Anticyclotomic p-adic L-functions and Ichino's formula

Abstract

We investigate the anticyclotomic p-adic L-function that interpolates the central value of the Rankin-Selberg L-function L(f✕χ,s), where f is a fixed classical modular form and χ a Hecke character of an imaginary quadratic field varying in an anticyclotomic family. Such p-adic L-functions were first constructed by Bertolini-Darmon-Prasanna using toric integrals. We give an alternative construction using Ichino's triple-product formula, applied to a fixed modular form and a pair of CM forms. Using two CM forms gives us flexibility to obtain precise results in cases that are difficult to analyze from other constructions, such as when the family of Hecke characters leads to residually reducible Galois representations after induction. In particular our results and techniques apply to the family passing through the trivial character, and thus are suited for future arithmetic applications towards elliptic curves.

The first step of the computation is to make Ichino's formula completely explicit in a classical context, resulting in an equation relating a classical Petersson inner product to a product of two Rankin-Selberg L-values. To work out the constant exactly, we need to evaluate certain local integrals at finite places that arise from Ichino's formula. To handle these, we collect various results and techniques from the literature and make some computations ourselves. Once we have an explicit equation, we simultaneously vary the two CM forms in Hida families. By combining some Hida theory with the Main Conjecture of Iwasawa Theory for imaginary quadratic fields (a theorem of Rubin) we show that the Petersson inner products on one side of our equation vary p-adically analytically. From this we conclude that the L-values we're interested in vary p-adically, and moreover obtain a formula relating the p-adic family interpolating Petersson inner products to a product of our p-adic L-function and a fixed L-value.