傅里叶谱互反性与典范赫克L-函数
Fourier Spectral Reciprocity and Canonical Hecke $L$-Functions
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中文总结 AI 辅助
本文证明数域上的傅里叶型环面谱互反公式，将其应用于CM域的典范赫克L-函数，得到含中心值与导数的一致一阶矩公式，结合次凸性界给出相关定量非零及Mordell-Weil秩1结果。
中文摘要 AI 辅助
我们证明了数域上的傅里叶型环面谱互反公式，将二次扩域上的Tate积分与基域上的对偶赫克周期族关联起来。该公式保留谱参数自由，并允许适配算术应用的测试函数。我们将此互反公式应用于CM域上的典范赫克L-函数，得到显式的一阶矩公式，包含中心值与中心导数。这些公式对权和扭导体均一致，且无需Heegner型分裂假设。结合次凸性界，它们给出典范赫克L-函数及其导数的定量非零结果。在权1情形，这给出Masri-Yang算术应用的秩1类比，得到相关CM阿贝尔簇二次扭的定量Mordell-Weil秩1结果。
英文摘要
We prove a Fourier-type toric spectral reciprocity formula over number fields, relating Tate integrals over a quadratic extension to a dual family of Hecke periods over the base field. The formula keeps the spectral parameter free and allows test functions adapted to arithmetic applications. We apply this reciprocity formula to canonical Hecke $L$-functions over CM fields and obtain explicit first moment formulas, including both central values and central derivatives. These formulas are uniform in the weight and the twisting conductor, and require no Heegner-type splitting hypothesis. Together with subconvexity bounds, they yield quantitative nonvanishing results for canonical Hecke $L$-functions and their derivatives. In weight one, this gives a rank-one analogue of the arithmetic applications of Masri--Yang, yielding quantitative Mordell--Weil rank-one results for quadratic twists of the associated CM abelian varieties.
