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arXiv · 2412.00688

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

Tuoping Du
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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.

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Tuoping Du. 2024-12-01. On the arithmetic inner product formula and central derivatives of L-functions. https://arxiv.org/abs/2412.00688

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