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叉学科。它旨在研究与还原群相关的几何对象的算术，例如Shimura簇及其在函数域上的类似物，遵循有限域上的数域和函数域之间的著名类比。我从两个有影响力的项目中获得灵感或直接线索：库德洛和朗兰兹的项目，其中志村品种发挥了核心作用。为了说明问题，前者将爱森斯坦级数在特殊值处的导数与志村簇上的代数圈联系起来。此外，它还旨在推广Hilbert模曲面上的Hirzebruch-Zagier定理，我们认为这是Langlands泛函原理在真实的二次基变换情况下的几何变体。 I. Kudla程序的p-adic变体。通过类比，我们可以在模形式的p-adic族的上下文中用经典模形式的显式theta提升来代替Eisenstein级数;并且经典导数被权随p-adic变化的p-adic导数代替。这对于GL（2）是富有成效的，并产生了良好的博士学位。近年来的论文问题。对于其他约化群，一个很大的障碍是目前缺乏高维志村簇中代数循环的p-adic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改进的Böcherer猜想的p-adic变体，该猜想将亏格2的Siegel模形式相关的旋量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定理的类似物。这个Böcherer猜想确实也符合Gan-Gross-Prasad全球周期图。 二.功能性。多年来吸引我的一个独特的主题是建立功能性的强版本，例如，Jacquet-Langlands对应，通过使用生活在志村变种的特殊或通用纤维中的代数循环。在特征零，我计划推广部分工作的一野Prasanna单一志村品种。在积极的特点，我计划补充工作的Xiao-Zhu处理某些西格尔模3倍的paramodular水平结构，不服从他们的方法的基础上的几何佐竹同构。 三.在功能领域：德林费尔德模块化品种。 a）我在这个领域的第一步是建立GL（n）的Drinfeld模形式族的存在性，G。罗索。第二个项目是证明GL（n）的Eichler-Shimura同余关系及其在Galois表示构造中的应用。 B）第二条调查路线涉及达尔蒙的斯塔克-希格纳点理论的类似物，由于I。隆吉我们共同感兴趣的一个变种的志村互易猜想的功能领域GL（2），刺激最近的工作Darmon-Vonk。

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.