Abstract of Eric Urban's award DMS-0401131. "p-adic deformation of Eisenstein series"
Urban will pursue his research on the construction and the study of congruences between automorphic forms in order to attack some of the Bloch-Kato conjectures and the Iwasawa main conjectures relating critical L-values and the size of Selmer groups. He will especially work on such congruences between Eisenstein series and cusp forms as in the works by Mazur-Wiles or in his work on the main conjecture for the symmetric square of an ordinary elliptic curve. In that direction, the PI will continue his joint work in progress with C. Skinner proving the Iwasawa main conjecture for elliptic curves via the study of the Eisenstein ideal for U(2,2). He also plans to guide some Ph-D students on a similar topic for some other unitary groups. The PI will also work on the theory of p-adic families of automorphic forms for general reductive groups by both geometric and topological approaches. This work will be undertaken for its own interest but also because the study of such congruences for Eisenstein series has numerous powerful applications. By these means, Urban plans to obtain some new and general constructions of certain arithmetical fundamental objects as for instance $p$-adic Euler systems that will be an essential tool to bound above the size of some Selmer groups.
Urban's research field is the arithmetic theory of automorphic forms. Those are certain holomorphic functions having many symmetries and whose deep arithmetical properties are read from their Fourier coefficients. These objects play a fundamental role in number theory as their study has been very fruitful in the last two decades, leading A. Wiles to the proof of the Fermat's last theorem. The ultimate goal of the inverstigator's research is to relate two apparently unrelated objects which are the special values of L-functions (a meromorphic function on the complex planes associated to an automorphic form) and the order of certain Selmer groups (a generalization of the class group of a number field) via the study of congruences between automorphic forms. One of the main tools that will be used in Urban's investigation is the theory of p-adic modular forms, a theory that encodes all the congruences between modular forms modulo arbitrary high powers of a given prime number p. He will develop new aspects of this theory for general reductive groups and apply it to relevant modular forms called Eisenstein series whose constant terms carry information on critical L-values. Among other things, the results of this research will also have some important implications to the p-adic and classical conjecture of Birch and Swinnerton-Dyer that are major conjectures on the set of solutions of cubic equations in two variables
