Automorphic Forms and Arithmetic

Let $Y$ be a locally symmetric space associated to an even dimensional rational quadratic space $(V,Q)$ of signature $(p,q)$. The Kudla-Millson lift is a lift from the $q$-th homology of $Y$ to modular forms of weight $\frac{p+q}{2}$.

A natural way of constructing a homology class is by embedding an algebraic torus in the orthogonal group of $V$. I will discuss the Kudla-Millson lift of such cycles, and in particular show that it is the diagonal restriction of a Hilbert modular form. In low rank, one can recover a result of Darmon-Pozzi-Vonk and a trace identity due to Darmon-Harris-Rotger-Venkatesh.

Asai L-functions for GLn are related to arithmetic of Asai motives. The twisted Gan-Gross-Prasad (GGP) conjecture opens a way of studying (a twist of) central Asai L-values via descents and period integrals. Firstly, I will prove new cases of twisted GGP conjecture (joint work with Weixiao Lu and Danielle Wang), based on the relative trace formula approach in the thesis work of Wang. Secondly, I will formulate an arithmetic twisted GGP conjecture on central derivatives. As a key ingredient, I will formulate and prove a twisted arithmetic fundamental lemma. For the proof, I will introduce new special cycles and Rapoport-Zink spaces related to mirabolic and parabolic groups.

Given a smooth proper surface $X$ over a $p$-adic field, we study the monodromy action on its $\ell$-adic cohomology when $X$ degenerates to a surface in characteristic $p$ with simple singularities, otherwise known as rational double points. This class of singularities is a generalization of ordinary double points and has natural incarnations in arithmetic geometry and in Lie theory. We will use a mixed-characteristic version of the Grothendieck-Brieskorn resolution to investigate reduction properties of models of $X$, and we will describe the associated local monodromy via certain Springer representations attached to an appropriate nearby cycles sheaf. Time permitting, we may see some applications on derived equivalences of K3 surfaces.

The Fontaine-Mazur conjecture (proved by Kisin and Emerton) says that (under certain technical hypotheses) a Galois representation $\rho:\mathrm{Gal}_{\mathbb{Q}}\rightarrow \mathrm{GL}_{2}(\overline{\mathbb{Q}}_{p})$ is modular if it is unramified outside finitely many places and de Rham at $p$. I will talk about what this means, and I will discuss an analogous modularity result for Galois representations $\rho:\mathrm{Gal}_{\mathbb{Q}}\rightarrow \mathrm{GL}_{2}(L)$ when $L$ is instead a non-archimedean local field of characteristic $p$.

In this talk we'll explain a strategy to deduce general relative Poincare duality in p-adic geometry (previously conjectured by Bhatt--Hansen) in a diagramatic manner, whose special cases were previously obtained respectively by Lan--Liu--Zhu, Gabber--Zavyalov, Mann. This is a joint work in preparation with Emanuel Reinecke and Bogdan Zavyalov.

A fundamental problem in the theory of meromorphic connections on P^1 is to understand the space of such systems with given local behavior. Here, the local behavior of a connection at a singular point means the "formal type" there--the isomorphism class of the induced formal connection. Given a collection of singular points and corresponding formal types, there are several natural questions one might ask:

1) Does there exist a connection with these formal types?
2) If such a connection exists, is it unique up to isomorphism?
3) Can one construct an explicit moduli space of such connections?

Classically, these questions were studied under the assumption that all singularities are regular singular (i.e. simple poles). For example, in 2003, Crawley-Boevey solved the Deligne-Simpson problem for Fuchsian connections (a variant of question 1) by reinterpreting the problem in terms of quiver varieties. Later, mathematicians including Boalch, Hiroe, and Yamakawa investigated these questions when "unramified" irregular singularities are allowed. (Unramified means that the formal types can be expressed in upper triangular form without introducing roots of the local parameter.) In recent years, there has been increasing interest in meromorphic connections (and G-connections where G is a reductive group) with ramified singularities due to developments in the geometric Langlands program. In this talk, I will give an overview of recent progress on the ramified version of these problems due to myself and various collaborators. Time permitting, I will also talk about some related work of myself and Kamgarpour on differential Galois groups of G-connections.

Classical ramification theory deals with complete discrete valuation fields $k((X))$ with perfect residue fields $k$. Invariants such as the Swan conductor capture important information about extensions of these fields. Many fascinating complications arise when we allow non-discrete valuations and imperfect residue fields $k$. Particularly in positive residue characteristic, we encounter the mysterious phenomenon of the defect (or ramification deficiency). The occurrence of a non-trivial defect is one of the main obstacles to long-standing problems, such as obtaining resolution of singularities in positive characteristic.
Degree $p$ extensions of valuation fields are building blocks of the general case. In this talk, we will present a generalization of ramification invariants for such extensions and discuss how this leads to a better understanding of the defect. If time permits, we will briefly discuss their connection with some recent work (joint with K. Kato) on upper ramification groups.

I'll discuss recent progress on the problem of classifying all unitary representations of a real reductive Lie group, particularly the exceptional groups. The talk will focus on applications of techniques/intuition from string theory, automorphic forms, and intertwining operators (joint work with: Michael Green and Pierre Vanhove; Joseph Hundley; and Jeff Adams, Marc van Leeuwen, and David Vogan.)

I will give an overview (and some details) of my proof with George Boxer, Frank Calegari, and Vincent Pilloni of the modularity of a positive proportion of curves over Q of genus 2.

I'll explain a construction of p-adic L-functions for $\mathrm{GSp}(4)\times \mathrm{GL}(2)$ by using Furusawa's integral and the proof of its interpolation formula. I'll describe how local functional equations are used to compute the zeta intgerals at p and how the archimedean integrals are computed by using Yoshida lifts together with p-adic Rankin-Selberg L-function and p-adic standard L-function of $\mathrm{Sp}(4)$. I'll also discuss its applications in studying congruences for Yoshida lifts.

Lusztig's theory of character sheaves for connected reductive groups is one of the most important developments in representation theory in the last few decades. I will give an overview of this theory and explain the need, from the perspective of the representation theory of p-adic groups, of a theory of character sheaves on jet schemes. Recently, R. Bezrukavnikov and I have developed the "generic" part of this desired theory. In the simplest non trivial case, this resolves a conjecture of Lusztig and produces perverse sheaves on jet schemes compatible with parahoric Deligne--Lusztig induction. This talk is intended to describe in broad strokes what we know about these generic character sheaves, especially within the context of the Langlands program.

Over the last decades, following works around the Pila-Wilkie counting theorem in the context of o-minimality, there has been a surge in interest around functional transcendence results, in part due to their connection with special points conjectures. A prime example is the Ax-Lindemann-Weierstrass (ALW) Theorem and its role in his proof of the Andre-Oort conjecture.

In this talk we will discuss how an entirely new approach, using the model theory of differential fields as well as other differential tools, can be used to prove functional transcendence results (including ALW) for Fuchsian automorphic functions and other covering maps. We will also explain how cases of the Andre-Pink conjecture can be obtained using this new approach. This is joint work with D. Blazquez-Sanz, G. Casale and J. Freitag.

For a smooth projective variety Y over complex numbers, one has the notion of Hodge structure. Associated to the Hodge structure is a Q reductive group MT(Y), called the Mumford-Tate group. If the Y is defined over a number field, then its p-adic etale cohomology is a Galois representation. There is a notion of p-adic etale monodromy group G_p(Y). The Mumford-Tate conjecture claims that the base change to Q_p of MT(Y) has the same neutral component as G_p(Y). In my talk, I will formulate a mod p analogue of the conjecture and sketch a proof for orthogonal Shimura varieties. Important special cases of orthogonal Shimura varieties include moduli spaces of polarized Abelian and K3 surfaces. The result has an interesting application to a mod p analogue of the Andre-Oort conjecture: if a subvariety of a Shimura variety contains a Zariski dense collection of special curves, then the subvariety is "almost" a Shimura subvariety.

Breuil and Mezard conjecture that the Hilbert-Samuel multiplicities of deformation rings of rank n representations of a p-adic field K with p-adic Hodge theoretic conditions are controlled by certain decomposition numbers the group GL_n(O_K). More recently, as part of the categorical p-adic Langlands program, Emerton and Gee gave a geometric interpretation of this phenomena as the (conjectural) existence of highly constrained Breuil-Mezard cycles in the Emerton-Gee stack. I will explain how the equivariant homology of certain affine Springer fibers gives a proposal for these cycles (at least in a generic regime), and how it elucidates their internal structures. This is based on joint work with Tony Feng, and work in progress with Zhongyipan Lin.