论文标题
Maaß表单$ L $ functions的添加曲折的中心价值
Central values of additive twists of Maaß forms $L$-functions
论文作者
论文摘要
在本文中，我们研究了Maaß形式$ L $ series的添加曲折的核心价值。在模块化组的情况下，我们表明添加剂曲折（平均分母时）在渐变上是正态分布的。这补充了Petridis-Risager的最新工作，该作品定居于Mazur-Rubin关于模块化符号的平均版本。本文的方法结合了由于Bettin而引起的动态输入，而第一篇命名作者的新事实是，添加剂的曲折以Zagier的意义定义了量子模块化形式。后一种属性显示为一般离散的，具有牙齿的共同限定组。我们的结果还具有许多算术应用。对于Hecke一致性组，量子模块化意味着对于扭曲的$ {\ rm gl} _2 $ -automorphic $ l $ runctions的扭曲时刻的某些互惠关系，扩大了Conrey和第二名作者的结果。对于模块化组的CuspidalMaaß形式，我们还可以计算出Maaß形式的$ L $功能的某些宽度矩。
In the present paper we study the central values of additive twists of Maaß forms $L$-series. In the case of the modular group, we show that the additive twists (when averaged over denominators) are asymptotically normally distributed. This supplements the recent work of Petridis--Risager which settled an averaged version of a conjecture of Mazur--Rubin concerning modular symbols. The methods of the present paper combine dynamical input due to Bettin and the first named author with the new fact that the additive twists define quantum modular forms in the sense of Zagier. This latter property is shown for a general discrete, co-finite group with cusps. Our results also has a number of arithmetic applications; in the case of Hecke congruence groups the quantum modularity implies certain reciprocity relations for twisted moments of twisted ${\rm GL}_2$-automorphic $L$-functions, extending results of Conrey and the second named author. In the case of cuspidal Maaß forms for the modular group, we also obtain a calculation of certain wide moments of twists of the $L$-function of the Maaß form.
