On standardp-adic L-functions of families of cusp forms

Abstract

Let$$F(X) = {\sum\limits\sbsp{n}{\infty}} A\sb n (X)q\sp n\in {\bf Z}\sb p\lbrack\lbrack X\rbrack\rbrack \lbrack\lbrack q\rbrack\rbrack$$be a $p$-adic analytic family of cusp forms. We construct a $p$-adic analytic function of two variables which interpolates the special values of standard $L$-functions of the members of the family, generalizing the work of Mazur. We investigate the $p$-adic interpolation property of the $L$-function in connection with the Hecke module structure of the cohomology groups of modular curves.

We also prove the functional equation of the two variable $p$-adic $L$-function.