A p-adic integral for the reciprocal of L-functions

摘要

We introduce an analog of part of the Langlands-Shahidi method to the p-adic setting, constructing reciprocals of certain p-adic L-functions us- ing the nonconstant terms of the Fourier expansions of Eisenstein series. We carry out the method for the group SL_2, and give explicit p-adic measures whose Mellin transforms are reciprocals of Dirichlet L-functions. The formu- las for these measures involve Fourier coefficients of Eisenstein series, plus a delicately chosen multiple of Haar measure necessary for boundedness. Key words and phrases. p-adic L-functions, Mazur measure, Iwasawa algebra, Riemann zeta- function, Eisenstein series, Langlands-Shahidi method.