The purpose of this project is develop further the theory of p-adic automorphic representations and to give applications of this theory to arithmetic questions like the Bloch-Kato conjectures on Selmer groups. Urban proposes to study the p-adic automorphic spectrum for any reductive group in a systematic way. In particular, he will make the p-adic Eisenstein series play a central role in this theory similar to the one played by the classical theory of Eisenstein series of Langlands, Shahidi and others. He will investigate its consequences for the study of p-adic L-functions as well as the consequences of functional equations for p-adic Eisenstein series. In this aspect, the theory of companion forms for general groups will be considered with potential applications to certain reciprocity laws and to the study of high order of vanishing of p-adic L-function at central critical values.
The domain of research of this project is the arithmetic of automorphic forms. Automorphic forms are holomorphic functions that satisfy nice transformation properties. The arithmetic theory of automorphic forms is the study of congruences between the Fourier coefficients of such forms. It has yielded significant advances in the past few years: a proof Fermat's Last Theorem and a proof the Sato-Tate conjecture, to name two. Urban`s proposal is a continuation on his work related to the construction and the study of congruences between specific automorphic forms, the Eisenstein series, and cuspidal automorphic forms of various weights and levels and their links with p-adic L-functions and certain arithmetically defined groups called Selmer groups. These objects play a fundamental role in the theory of (motivic) Galois representations and in number theory. Urban proposes to build some of the foundations of the theory of the p-adic automorphic forms and p-adic Eisenstein series for the unitary and symplectic groups with the goal of applying them to attack the p-adic Bloch-Kato conjecture. In particular, it should shed some new light on the Birch and Swinnerton-Dyer conjecture which relates the L-function of an elliptic curve to its set of rational solutions which is one of most famous problems in number theory.
