Hyperelliptic jacobians and \U₃(2^m)

2001/03/13 by Yuri G. Zarhin, Zarhin, Yuri G.
Mathematics · #11G10 #11G30 #14H40 #14K05 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #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/0103082

openalex publication_date 2001/03/13 · arxiv created 2001/08/30 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 this paper we do the case of Gal(f)=\U₃(2^m) and n=2^3m+1.

Related

• Hyperelliptic jacobians without complex multiplication and Steinberg representations in positive characteristic
• Hyperelliptic jacobians and modular representations
• Hyperelliptic jacobians without complex multiplication, doubly transitive permutation groups and projective representations
• Hyperelliptic jacobians and projective linear Galois groups
• Non-supersingular Hyperelliptic jacobians