-adic -functions for

Abstract

Abstract. Since Rob Pollack and Glenn Stevens used overconvergent modular symbols to construct p-adic L-functions for non-critical slope rational modular forms, the theory has been extended to construct p-adic L-functions for non-critical slope automorphic forms over totally real and imaginary quadratic elds by the rst and second authors respectively. In this paper, we give an analogous construction over a general number eld. In particular, we start by proving a control theorem stating that the specialisation map from overconvergent to classical modular symbols is an isomorphism on the small slope subspace. We then show that if one takes the modular symbol attached to a small slope cuspidal eigenform, then one can construct a ray class distribution from the corresponding overconvergent symbol, that moreover interpolates critical values of the L-function of the eigenform. We prove that this distribution is independent of the choices made in its construction. We de ne the p-adic L-function of the eigenform to be this distribution.

Parabolic eigenvarieties via overconvergent cohomology

“…It was also used in [7] to construct p-adic L-invariants and prove an exceptional zero conjecture for Bianchi modular forms. -Moreover, versions of the refined slope conditions given by Theorem A were used in [8,44] to construct p-adic L-functions attached to automorphic forms for GL 2 .…”

“…From the definition, we find that for f ∈ V G λ and t ∈ T + , we have ), and the * -action can be viewed as an 'optimal' integral normalisation of it. (iii) For GL 2 , it is easy to write down the •-action on V G,∨ λ explicitly, and one easily sees that this explicit action extends to distributions; this is done, for example, in [2,8,35]. We warn the reader, however, that this does not give the * -action of T + on distributions defined here: in particular, it does not preserve integrality (see [8, Sect.…”

“…In the remainder of the paper, we show how to carry out the last part of the strategy described above in a new situation: producing families of p-adic L-functions for GL 2 over CM fields. In this case, the construction for a single form (valid over arbitrary number fields) was given in our earlier paper [BSW19], to which we refer the reader for further details on our conventions.…”

“…Let π + be an automorphic representation for G + (A), which we will always assume does not have CM by E, and let π be its base-change to G(A). To maintain consistency with [BSW19], rather than using the more conceptual language of automorphic representations, we instead switch to the language of automorphic forms. In particular, throughout f + will be a cuspidal automorphic eigenform in π + -a Hilbert modular form -and f its base-change, that is, a choice of automorphic eigenform in π with the correct Hecke eigenvalues.…”

Stark–Heegner cycles attached to Bianchi modular forms

“…Proof In the inert case, this is [48, Corollary 5.9], noting that as is new at , we have (see, for example, [39, Corollary 4.8]). In the split case, this follows more or less immediately from the results of [48, § 6] noting that in any case, this locally analytic, partially overconvergent formulation is [38, Lemma 9.8].…”