I will present an extension of Wei Zhang's Arithmetic Fundamental Lemma conjecture to the whole Hecke algebra, and present some evidence for it. Joint work with Chao Li and Wei Zhang.

This talk considers triple-product p-adic L-functions corresponding to various regions of classical interpolation. It describes their relations with rational points and Selmer classes on modular elliptic curves, obtained from explicit reciprocity laws for appropriate diagonal classes.

Let $K=\Q(\sqrt d)$ be an imaginary quadratic field. Let $L$ be a unimodular $O_K$-lattice of signature $(n-1, 1)$, and let $\mathcal X$ be the compactified Shimura variety over $O_K$ associated at $L$.  In this talk, we describe roughly how to construct an arithmetic theta series (arithmetic theta  kernel), and prove that it is a modular form of level $d$, weight $n$, character $\chi_d^n$ with values in  $\widehat{CH}^1(\mathcal X^*)$.  Using this arithmetic theta kernel, we can produce classical modular forms from arithmetic $1$-cycles (Faltings’ height pairing) and produce arithmetic divisors from classical cusp forms.  This is joint work with Bruinier, Howard, Kudla, and Rapoport for $n \ge 3$. The case $2$ is a joint work in progress with Qiao He and Yousheng Shi.

The Kudla-Rapoport conjecture, proved jointly with Wei Zhang, is a precise identity relating arithmetic intersection numbers of special cycles on unitary Shimura varieties with good reduction and central derivatives of Siegel Eisenstein series. We discuss how to formulate and prove an analogous identity for certain unitary Shimura varieties with bad reduction (Krämer models at ramified places). We will motivate these conjectures and highlight interesting new phenomena in the presence of bad reduction. This is joint work with Qiao He, Yousheng Shi and Tonghai Yang.

We discuss 2-Selmer groups and L-values of elliptic curves under quadratic twists. This is based on joint works with Pan and with He-Xiong.

In this talk, we will discuss the proof of irrationality of 2-adic zeta value at 5. It uses an arithmetic holonomicity theorem, which can be viewed as a refinement of André’s algebraicity criterion and has also been studied in the recent work of Bost and Charles. We will also discuss some other linear independence problems.

This is joint work with Frank Calegari and Vesselin Dimitrov.

In earlier work we defined a theta series of algebraic cycles on the moduli space of unitary Shtukas with an arbitrary number of legs. This can be viewed as a generalization of theta series and arithmetic theta series for function fields. A main open problem after this construction is the modularity of the higher theta series. In this talk, I will explain a proof of the modularity of higher theta series when restricted to the generic fiber. This is joint work with Tony Feng and Wei Zhang.

I will discuss developments for algebraic and p-adic aspects of L-functions, with a view toward my ongoing joint work with G. Rosso and S. Shah on spin L-functions for GSp_6.  I will emphasize how these developments fit into a broader context, highlighting common ingredients among constructions during the past several decades, while also indicating where new technical challenges arise.  All are welcome, and expertise in spin L-functions is unnecessary to follow this lecture.

We use the doubling method to construct p-adic L-functions for finite-slope families of automorphic forms on symplectic and unitary groups. A key construction is the p-adic deformation of iterations of Maass-Shimura differential operators on nearly overconvergent families. We will explain how to generalize the construction for GL(2) by Andreatta and Iovita, and how to bypass explicit computations of formulas of iterations of differential operators on q-expansions by using dynamics of U_p operators.

Given an automorphic representation of a unitary group, one can define an arithmetic automorphic period as the Petersson inner product of a deRham rational form. Here the deRham rational structure comes from the cohomology of Shimura varieties. When the form is holomorphic, the period can be related to special values of L-functions and is better understood. In this talk, we formulate a conjecture on relations among general arithmetic periods of representations in the same L-packet and explain a conditional proof.

Certain Rankin-Selberg motives of rank n(n+1) are endowed with algebraic cycles arising from maps of unitary Shimura varieties. Gan-Gross-Prasad conjectured that these cycles are analogous to Heegner points, in the sense that their nontriviality should be detected by derivatives of L-functions.

I will propose another nontriviality criterion, based on p-adic L-functions. Under some assumptions, this variant can be established in a refined quantitative form, via a comparison of p-adic relative-trace formulas. Together with the recent LTXZZ Euler system, this gives cases of the p-adic Beilinson-Bloch-Kato conjecture of Perrin-Riou. (Joint work with Wei Zhang.)