m(2^r)$ on the projective space.