Fixed points and homology of superelliptic Jacobians

2013/09/02 by Haining Wang, Jiangwei Xue, Wang, Haining +4
Mathematics · #14F25 #14H40 #Advanced Algebra and Geometry #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #math.AG #msc:14F25 #msc:14H40

paper · pdf · doi:10.48550/arxiv.1309.0295

21 pages. Fixed some typos. Statement of Theorem 1.3 improved. Results unchanged

openalex publication_date 2013/09/02 · arxiv created 2013/09/11 · arxiv updated 2013/09/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let η: C_f,N→ ℙ¹ be a cyclic cover of ℙ¹ of degree N which is totally and tamely ramified for all the ramification points. We determine the group of fixed points of the cyclic group mu_N≅ ℤ/Nℤ acting on the Jacobian J_N:=\Jac(C_f,N). For each ℓ distinct from the characteristic of the base field, the Tate module T_ℓ J_N is shown to be a free module over the ring ℤ_ℓ[T]/(∑_i=0^N-1Tⁱ). We also calculate the degree of the induced polarization on the new part J_N^new of the Jacobian.

Citations

• The endomorphism rings of jacobians of cyclic covers of the projective line
• Endomorphism algebras of factors of certain hypergeometric Jacobians

Related

• Cyclic covers and Ihara's Question
• Cyclic covers of prime power degree, jacobians and endomorphisms
• Endomorphisms of ordinary superelliptic jacobians
• Superelliptic jacobians and central simple representations
• Simplicity of some Jacobians with many automorphisms