On the arithmetic inner product formula and central derivatives of L-functions

TL;DR

This paper advances the Kudla program by defining the arithmetic theta lift, establishing an inner product formula, and exploring new representations and conjectures related to derivatives of L-functions and harmonic weak Maass forms.

Contribution

It provides a formal definition of the arithmetic theta lift, proves the arithmetic inner product formula, and introduces a new arithmetic representation for central derivatives of L-functions.

Findings

Arithmetic inner product formula equivalent to Gross-Zagier formula

New arithmetic representation for derivatives of L-functions

Conjecture linking L-function derivatives to harmonic weak Maass forms

Abstract

This research provides a formal definition of the arithmetic theta lift for cusp forms of weight $3/2$ and establishes the arithmetic inner product formula, thereby completing the Kudla program on modular curves. This formula is demonstrated to be equivalent to the Gross-Zagier formula, for which we provide a new proof. Additionally, the authors introduce a new arithmetic representation for the central derivatives of L-functions associated with cusp forms of higher weight. Although this representation differs from Zhang's higher weight Gross-Zagier formula, it maintains a significant connection to it. This study also proposes a conjecture indicating that the vanishing of derivatives of L-functions is determined by the algebraicity of the coefficients of harmonic weak Maass forms. A consistent approach is employed to study both parts of this work.