2002/01/20 by Yuri G. Zarhin, Zarhin, Yuri G.
Mathematics · #11G10 #11G30 #14H40 #14K05 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT) #math.AG #math.NT #msc:11G10 #msc:11G30 #msc:14H40 #msc:14K05
paper · pdf · doi:10.48550/arxiv.math/0201185
LaTeX, 15 pages
arxiv created 2002/01/20 · openalex publication_date 2002/01/20 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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: y2=f(x) has only trivial endomorphisms over an algebraic closure Ka of the ground field K if the Galois group Gal(f) of the irreducible polynomial f(x) ∈ K[x] is either the symmetric group Sn or the alternating group An. 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=2r+1, Gal(f)=L2(2r). The case of small Mathieu groups Mn (with n=11,12) was also treated. In this paper we do the case of large Mathieu groups Mn (with n=22,23,24). We also treat the infinite series Gal(f)=Lm(2r) (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)(F2r) over the finite field F2r of characteristic 2 in such a way that the Galois action on R becomes the natural action of Lm(2r) on the projective space.