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

我的研究项目是几何和数论的交叉。它的目的是研究与约化群相关的几何对象的算法，例如函数域上的志村变异及其类似物，遵循著名的数域和有限域上的函数域之间的类比。我从两个有影响力的项目中获得灵感或直接线索：库德拉和朗兰兹的项目，志村品种在其中发挥了核心作用。为了说明这一点，前者在Shimura变体上具有代数循环的Eisenstein级数的特殊值处连接导数。此外，它还旨在推广Hilbert模曲面上的Hirzebruch-Zagier定理，我们将这一结果视为Langlands泛函原理在实二次基变化情况下的几何变体。I. Kudla程序的p进式变体。类比地说，在模形式的p进族中，我们可以用经典模形式的显式提升来代替爱森斯坦级数；经典的导数被p进的权值的p进的导数取代。这对于GL(2)来说是富有成效的，并在近年来产生了很好的博士论文问题。对于其他约化群，一个很大的障碍是目前缺乏高维Shimura变代数循环的p进族的例子，因此需要新的想法。因此，对于组GSp(4)，我的项目是建立一个p进变型的改进Böcherer猜想，该猜想与与格2的西格尔模形式相关的旋量l函数的二次扭曲的中心值及其傅里叶系数有关，被视为Waldspurger定理的类比。这个Böcherer猜想确实也符合甘-格罗斯-普拉萨德全球周期猜想。2。Functoriality。多年来让我着迷的一个独特主题是通过使用志村品种的特殊或一般纤维中的代数循环来建立强大版本的功能，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的志村变种。在积极特征方面，我计划通过处理某些不适合他们基于朱的几何Satake同构的方法的旁模水平结构的Siegel模3折叠来补充Xiao-Zhu的工作。3。在函数域上：Drinfeld模块变种。a)我在这个领域的第一步是与G. Rosso建立了GL(n)的Drinfeld模形式族的存在性。第二个课题是证明GL(n)的Eichler-Shimura同余关系，并将其应用于伽罗瓦表示的构造。b)由于I. Longhi的研究，第二条探究线涉及到Darmon的Stark-Heegner点理论的类比。我们共同对GL(2)的函数域上的Shimura互反猜想的一个变体感兴趣，该猜想是由Darmon-Vonk最近的工作所激发的。

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.