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

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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.