Cyclic covers of prime power degree, jacobians and endomorphisms

2003/12/27 by Yuri G. Zarhin, Zarhin, Yuri G.
Mathematics · #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:11G30 #msc:14H40 #msc:14K05

paper · pdf · doi:10.48550/arxiv.math/0312471

33 pages, LaTeX2e

arxiv created 2003/12/27 · openalex publication_date 2003/12/27 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Suppose that K is a field of characteristic 0, K_a is its algebraic closure, p is a prime, q=p^r is a power prime. Suppose that f(x) ∈ K[x] is a polynomial of degree n > 4 without multiple roots. Let us consider the superelliptic curve C: y^q=f(x) and its jacobian J(C). We study the endomorphism algebra End⁰(J(C)) of all K_a-endomorphisms of J(C). We prove that End⁰(J(C)) is "as small as possible" if the Galois group of f over K is either the full symmetric group S_n or the alternating group A_n.

Citations

• The endomorphism rings of jacobians of cyclic covers of the projective line

Related

• The endomorphism rings of jacobians of cyclic covers of the projective line
• Endomorphisms of ordinary superelliptic jacobians
• Cyclic covers of the projective line, their jacobians and endomorphisms
• Endomorphism algebras of Jacobians of certain superelliptic curves
• The endomorphism rings of jacobians of cyclic covers of the projective line