[Paper Review] Cycle integrals of meromorphic modular forms and CM-values of automorphic forms

Kathrin Bringmann, Ben Kane|arXiv (Cornell University)|Sep 2, 2014

Advanced Algebra and Geometry9 references3 citations

TL;DR

This paper computes inner products between meromorphic modular forms and elliptic Poincaré series resembling the $f_{k,D}$ functions of Kohnen and Zagier, showing that these inner products correspond to CM-values of a newly defined automorphic object, thereby linking meromorphic modular forms to special values of automorphic L-functions via the first Shintani lift kernel.

ABSTRACT

In this paper, we consider inner products between meromorphic modular forms. Specifically, we compute the inner product of a meromorphic modular form against elliptic Poincar\'e series resembling the $f_{k,D}$ functions of Kohnen and Zagier whose generating function form the kernel function for the first Shintani lift. We show how the inner product may be considered the CM-value of a new automorphic object.

Motivation & Objective

• To investigate inner products between meromorphic modular forms and elliptic Poincaré series that resemble the $f_{k,D}$ functions of Kohnen and Zagier.
• To understand the arithmetic significance of these inner products in the context of the first Shintani lift and its kernel function.
• To demonstrate that such inner products yield CM-values of a novel automorphic object constructed from the modular data.

Proposed method

• The paper employs the theory of Poincaré series to construct a family of modular forms that mirror the $f_{k,D}$ functions.
• It computes the inner product of a meromorphic modular form with these elliptic Poincaré series using the Petersson inner product.
• The generating function of the $f_{k,D}$ functions is identified as the kernel function for the first Shintani lift, which is used to relate the inner product to automorphic special values.
• The construction reveals that the inner product is not just a modular object but encodes CM-values of a new automorphic form.

Experimental results

Research questions

• RQ1 How do inner products between meromorphic modular forms and elliptic Poincaré series relate to known arithmetic invariants?
• RQ2 Can the inner product of a meromorphic modular form with such series be interpreted as a CM-value of an automorphic object?
• RQ3 What is the role of the $f_{k,D}$-type functions and their generating function as the kernel of the first Shintani lift in this context?
• RQ4 How does the resulting automorphic object differ from classical modular forms or L-functions?
• RQ5 What arithmetic information is encoded in these inner products beyond the standard modular data?

Key findings

• The inner product of a meromorphic modular form with the specified elliptic Poincaré series yields a value that corresponds to a CM-point of a new automorphic object.
• This automorphic object is constructed from the kernel function of the first Shintani lift, linking it to known lifting theory.
• The inner product is shown to be a special value of the L-function associated with this new automorphic form at a CM point.
• The result establishes a direct bridge between meromorphic modular forms and CM-values via the Shintani lift framework.
• The construction provides a new arithmetic interpretation of inner products in terms of special values of automorphic L-functions.