Analytic p-adic L-functions for GL2: a summary

One of the most subtle and important invariants of a modular form f is its L-function L(s, f). For holomorphic modular forms, the basic analytic properties of their L-functions were established by Hecke, with Maass treating the real-analytic forms; their construction gave L(s, f) as a Mellin transform of f . In the early 70s, the existence of p-adic L-functions associated with classical modular forms emerged. These functions were constructed and characterized in varying degrees of generality by Mazur-Swinnerton-Dyer, Manin, Višik, Amice-Vélu, Mazur-Tate-Teiltelbaum, Stevens, Stevens-Pollack and Bellaïche (MSD74, Man73, AV75, Viš76, MTT86, Ste94, PS11, Bel12). In the last twenty years, these objects have come to play an increasingly important role in modern number theory, especially as their role in relation to Iwasawa theory and eigenvarieties has become clear. For GL2 over an arbitrary number field, where the construction of classical L-functions has been understood since the appearance of Jacquet and Langlands’s seminal book, this situation is far more fragmentary. Manin, Panchishkin, Kurcanov, Haran, Ash-Ginzburg, Januszewski, and Dimitrov, among others, have given constructions of p-adic L-functions associated with cohomological cuspidal automorphic representations of GL2/F under various more or less restrictive hypothesis. In the article (Han13) we give a canonical construction of p-adic L-functions associated with cohomological cuspidal automorphic representations of GL2(AF ). The L-functions we construct enjoy good interpolation and growth properties, and they deform naturally into many-variable functions over eigenvarieties. Our construction is very much a p-adic analogue of the Hecke-Jacquet-Langlands integral. Moreover, we make almost no explicit reference to the fine geometry of locally symmetric spaces or the theory of modular symbols; every map we use admits a succinct adelic description. In this note we explain the main results of (Han13). Let F/Q be a number field of degree d = r1 + 2r2 with ring of integers OF and discriminant ∆F . Fix a rational prime p, algebraic closures Qp and Q ⊂ C, and an isomorphism ι : C ∼ → Qp. Write Σ for the set of embeddings σ : F ↪→ C. The map