CM cycles on Shimura curves, and p-adic L-functions

TL;DR

This paper constructs algebraic cycles on Chow motives related to p-adic L-functions of modular forms over quadratic imaginary fields, extending previous results to non-central critical values for weights k>=4.

Contribution

It introduces a new construction of algebraic cycles encoding derivatives of p-adic L-functions at critical points, generalizing prior work for central values.

Findings

Constructed Chow motives with algebraic cycles linked to p-adic L-function derivatives.

Extended the formula for p-adic L-functions to non-central critical values.

Provided a different approach from Iovita-Spiess for the case s=k/2.

Abstract

Let f be a modular form of weight k>=2 and level N, let K be a quadratic imaginary field, and assume that there is a prime p exactly dividing N. Under certain arithmetic conditions on the level and the field K, one can attach to this data a p-adic L-function L_p(f,K,s), as done by Bertolini-Darmon-Iovita-Spiess. In the case of p being inert in K, this analytic function of a p-adic variable s vanishes in the critical range s=1,...,k-1, and therefore one is interested in the values of its derivative in this range. We construct, for k>=4, a Chow motive endowed with a distinguished collection of algebraic cycles which encode these values, via the p-adic Abel-Jacobi map. Our main result generalizes the result obtained by Iovita-Spiess, which gives a similar formula for the central value s=k/2. Even in this case our construction is different from the one found by Iovita-Spiess.