Frobenius Distribution for Quotients of Fermat Curves of Prime Exponent

Fité, Francesc ; González, Josep ; Lario, Joan-Carles

Canadian journal of mathematics, Tome 68 (2016) no. 2, pp. 361-394

Résumé

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.

Détail

DOI : 10.4153/CJM-2015-028-x

Mots-clés : 11D41, 11M50, 11G10, 14G10, Sato–Tate group, Fermat curve, Frobenius distribution