P-adic Galois Representations Attached to Modular Forms

University Of Washington, Seattle WA

Investigators

Abstract

Abstract, The proposer intends to study the local structure of p-adic Galois representations attached to modular forms. This study will be done via the theory of Fontaine. The results of this study are needed in order to prove formulas relating the central values of the derivatives of the anticyclotomic p-adic L-functions attached to modular forms, defined previously by the proposer and his collaborators, to global objects namely the image under the p-adic Abel-Jacobi map of the suitable Heegner cycle. Galois representations have proven to be extraordinarily powerful tools for the study of Diophantine equations and their connections with geometry. The most striking example of this power is to be seen in the recent work of Andrew Wiles, namely his proof of ``Fermat's Last Theorem". The study and understanding of Diophantine equations is essential for applications in cryptography, computer science and other areas of general interest.