Automorphic Forms and Arithmetic

I will describe how the orbit method can be developed in a quantitative form, along the lines of microlocal analysis, and applied to local problems in representation theory and global problems concerning automorphic forms. The local applications include asymptotic expansions of relative characters. The global applications include moment estimates and subconvex bounds for L-functions. These results are the subject of two papers, the first joint with Akshay Venkatesh:
https://arxiv.org/abs/1805.07750
https://arxiv.org/abs/2012.02187

In the talk I shall report on recent joint work with Tong Liu on integral p-adic Hodge theory. Using newly developed cohomology theory we extend a result of Caruso, stating roughly that, in nice situations, certain natural structure on the crystalline cohomology of a variety is a Breuil module related to its étale cohomology.

Petersson inner products of classical cusp forms contain important arithmetic information, such as congruences of modular forms. When the cusp forms are replaced by meromorphic modular form, the Petersson inner product can still be defined and calculated after suitable regularization. It turns out these regularized inner product also carry interesting arithmetic information, such as special values of derivatives of L-function. In this talk, we will recall some of these results, and discuss a joint work with Markus Schwagenscheidt from ETH, where we obtained an integrality result of such regularized inner products involving unary theta functions.

In 1979 D. Goldfeld conjectured: 50% of the quadratic twists of an elliptic curve over the rationals have analytic rank zero. We present the first instance - the congruent number elliptic curves (joint with Y. Tian).

The classical Siegel-Weil formula relates an integral of a theta function along one classical group H to special values of the Siegel-Eisenstein series on another classical group G. Kudla proposed an arithmetic analogue of it that relates a generating series of algebraic cycles on the Shimura variety for H to the first derivative of the Siegel-Eisenstein series for G, which has become a very active program. We propose to go further in the function field case, relating a generating series of algebraic cycles on the moduli of H-Shtukas with multiple legs to higher derivatives of the Siegel-Eisenstein series for G, when H and G are unitary groups. We prove such a formula for nonsingular Fourier coefficients, relating their higher derivatives to degrees of zero cycles on the moduli of unitary Shtukas. The proof ultimately relies on an argument from Springer theory. This is joint work with Tony Feng and Wei Zhang.

The stacks of shtukas play an important role in the Langlands correspondence for function fields. In this talk, we will recall the definition of cohomology sheaves of stacks of shtukas and review the partial Frobenius morphisms and Drinfeld's lemma. Then we will talk about the smoothness property of the cohomology sheaves and some applications.

In this talk, we will discuss the theory of twisted automorphic descents, which is an extension of the automorphic descent of Ginzburg-Rallis-Soudry. One of our goals is to construct cuspidate automorphic modules in the generic global Arthur packets by using Fourier coefficients of automorphic representations. Moreover, we will discuss our approach for one direction of Gan-Gross-Prasad Conjecture for the Bessel-Fourier models and some connections between Fourier coefficients and spherical varieties. This is a joint work with Dihua Jiang.

I'll discuss 4 different types of single-valued integration on algebraic varieties -- one in the complex setting, one in the tropical setting, and two in the p-adic setting, and the relationships between them. In particular, I'll explain how to compute Vologodsky's "single-valued" iterated integrals on curves of bad reduction in terms of Berkovich integrals, and how to give a single-valued integration theory on complex varieties. Time permitting, I'll explain some potential arithmetic applications. This is a report on joint work in progress with Sasha Shmakov (in the complex setting) and Eric Katz (in the p-adic setting).