Hyperelliptic jacobians and projective linear Galois groups

2000/09/12 by Yuri G. Zarhin, Zarhin, Yuri G.
Mathematics · #Finite Group Theory Research #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory

paper · pdf · doi:10.48550/arxiv.math/0009123

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: y²=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) ∈ 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)=L₂(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.

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