Frobenius distribution for quotients of Fermat curves of prime exponent
Let C denote the Fermat curve over Q of prime exponent l. The Jacobian Jac(C) of C splits over Q as the product of Jacobians Jac(C_k), 0< k < l-1, where C_k are curves obtained as quotients of C by certain subgroups of automorphisms of C. It is well known that Jac(C_k) is the power of an absol...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2015 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/81570 |
| Acceso en línea: | https://hdl.handle.net/2117/81570 https://dx.doi.org/10.4153/CJM-2015-028-x |
| Access Level: | acceso abierto |
| Palabra clave: | Sato-Tate group Fermat curve Frobenius distribution Classificació AMS::11 Number theory::11D Diophantine equations Àrees temàtiques de la UPC::Matemàtiques i estadística |
| Sumario: | Let C denote the Fermat curve over Q of prime exponent l. The Jacobian Jac(C) of C splits over Q as the product of Jacobians Jac(C_k), 0< k < l-1, where C_k are curves obtained as quotients of C by certain subgroups of automorphisms of C. It is well known that Jac(C_k) 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 CM type. For a non-degenerate pair (l,k), we compute the Sato-Tate group of Jac(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 (l,k) being degenerate or not, 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. |
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