The research in this grant concerns the theory of p-adic Hecke algebras. This theory attempts to give a systematic way to construct p-adic L-functions associated with p-adic or complex cusp forms and their Galois representations over power series rings with coefficients in the p-adic integers. Also the principal investigator hopes to find arithmetic relations between these Galois representations and the constructed p-adic L- functions. This theory has been developed for the rational numbers but much remains to be done over general number fields. There will be applications to the Iwasawa theory of imaginary quadratic fields. The p-adic theory of modular forms is the subject of this research. Modular forms are functions that transform in certain ways and are at the center of much research in number theory today. The p-adic theory replaces the usual real or complex numbers with the number theoretic p-adic numbers. Obtaining analogs with the usual theory is a major focus of this research.
