Main Conjectures for Families of Automorphic Forms

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My research project aims at solving problems in two areas of number theory which are currently experiencing an impressive development and are closely related: Iwasawa theory and p-adic families of automorphic forms. My project has possibilities for many ramifications which would provide interesting problems for Masters' and PhD students.One of the most important goals of number theory is the study of integral solutions of certain sets of polynomial equations; this is done using methods from algebraic geometry and analysis. It has been known that to most of "motives" one can associate two objects: the L-function and the Selmer group (which generalizes the concept of rational points of a variety). A series of conjectures state that information on the vanishing order and the leading coefficient of the L-function can be translated to information about the Selmer group. A very fruitful approach to these conjectures is via p-adic deformations. Given a motive, one can study how it varies over certain towers of field extensions, such as the tower generated by the p^n-th roots of unity. In many cases, one can define a p-adic L-function and a "big" Selmer group over the tower of extensions. Conjectures by Iwasawa, Greenberg, and Benois state that the "big" Selmer group is (co)-torsion over the ring of formal series with Z_p coefficients, and that its characteristic ideal is generated by the p-adic L-function. This often implies the Bloch--Kato conjecture. The first objective of my research program, in a joint project with Z. Liu (McGill whenever the constant term (i.e. the p-adic L-function) vanishes, one can use ideas of Ribet to construct co-cycles in the Selmer group which can be used to bound its size, and hence proving an inclusion between the two ideals. This shows that p-adic families are necessary for Main Conjectures but so far families have been constructed only when the p-ordinary locus of the Shimura variety is not empty, which excludes many cases. This leads us to the second objective of my project. With R. Brasca (Universite Paris 7) we are developing Hida theory for PEL Shimura varieties without ordinary locus. The idea is to substitute the multiplicative part of the universal p-divisible group with its whole filtered connected part. This approach uses mainly properties of the Dieudonne modules with its filtration. Hence, our subsequent project is to generalize this to Shimura varieties which are not of PEL type but with a map to the stack of G-zips, "Dieudonne modules with extra structure", such as the orthogonal Shimura varieties. Application to the construction of p-adic L-functions are expected.