Toric periods, modular forms, and number theory

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，我提出了遍历理论和$p$进数论之间的一个意想不到的联系；这一基本见解仍然是活跃而富有成果的研究主题，我现在建议通过引入$p$-adic朗兰兹纲领、表征理论和特征$p$中的朗兰兹纲领来扩展这种联系。******最重要的主题是环面周期，即复值自同构形式在四元数代数上沿嵌入的最大环面轨道的积分。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视角来看待环面周期：模和p进表示理论的视角，也就是说，特征p的表示理论和p进域的系数。这个计划的萌芽包含在论文[Vat17]和[Vat18]中，但新的方向是意想不到的和有趣的。******我提出的主要方向是发展GL_2（Q_p）的特征p和p进表示的环周期理论。我已经在我关于测试向量的工作中发展了这个理论的一部分，并且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）的元模覆盖联系起来。长期以来，人们一直在推测这种对应关系，感到自己即将实现这种对应关系是令人兴奋和充满活力的。类似地，我建议考虑GL_2（Q_p）的p进表示的几何，并使用德林菲尔德上半平面的<s:1>覆盖层的几何，将Bertolini和Darmon对特殊表示（出现在塔底）的周期积分计算扩展到分支超尖表示的情况。* * * * *

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. *****