[Paper Review] Special values of anticyclcotomic Rankin-Selberg L-functions

Ming-Lun Hsieh|arXiv (Cornell University)|Dec 7, 2011

Advanced Algebra and Geometry19 references3 citations

TL;DR

This paper establishes an explicit Waldspurger formula for toric Hilbert modular forms and constructs anticyclotomic p-adic Rankin-Selberg L-functions for Hilbert modular forms, generalizing prior work in the elliptic case. It proves a necessary and sufficient condition for the vanishing of the Iwasawa μ-invariant and establishes non-vanishing modulo p of central L-values with anticyclotomic twists under specific hypotheses on CM types and root numbers.

ABSTRACT

In this article, we prove an explicit Waldspurger formula for the toric Hilbert modular forms. As an application, we construct a class of anticyclotomic p-adic Rankin-Selberg L-functions for Hilbert modular forms, generalizing the construction of Bertolini, Darmon and Prasanna in the elliptic case. Moreover, building on works of Hida, we give a necessary and sufficient condition when the Iwasawa mu-invariant of this p-adic L-function vanishes and prove a result on the non-vanishing modulo $p$ of central Rankin-Selberg L-values with anticyclotomic twists.

Motivation & Objective

• To construct anticyclotomic p-adic Rankin-Selberg L-functions for Hilbert modular forms, extending the elliptic case construction by Bertolini, Darmon, and Prasanna.
• To establish a necessary and sufficient condition for the vanishing of the Iwasawa μ-invariant of these p-adic L-functions.
• To prove the non-vanishing modulo p of central Rankin-Selberg L-values under anticyclotomic twists, under specified local root number and p-ordinariness conditions.
• To generalize Waldspurger's formula to the setting of toric Hilbert modular forms.

Proposed method

• Derives an explicit Waldspurger formula for toric periods of Hilbert modular forms using period integrals and p-adic interpolation.
• Constructs p-adic L-functions via p-adic interpolation of central L-values of Rankin-Selberg products with anticyclotomic twists.
• Applies Hida's theory of p-adic families and ordinary automorphic forms to analyze the μ-invariant of the constructed p-adic L-functions.
• Uses the local root number hypothesis and p-ordinariness condition to ensure the p-adic L-function is well-defined and has good control over its μ-invariant.
• Employs Galois-theoretic arguments and Hecke eigenvalue congruences to verify non-vanishing modulo p of toric periods.
• Relies on the identification of CM types and the structure of the anticyclotomic Z_p^{[F:Q]}-extension to define the p-adic L-function and analyze its Iwasawa invariants.

Experimental results

Research questions

• RQ1 Under what conditions does the Iwasawa μ-invariant of the anticyclotomic p-adic Rankin-Selberg L-function vanish for Hilbert modular forms?
• RQ2 How can an explicit Waldspurger formula be established for toric Hilbert modular forms?
• RQ3 What is the precise relationship between the central L-values of twisted Rankin-Selberg products and toric period integrals in the anticyclotomic setting?
• RQ4 When do central L-values with anticyclotomic twists not vanish modulo p, under the given hypotheses?
• RQ5 How does the p-ordinariness condition on the CM type affect the construction and properties of the p-adic L-function?

Key findings

• The paper proves a necessary and sufficient condition for the vanishing of the Iwasawa μ-invariant of the constructed anticyclotomic p-adic L-function, based on the vanishing of certain local p-adic L-invariants.
• An explicit Waldspurger formula is established for toric Hilbert modular forms, relating central L-values to toric period integrals via the p-adic interpolation of automorphic forms.
• The constructed p-adic L-function interpolates central values of Rankin-Selberg L-functions twisted by anticyclotomic Hecke characters, generalizing the elliptic case to Hilbert modular forms.
• The non-vanishing modulo p of central L-values with anticyclotomic twists is proven for almost all Hecke characters under the assumption that the local root number is +1 at all places dividing the conductor of the twist.
• The p-adic L-function is shown to be well-defined and p-adically analytic under the p-ordinariness hypothesis, ensuring the existence of a canonical p-adic measure on the anticyclotomic tower.
• The proof of non-vanishing modulo p relies on verifying a Galois-theoretic condition (H′) on Hecke eigenvalues, using congruences in the Hecke algebra modulo the maximal ideal.