Shimura Varieties, Families of Modular Forms and Analogues over Function Fields

我的研究项目是几何学和数论的交汇点。它的目的是研究与约化群有关的几何对象的算法，例如函数域上的Shimura簇及其类似物，遵循有限域上数域和函数域之间著名的类比。我从两个有影响力的节目中获得灵感或直接的线索：库德拉和朗兰兹的节目，在这些节目中，下村品种扮演着核心角色。作为说明，前者将Eisenstein级数特殊值处的导数与Shimura簇上的代数环联系起来。此外，还推广了Hilbert模曲面上的Hirzebruch-Zagier定理，在实二次基变化的情况下，我们将该结果视为朗兰兹函数性原理的几何变体。I.库德拉方案的p进变种。以此类推，我们可以用经典模型族的显式theta提升来代替Eisenstein级数；而经典导数则可以用p-adadly变权的p-进导数来代替。这对GL(2)来说是卓有成效的，并在最近几年引发了很好的博士论文问题。对于其他约化群，一个很大的绊脚石是目前在高维Shimura簇中缺乏p-进数族代数圈族的例子，因此需要新的想法。因此，对于群GSP(4)，我的项目是建立一个精化的Böcher er猜想的p-进变体，该猜想将与亏格2的Siegel模形式有关的旋量L函数的二次扭曲的中心值与其傅立叶系数联系起来，被视为Waldspurger定理的类似。这一Böcher er猜想确实也符合Gan-Gross-Prasad全球时期的猜想。二、功能定位。多年来一直吸引我的一个独特的主题是通过使用生活在下村变种的特殊或普通纤维中的代数循环来建立功能的强版本，例如雅克-朗兰兹对应。在特征零中，我计划将Ichino-Prasanna的部分工作推广到酉Shimura变种。在正特征中，我计划通过处理某些Siegel模三重的准模块层次结构来补充小朱的工作，这些结构不符合他们基于朱的几何Satake同构的方法。三、超功能领域：Drinfeld模块化品种。A)我在这个领域的第一步是与G.Rosso一起建立了GL(N)的Drinfeld模型族的存在性。第二个项目是证明GL(N)的Eichler-Shimura同余关系，并将其应用于Galois表示的构造。B)第二条问题是关于达蒙的斯塔克-海格纳点理论的类比，这是由于I·朗希。受Darmon-Vonk最近的工作启发，我们共同对GL(2)的函数域上的Shimura互易猜想的一个变体感兴趣。

My research project is at the intersection of geometry and number theory. It aims to study the arithmetic of geometric objects associated to reductive groups, for example Shimura varieties and their analogues over function fields, following the celebrated analogy between number fields and function fields over finite fields. I draw inspiration or direct cues from two influential programs: Kudla's and Langlands' programs in which Shimura varieties play a central role. For illustration, the former links derivatives at special values of Eisenstein series with algebraic cycles on Shimura varieties. Moreover, it also aims at generalizing the Hirzebruch-Zagier theorem on Hilbert modular surfaces, a result that we view as a geometric variant of Langlands' functoriality principle in the case of real quadratic base change. I. p-adic variant of the Kudla program. By analogy, we may replace Eisenstein series with explicit theta lifts of classical modular forms in the context of p-adic families of modular forms; and the classical derivative is replaced by the p-adic derivative of the weight varying p-adically. This is fruitful for GL(2), and gave rise to good Ph.D. thesis problems in recent years. For other reductive groups, a big stumbling block is the current lack of examples of p-adic families of algebraic cycles in higher dimensional Shimura varieties, so new ideas are needed.  For the group GSp(4), my project is therefore to establish a p-adic variant of a refined Böcherer conjecture relating the central value of the quadratic twists of the spinor L-function associated to Siegel modular forms of genus two to the Fourier coefficients thereof, viewed as an analogue of a theorem of Waldspurger. This Böcherer conjecture indeed also fits with the Gan-Gross-Prasad global period conjectures. II. Functoriality. A distinct theme which has fascinated me for years is establishing strong versions of functoriality e.g., the Jacquet-Langlands correspondence, by using algebraic cycles living in special or generic fibers of Shimura varieties. In characteristic zero, I plan to generalize parts of the work of Ichino-Prasanna to unitary Shimura varieties. In positive characteristic, I plan to complement the work of Xiao-Zhu by treating certain Siegel modular 3-folds of paramodular level structure that are not amenable to their method based on the geometric Satake isomorphism of Zhu. III. Over function fields: Drinfeld modular varieties. a) My first step in this realm was to establish the existence of families of Drinfeld modular forms for GL(n) with G. Rosso. A second project is to prove an Eichler-Shimura congruence relation for GL(n) with applications to construction of Galois representations. b) A second line of inquiry concerns the analogue of the theory of Stark-Heegner points of Darmon, due to I. Longhi. We are jointly interested in a variant of a Shimura reciprocity conjecture over function fields for GL(2), stimulated by recent work of Darmon-Vonk.