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Arithmetic of L functions and Galois structures

Total activity: 10
Type of activity
Competitive project
Funding entity
Funding entity code
111.150,00 €
Start date
End date
formas modulares, funciones L, módulos de Galois, variedades abelianas
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
Adm. Estat
Plan Nacional de Investigación Científica, Desarrollo e Innovación Tecnológica 2008-2011
Call year
Funcding program
Funding subprogram
Funding call
Proyectos de investigación y acciones complementarias-MTM
Grant institution
Gobierno De España. Ministerio De Economía Y Competitividad, Mineco


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