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

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

2002-01-20

arxiv.org/abs/math/0201185v1

Mathematics

Algebraic Geometry

LaTeX, 15 pages

Scientific paper

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.

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