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Frobenius Distribution for Quotients of Fermat Curves of Prime Exponent

Published online by Cambridge University Press:  20 November 2018

Francesc Fité ,
Josep González  and
Joan-Carles Lario
Francesc Fité
Affiliation:
Institut für Experimentelle Mathematik/Fakultät für Mathematik, Universität Duisburg-Essen, D-45127 Essen, Germany e-mail: francesc.fite@gmail.com
Josep González
Affiliation:
Departament de Matemàtica Aplicada IV, Universitat Politècnica de Catalunya, Av. Víctor Balaguer s/n., E-08800 Vilanova i la Geltrύ, Catalonia e-mail: josepg@ma4.upc.edu
Joan-Carles Lario
Affiliation:
Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Edifici Omega-Campus Nord, Jordi Girona 1-3, E-08034 Barcelona, Catalonia e-mail: joan.carles.lario@upc.edu
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Abstract

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Let $C$ denote the Fermat curve over $\mathbb{Q}$ of prime exponent $l$ . The Jacobian $\text{Jac(}C\text{)}$ of $C$ splits over $\mathbb{Q}$ as the product of Jacobians $\text{Jac(}{{C}_{k}})$ , $1\,\le \,k\,\le \,\ell \,-\text{2}$ , where ${{C}_{k}}$ are curves obtained as quotients of $C$ by certain subgroups of automorphisms of $C$ . It is well known that $\text{Jac(}{{C}_{k}}\text{)}$ is the power of an absolutely simple abelian variety ${{B}_{k}}$ with complex multiplication. We call degenerate those pairs $(l,\,k)$ for which ${{B}_{k}}$ has degenerate $\text{CM}$ type. For a non-degenerate pair $(l,\,k)$ , we compute the Sato–Tate group of $\text{Jac(}{{C}_{k}}\text{)}$ , prove the generalized Sato–Tate Conjecture for it, and give an explicit method to compute the moments and measures of the involved distributions. Regardless of whether $(l,\,k)$ is degenerate, we also obtain Frobenius equidistribution results for primes of certain residue degrees in the $l$ -th cyclotomic field. Key to our results is a detailed study of the rank of certain generalized Demjanenko matrices.

Keywords

11D4111M5011G1014G10Sato–Tate groupFermat curveFrobenius distribution
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 68 , Issue 2 , 01 April 2016 , pp. 361 - 394
Copyright
Copyright © Canadian Mathematical Society 2016

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