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

TL;DR

This paper constructs p-adic L-functions for Hilbert modular eigenforms in specific extensions, providing explicit formulas and linking to Iwasawa theory, thus advancing understanding of their arithmetic properties.

Contribution

It introduces a new adelic construction of p-adic L-functions for Hilbert modular forms in dihedral and anticyclotomic extensions, extending previous work and deriving explicit interpolation formulas.

Findings

Derived a Waldspurger type interpolation formula.

Established a formula for the dihedral mu-invariant.

Connected nonvanishing criteria to Iwasawa main conjecture.

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 formula, as well as a formula for the dihedral mu-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.