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摘要翻译：
最近的工作，主要与Ramanujan的模拟θ函数有关，利用了调和弱Maass形式可以是组合母函数的事实。推广Waldspurger、Kohnen和Zagier的工作，我们证明了这类形式也是权重为2的二次扭曲模$L$-函数的中心值和导数的“母函数”。为了得到这些结果，我们通过适当地将Borcherds升程推广到调和弱Maass形式，构造了具有扭曲Heegner因子的第三类微分。利用Scholl、Waldschmidt、Gross和Zagier的工作，分析了这些微分的性质，得到了它们与周期、Fourier系数、$L$-函数的导数以及模曲线雅可比点之间的联系。
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英文标题：
《Heegner divisors, $L$-functions and harmonic weak Maass forms》
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作者：
Jan H. Bruinier and Ken Ono
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最新提交年份：
2008
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分类信息：
一级分类：Mathematics 数学
二级分类：Number Theory 数论
分类描述：Prime numbers, diophantine equations, analytic number theory, algebraic number theory, arithmetic geometry, Galois theory
素数，丢番图方程，解析数论，代数数论，算术几何，伽罗瓦理论
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一级分类：Mathematics 数学
二级分类：Algebraic Geometry 代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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英文摘要：
Recent works, mostly related to Ramanujan's mock theta functions, make use of the fact that harmonic weak Maass forms can be combinatorial generating functions. Generalizing works of Waldspurger, Kohnen and Zagier, we prove that such forms also serve as "generating functions" for central values and derivatives of quadratic twists of weight 2 modular $L$-functions. To obtain these results, we construct differentials of the third kind with twisted Heegner divisor by suitably generalizing the Borcherds lift to harmonic weak Maass forms. The connection with periods, Fourier coefficients, derivatives of $L$-functions, and points in the Jacobian of modular curves is obtained by analyzing the properties of these differentials using works of Scholl, Waldschmidt, and Gross and Zagier.
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PDF链接：
https://arxiv.org/pdf/0710.0283
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