$p$-adic analogue of modular forms, upper half-plane, and $L$-functions

Question

In the classical picture, there is the (complex) modular form, defined on the (complex) upper half plane, which is related to the (complex) $L$-function via the Mellin transform. As I have recently been aware of the notions of $p$-adic modular forms, $p$-adic upper half plane, and $p$-adic $L$-functions, I was thus wondering if there is a $p$-adic analogue of this classical picture which relates the three objects. If not, what would be a 'good/correct/conjectural/etc.' analogue in the $p$-adic setting? Any comments would be appreciated.

Answer 1

The subjects of "p-adic L-functions" and "p-adic modular forms" are so closely intertwined that it's virtually impossible to talk about either one without immediately running into the other. Applications to p-adic L-functions have been baked into the theory of p-adic modular forms from the start (cf. Serre's paper Formes modulaires et fonctions zêta p-adiques in the 1972 Antwerp proceedings, which introduces the definition of a p-adic modular form, and then immediately uses it to give a new construction of the Kubota–Leopoldt p-adic zeta function). There are numerous works (notably Mazur–Kitagawa, Greenberg–Stevens, multiple papers of Hida, etc.) devoted to this interplay between p-adic modular forms and p-adic L-functions.

The p-adic upper half-plane (in Drinfeld's sense) is a little less immediately related — p-adic modular forms are not defined as functions on it — but it does come up in the theory from time to time; Darmon's paper "Integration on $\mathcal{H}_p \times \mathcal{H}$ and arithmetic applications" is perhaps the first paper that comes to mind in this direction.