Toric periods, modular forms, and number theory

我提出的未来5年的研究计划代表着我探索了近20年的主题的演变。在本世纪初，我在遍历理论和$p$-进制数理论之间引入了一种意想不到的联系；这一基本见解仍然是积极和富有成果的研究的主题，现在我提议通过引入$p$-进制朗兰兹方案、表示理论和特征$p$中的朗兰兹方案的思想来扩展这种联系。 最主要的主题是环周期，即四元数代数上的复值自同构形沿嵌入的极大环的轨道的积分。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Ratner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环面周期：模和p-进表示理论的那些，也就是说，特征p中的表示理论和$p$-进数域中的系数。这一计划的萌芽包含在文件[Vat17]和[Vat18]中，但新的方向是出人意料的和耐人寻味的。 我打算追求的主要方向是发展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一理论，完成这一理论将导致特征p中的theta对应，并将PGL_2(Q_P)的模表示与SL_2(Q_P)的亚可解覆盖联系起来。这样的通信长期以来一直被猜测，感觉自己即将实现这一点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，并利用Drinfeld上半平面的étale覆盖的几何，将Bertolini和Darmon对特殊表示(发生在塔底)所做的周期积分计算推广到分支的超尖球面表示的情况。

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.