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Frobenius Distribution for Quotients of Fermat Curves of Prime Exponent

  Published:2016-02-05
 Printed: Apr 2016
  • Francesc Fité,
    Institut für Experimentelle Mathematik/Fakultät für Mathematik, Universität Duisburg-Essen, D-45127 Essen, Germany
  • Josep González,
    Departament de Matemàtica Aplicada IV, Universitat Politècnica de Catalunya, Av. Víctor Balaguer s/n., E-08800 Vilanova i la Geltrú, Spain
  • Joan Carles Lario,
    Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Edifici Omega-Campus Nord, Jordi Girona 1-3, E-08034 Barcelona, Spain
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Abstract

Let $\mathcal{C}$ denote the Fermat curve over $\mathbb{Q}$ of prime exponent $\ell$. The Jacobian $\operatorname{Jac}(\mathcal{C})$ of~$\mathcal{C}$ splits over $\mathbb{Q}$ as the product of Jacobians $\operatorname{Jac}(\mathcal{C}_k)$, $1\leq k\leq \ell-2$, where $\mathcal{C}_k$ are curves obtained as quotients of $\mathcal{C}$ by certain subgroups of automorphisms of $\mathcal{C}$. It is well known that $\operatorname{Jac}(\mathcal{C}_k)$ is the power of an absolutely simple abelian variety $B_k$ with complex multiplication. We call degenerate those pairs $(\ell,k)$ for which $B_k$ has degenerate CM type. For a non-degenerate pair $(\ell,k)$, we compute the Sato-Tate group of $\operatorname{Jac}(\mathcal{C}_k)$, 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 $(\ell,k)$ being degenerate or not, we also obtain Frobenius equidistribution results for primes of certain residue degrees in the $\ell$-th cyclotomic field. Key to our results is a detailed study of the rank of certain generalized Demjanenko matrices.
Keywords: Sato-Tate group, Fermat curve, Frobenius distribution Sato-Tate group, Fermat curve, Frobenius distribution
MSC Classifications: 11D41, 11M50, 11G10, 14G10 show english descriptions Higher degree equations; Fermat's equation
Relations with random matrices
Abelian varieties of dimension $> 1$ [See also 14Kxx]
Zeta-functions and related questions [See also 11G40] (Birch-Swinnerton-Dyer conjecture)
11D41 - Higher degree equations; Fermat's equation
11M50 - Relations with random matrices
11G10 - Abelian varieties of dimension $> 1$ [See also 14Kxx]
14G10 - Zeta-functions and related questions [See also 11G40] (Birch-Swinnerton-Dyer conjecture)
 

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