On the quaternionic p-adic L-functions associated to Hilbert modular eigenforms

Abstract

We construct p-adic L-functions associated to cuspidal Hilbert modular eigenforms of parallel weight two in certain dihedral or anticyclotomic extensions via the Jacquet-Langlands correspondence, generalizing works of Bertolini-Darmon, Vatsal and others. The construction given here is adelic, which allows us to deduce a precise interpolation formula from a Waldspurger type theorem, as well as a formula for the dihedral µ-invariant. We also make a note of Howard's nonvanishing criterion for these p-adic L-functions, which can be used to reduce the associated Iwasawa main conjecture to a certain nontriviality criterion for families of p-adic L-functions. Contents 1. Introduction 1 2. Some preliminaries 5 3. Modular forms on totally definite quaternion algebras 8 4. Construction of measures 17 5. Howard's criterion 28 References 30denote the specialization of λ to ρ. Let P denote the maximal ideal of O. We define the µ-invariant µ = µ(λ) associated to an element λ ∈ Λ to be the largest exponent c such that λ ∈ P c Λ. Let π = π f denote the cuspidal automorphic representation of GL 2 over F associated to f , withthe L-series of the adjoint representation of π. LetWe can state the criterion of Conjecture 5.1 in the following more explicit way. Let us now assume for simplicity that N is prime to the relative discriminant of K over F . Fix a positive integer k. Let us define a set of admissible primes L k of F , all of which are inert in K, with the condition that for any ideal n in the set S k of squarefree products of primes in L k , there exists a nontrivial eigenform f (n) of level nN such that we have the following congruence on Hecke eigenvalues: