Abstract view
Frobenius Distribution for Quotients of Fermat Curves of Prime Exponent


Published:20160205
Printed: Apr 2016
Francesc Fité,
Institut für Experimentelle Mathematik/Fakultät für Mathematik, Universität DuisburgEssen, D45127 Essen, Germany
Josep González,
Departament de Matemàtica Aplicada IV, Universitat Politècnica de Catalunya, Av. Víctor Balaguer s/n., E08800 Vilanova i la Geltrú, Spain
Joan Carles Lario,
Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Edifici OmegaCampus Nord, Jordi Girona 13, E08034 Barcelona, Spain
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
$\operatorname{Jac}(\mathcal{C}_k)$, $1\leq k\leq \ell2$, 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 nondegenerate pair $(\ell,k)$, we compute the SatoTate
group of $\operatorname{Jac}(\mathcal{C}_k)$, prove the generalized
SatoTate 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.