Single-valued periods of meromorphic modular forms and a motivic interpretation of the Gross-Zagier conjecture

TL;DR

This paper provides a geometric and motivic interpretation of the Gross-Zagier conjecture, introduces new matrix-valued Green's functions as periods of motives, and offers a geometric proof for specific cases, advancing the understanding of special values of L-functions.

Contribution

It introduces a motivic framework for the Gross-Zagier conjecture, defines new matrix-valued Green's functions, and proves a geometric case for level 1 and weight 4.

Findings

Motivic interpretation of the Gross-Zagier conjecture.

Definition of matrix-valued higher Green's functions as periods.

Geometric proof for level 1, weight 4 case.

Abstract

A well-known conjecture of Gross and Zagier states that the values of the higher automorphic Green's function at pairs of points with complex multiplication in the upper half-plane are proportional to the logarithm of an algebraic number. It was recently settled in the case of congruence subgroups of the form $\Gamma_0(N)$ by analytic methods. In this paper we provide a geometric and motivic interpretation of the general conjecture, and show that it is a consequence of a standard conjecture in the theory of motives. In addition, we define a new class of matrix-valued higher Green's functions for both odd and even weight modular forms, and show that they are single-valued periods of a motive constructed from a suitable moduli stack of elliptic curves with marked points. The motive has the structure of a biextension involving symmetric powers of the motives of elliptic curves. This suggests a very general extension of the Gross-Zagier conjecture relating values of matrix-valued higher Green's functions at points which do not necessarily have complex multiplication to special values of $L$-functions. In particular, our motivic interpretation of the Gross-Zagier log-algebraicity conjecture enables us to give a completely geometric proof in level 1 and weight 4 by showing that the motive of the moduli stack $\mathcal{M}_{1,3}$ of elliptic curves with 3 marked points is mixed Tate. In the course of this paper we develop many new foundational results on: the theory of weak harmonic lifts, meromorphic modular forms, biextensions of modular motives, and their corresponding algebraic de Rham cohomology and single-valued periods, which may all be of independent interest.