The late twentieth century saw important advances in Number Theory and Arithmetic Geometry, such as the classification of the torsion of elliptic curves over the field of rational numbers (Mazur), the modularity of elliptic curves and abelian varieties of GL_2 type (Wiles and others) and the proof of Serre's conjecture on 2-dimensional Galois representations (Khare, Wintenberge and others). Along with these significant achievements there were no less important challenges, among which the conjecture of Birch and Swinnerton-Dyer and the conjecture of Sato-Tate with their generalizations due to Bloch-Kato and Serre. A very important part of the research currently being undertaken both in the more abstract field of Number Theory or in Cryptography with direct applications lies in the study and understanding of the number of points (and their algebraic structure) on algebraic varieties defined over numbers fields and finite fields. Under the assumption of the above mentioned conjectures, these properties are encoded in the arithmetic of their associated L-functions. Also, it is noteworthy that the increase of computational aspects in this area is a very important part of the work of a large number of groups and research centers worldwide. Our research group has spent nearly two decades interested in the arithmetic of modular and Shimura curves, abelian varieties and their associated L-functions and the Galois problems that appear naturally when considering these issues. More recently, we have approached towards partial results in the direction of the Birch and Swinnerton-Dyer conjecture and the Sato-Tate conjecture with excellent outcome. The objective of this project is to continue with this research: 1. First, to improve and deepen some aspects already addressed. One can highlight the study of: the quotients of modular abelian varieties, fields of definition and fields of moduli of hyperelliptic curves and curves of small genus, endomorphism algebras, Hilbert modular forms, Galois immersion problems, design and implementation of algorithms in arithmetic, etc. 2. We will also address new problems, some of which mean a significant step in both the objects of study and the objectives: p-adic variants of the conjecture of Birch and Swinnerton-Dyer, study of the Sato-Tate distributions under the influence of twist on curves of genus greater than two and certain hypersurfaces, obtention of Jacobians with Mordel-Weil group of high rank, among others.
Plan Nacional de Investigación Científica, Desarrollo e Innovación Tecnológica 2008-2011
Proyectos de investigación y acciones complementarias-MTM
Gobierno De España. Ministerio De Economía Y Competitividad, Mineco