p-adic Modular Forms

Given a totally real number field, we can define its Dedekind zeta function, which is a meromorphic function over C. We know that these functions, evaluated at any odd negative integer, naturally appear as constant terms of classical modular forms. By taking suitable p-adic limits of sequences of modular forms, one sees that the constant terms of these functions recover Kubota-Leopoldt L-functions. Various results on Serre's p-adic modular forms will be showed. Finally, we will consider the case of families of modular forms depending, in some sense, p-adic analytically on a parameter. In particular, one is brought to the definition of $\Lambda$-adic families of Eisenstein series, and later on, with the work of Hida, of cusp forms.