Toric periods, modular forms, and number theory

环面周期、模形式和数论

• 批准号: RGPIN-2019-03929
• 负责人: Vatsal, Vinayak
• 金额: $ 1.38万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758593
• 关键词: Toric; periods; modular; forms; number

The research program which I propose over the next 5 years represents an evolution  of the themes I have explored for almost 20 years. In the early 2000s, I introduced an unexpected connection between ergodic theory and $p$-adic number theory; this fundamental insight is still the topic of active and fruitful research, and I now propose to extend the connections by introducing ideas from the $p$-adic Langlands programme, representation theory, and the Langlands programme in characteristic $p$. The overarching theme is that of toric periods, namely, the integrals of complex valued automorphic forms on a quaternion algebra along the orbits of embedded maximal tori. In my earlier work, the point was to show that these period integrals are nonzero, and this was accomplished by using Ratner's theorems  from ergodic theory on uniform distribution. This time, however, I propose to look at toric periods through completely different lenses: those of modular and p-adic representation theory, which is to say, representation theory in characteristic p and with coefficients in $p$-adic fields. The germs of this program are contained in the papers [Vat17] and [Vat18], but the new directions are unexpected and intriguing. The principal directions I propose to pursue are to develop the theory of toric periods for characteristic p and p-adic representations of GL_2(Q_p). I have already developed pieces of this theory in my work on test vectors, and completing the theory would lead to a theta correspondence in characteristic p, relating modular representations of PGL_2(Q_p) and the metaplectic cover of SL_2(Q_p). Such a correspondence has long been speculated, and it is exciting and energizing to feel that one is close to achieving it. Analogously, I propose to consider the geometry of p-adic representations of GL_2(Q_p), and to use the geometry of étale covers of Drinfeld's upper half plane to extend the period integral calculations made by Bertolini and Darmon for the special representations (which occur at the bottom of the tower) to the case of ramified supercuspidal representations.

我提出的未来5年的研究计划代表了我近20年来探索的主题的演变。在21世纪初，我介绍了遍历理论和$p$-adic数论之间的一个意想不到的联系;这个基本的见解仍然是活跃和富有成效的研究主题，我现在建议通过引入$p$-adic朗兰兹纲领，表示论和朗兰兹纲领的思想来扩展这种联系。 首要的主题是环面周期，即，积分的复值自守形式的四元数代数沿着轨道嵌入的最大环面。在我早期的工作中，重点是证明这些周期积分是非零的，这是通过使用均匀分布遍历理论中的拉特纳定理来实现的。然而，这一次，我建议通过完全不同的透镜来看待环面周期：模和p-adic表示理论，也就是说，在特征p和系数在$p$-adic域的表示理论。这个程序的萌芽包含在论文[Vat 17]和[Vat 18]中，但新的方向是意想不到的和有趣的。 我建议追求的主要方向是发展GL_2（Q_p）的特征p和p-adic表示的环面周期理论。我已经在测试向量的工作中发展了这个理论的部分，完成这个理论将导致特征p中的theta对应，将PGL_2（Q_p）的模表示与SL_2（Q_p）的亚复盖联系起来。这样的对应关系早已被推测出来，当我们感觉到自己即将实现它时，我们会感到兴奋和充满活力。类似地，我建议考虑GL_2（Q_p）的p-adic表示的几何，并利用德林费尔德上半平面的三角形覆盖几何推广了Bertolini和Darmon对特殊表示的周期积分计算（发生在塔的底部）的情况下，分歧supercuspidal表示。