2024/12/01 by Tuoping Du, Du, Tuoping
Mathematics · #Advanced Algebra and Geometry #Analytic Number Theory Research
paper · pdf · doi:10.48550/arxiv.2412.00688
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.