Beijing Institute of Mathematical Sciences and Applications

The Riemann zeta function $\zeta(s)$ takes rational values at negative odd integers, and the Kummer congruence relations assert that these values satisfy congruences enabling them to be interpolated continuously in the $p$-adic topology. The $p$-adic $L$-function of Kubota-Leopoldt serves as the $p$-adic analog of the Riemann zeta function, incorporating the Kummer congruence relations. Coates conjectured that there exist $p$-adic $L$-functions attached to motives or algebraic automorphic representations, which interpolate the critical values of complex $L$-functions with explicit modifying factors at infinity and at $p$. Starting with basic notions on $p$-adic $L$-functions, we will discuss Coates' conjecture in the setting of automorphic representations. We verify it for symplectic-type representations. The talk is based on a joint work with Dongwen Liu.

Graded Hecke algebras, which were introduced by Lusztig in order to study the category of unipotent representations of a $p$-adic group, are graded analogues of affine Hecke algebras. Twisted graded Hecke algebras are slightly more general versions in which the group algebra of the spherical Weyl group $W$ is replaced by a twisted group algebra of an extension of $W$ by a finite abelian group. I will explain how twisted graded Hecke algebras naturally occur in both the general representation theory of $p$-adic groups and the spectral side of the local Langlands correspondence, and can be used to construct the later in a vaste range of situations.

The Aubert involution is a duality on representations of $p$-adic groups that is conjectured to preserve unitarity, making it a natural tool for constructing non-tempered unitary representations. It is well understood for connected classical groups, but much less so for non-linear covering groups. We determine explicitly the Aubert duals of strongly positive representations -- the building blocks of discrete series -- of the metaplectic group Mp(n) over a non-archimedean local field of characteristic different from two. Building on Mati\'c's classification and a careful analysis of Jacquet modules in the covering setting, we describe these duals in terms of precise inducing data. The results are the metaplectic counterpart of Mati\'c's description for classical groups, and the same method also yields an explicit description for odd general spin groups.

The category of smooth complex representations of a $p$-adic group decomposes into a product of indecomposable full subcategories, called Bernstein blocks. In joint work with Jeffrey D. Adler, Jessica Fintzen, and Manish Mishra, we proved, under explicit tameness assumptions, that every Bernstein block is equivalent to a special kind of Bernstein block, called a depth-zero block. Such blocks are closely related to representations of finite groups of Lie type and are therefore much more accessible than general Bernstein blocks. However, the resulting “reduction-to-depth-zero” functor is not unique. In this talk, I will explain the result that the functor can be chosen to preserve the generic representations in each block, which play an important role in the local Langlands correspondence. I will also explain that this requirement determines the functor ``outside the supercuspidal part''. The talk is based on joint work in progress with Cheng-Chiang Tsai.

It is well-known that $l$-modular blocks show great difference comparing to complex blocks of $p$-adic groups. In this talk, I will introduce some examples of depth zero blocks and all the $l$-modular blocks of $SL_n$ from the perspective of establishing nonsplit projective objects. We will explore the technical challenges in associating an $l$-modular block with a depth-zero block and consider a natural connection between them.

Period integrals of automorphic forms on reductive groups are of high interest. They may detect functorial transfer from other groups and their values may have arithmetic or automorphic significance. However, it is quite possible that the period over a reductive subgroup H may be identically zero on the cuspidal spectrum, and to discuss it on the residual spectrum requires regularization. In this talk I discuss new period integral of this type and construct new residual automorphic representations for symplectic groups. This is joint work with David Ginzburg and Omer Offen.

We will discuss explicit formulas for Whittaker functions associated with all irreducible generic representations of GL(4,R). As an application, we provide test vectors for the Bump–Friedberg integrals. We will also report on ongoing work concerning Shalika functions and Friedberg–Jacquet integrals. The former part is joint work with Miki Hirano and Tadashi Miyazaki.

Derivatives and exceptional poles are crucial in studying local L-functions over non-archimedean local fields. This approach contributes to the product formula and test vector problems for local L-functions. In this talk, we present the Archimedean analogue of exceptional poles of Rankin-Selberg L-functions and describe derivatives for irreducible principal series in general position. To be precise, we demonstrate that exceptional poles of type 1 and type 2 coincide for irreducible principal series in general position. This conjecture was originally proposed by Chai in his Ph.D. thesis. Unlike non-Archimedean cases, additional level structures emerge, and we present a strategy to get around this difficulty. This talk is based on my recent joint work with Santosh Nadimpalli and Akash Yadav. If time permits, we motivate the Archimedean test vector problem within Rankin-Selberg L-functions and explain the specific challenge posed by the G.C.D. definition of Archimedean L-functions.

This talk discusses the classification of anomaly-free hyperspherical Hamiltonian spaces. Starting from the basic framework of BZSV quadruples, we explain the role of the anomaly-free condition and describe how such spaces are classified when $G$ is a simple reductive group. We will also indicate how the resulting list suggests relative Langlands dual objects. This is a joint work with Guodong Tang at NUS and Chen Wan at Rutgers.

In joint work with Dorian Goldfeld and Eric Stade, we prove an aymptotic orthogonality relation for SL(n,Z) Maass forms. The main setup for the theorem is a Kuznetsov trace formula obtained by calculating the inner product of a Poincare series in two separate ways. I will discuss the trace formula and the ideas behind obtaining our applications. I will also contrast our work to results of Matz-Templier and Finis-Matz, which treat a much broader range of groups, but give weaker bounds.

I will present new proofs of results relating distinction of cuspidal representations of inner forms of GLn over p-adic fields and and that of their Jacquet-Langlands transfer, using local intertwining periods. In the case of the Guo-Jacquet and quaternionic symplectic models, this gives the missing p=2 case of conjectures of Prasad--Takloo-Bighash and Prasad.

The orbit method, pioneered by Kirillov, seeks to realize unitary representations of a Lie group as quantizations of its coadjoint orbits. For reductive Lie groups, however, this philosophy faces substantial obstacles, especially in the nilpotent setting. Among nilpotent orbits, Lusztig singled out a remarkable subclass—the special nilpotent orbits—and conjectured that the associated locally closed subvarieties of the nilpotent cone, known as special pieces, have only quotient singularities. In joint work with Juteau, Levy, and Sommers, we prove Lusztig’s conjecture in full generality. I will explain how this geometric result leads to new applications in the representation theory of real reductive groups. In particular, certain special unipotent representations, in the sense of Arthur and Adams–Barbasch–Vogan, can be realized as quantizations of special pieces, based on my previous two joint papers with Leung and Losev. Their unitarity then follows from mixed Hodge theory. Combined with the ongoing work of Adams-Ionov-Mason-Brown-Vogan, this yields a geometric proof of the unitarity of all special unipotent representations. The case of classical groups were previously established by Barbasch, Ma, Sun, and Zhu via theta correspondence. The approach presented here is of geometric nature and applies uniformly to both classical and exceptional groups.

The notion of a Whittaker model and the more general degenerate Whittaker model (which were introduced for p-adic groups by Moeglin-Waldspurger) is of fundamental importance to representation theory and automorphic forms. They also appear prominently in the GGP conjectures as Bessel and FJ models. The talk is meant to introduce the subject, discuss some of the less well-known examples, and talk about some open questions.

It is widely believed that the average analytic rank of a ``generic'' family of curves is 1/2. In this talk, we will discuss an upper bound of the average analytic rank in cases where the moduli space of the family of curves is a weighted projective space or the image of a morphism between weighted projective spaces. The main ingredients of the proof are the construction of the mod p reduction map on the rational points of the weighted projective space and the counting of rational points in the weighted projective space. We also study the asymptotic behavior of local conditions at primes. For example, we prove that the probability that a genus 2 curve over rationals with a rational Weierstrass point has a tacnode at a prime greater than 5 is $(p-1)/p^4(1-p^{-20})$.

Ramanujan graphs, as a typical class of the so-called expander graphs, are important in graph theory, number theory as well as computer science. Expander complexes and Ramanujan complexes, as their higher dimensional analogues, have gradually attracted interest in the past decades. In this talk, we give a new construction of Ramanujan complexes from unitary groups over number fields, based on the endoscopic classification of (non-quasisplit) unitary groups. Under the "golden" condition, our construction is explicit, meaning that we have an algorithm that is practically executable on a modern (or future) computer. This is joint work with Rahul Dalal and Alberto Minguez.

One of the most important open problems in mathematics is the Unitary Dual Problem: given a group, classify its irreducible unitary representations. The most common approach to classifying unitary representations is to study representations admitting non-degenerate invariant Hermitian forms (these are classified), compute the signatures of the forms, and then determine when the forms are definite. Signature character formulas are very complicated due to the recursive nature of the formulas. However, it turns out that the complexity may be encoded by well-known combinatorial objects: signature character formulas involve signed Kazhdan-Lusztig polynomials (which turn out to be classical Kazhdan-Lusztig polynomials evaluated at -q times a sign) and pieces of Hall-Littlewood polynomials (called Hall-Littlewood polynomial summands) evaluated at $q=-1$. We will discuss unitarizability of holomorphic discrete series representations and classifying unitary highest weight modules. Some of the material covered in this talk is joint work with Justin Lariviere.

The generalized Ramanujan conjecture (GRC) is one of the most significant open problems in modern number theory. Hypothesis H can often substitute for the GRC in analytic applications. In this talk, we will discuss our recent work that resolves this hypothesis. Furthermore, we achieve a stronger result than the original formulation. This directly yields immediate applications to other central problems.

In this talk we discuss the Birch and Swinnerton-Dyer formula modulo square of rational numbers for the quadratic twist family of a given elliptic curve over $\mathbb{Q}$. In particular we show the following: let $E$ be a semistable elliptic curve over $\mathbb{Q}$ with conductor $N$, whose analytic rank is at most one, then for any positive fundamental discriminant $D$ that is relatively prime to $N$, such that the quadratic twist $E^D$ again has analytic rank at most one, we have that the Birch and Swinnerton-Dyer formula modulo square of rational numbers holds for $E$ if and only if it holds for $E^D$. Joint work with Alexander Barrios.

Let $G$ be a finite group. Let $K/k$ be a Galois extension of number fields with Galois group isomorphic to $G$, and let $C \subseteq \mathrm{Gal}(K/k) \simeq G$ be a conjugacy invariant subset. It is well known that there exists an unramified prime ideal $\mathfrak{p}$ of $k$ with Frobenius element lying in $C$ and norm satisfying $\left | k_{\mathfrak{p}}\right | \ll |\mathrm{Disc}(K)|^{\alpha}$ for some constant $\alpha = \alpha(G,C)$. There is a rich literature establishing unconditional admissible values for $\alpha$, with most approaches proceeding by studying the zeros of $L$-functions. We give an alternative approach, not relying on zeros, that often substantially improves this exponent $\alpha$ for any fixed finite group $G$, provided $C$ is a union of rational equivalence classes. As a particularly striking example, we prove that there exist absolute constants $c_1,c_2 > 0$ such that for any $n\geq 2$ and any conjugacy class $C \subset S_n$, one may take $\alpha(S_n,C) = c_1 e^{-c_2n}$. Our approach reduces the core problem to a question in character theory.

Let $G$ be a reductive group defined over a local field of characteristic $0$ (real or $p$-adic). By Harish-Chandra’s regularity theorem, the character $\Theta_{\pi}$ of an irreducible representation ${\pi}$ of $G$ is given by a locally integrable function $f_{\pi}$ on $G$. In fact, $f_{\pi}$ has even better integrability properties, namely, it is locally $L^{1+r}$-integrable for some $r>0.$ This gives rise to a new singularity invariant of representations $e_{\pi}$ by considering the largest such $r$. We explore $e_{\pi}$, show it is bounded from below only in terms of the group $G$, and calculate it in the case of $p$-adic $GL_n$. To do so, we relate $e_{\pi}$ to the integrability of Fourier transforms of nilpotent orbital integrals appearing in the local character expansion of $\Theta_{\pi}$. As a main technical tool, we use explicit resolutions of singularities of certain hyperplane arrangements. We obtain several applications, including bounds on multiplicities of K-types in irreducible representations in the p-adic case, and on multiplicities of irreducible representations appearing in the $L^2$-space of a compact homogeneous space. Based on joint work with Itay Glazer and Julia Gordon.

Bessel functions are special matrix coefficients attached to irreducible generic representations of GL(n, F_q). They show up in computations of gamma factors and allow one to give explicit matrix realizations of irreducible cuspidal representations. In this talk I will explain some of my results expressing special values of Bessel functions (and some generalizations) in terms of Kloosterman sums/sheaves and how these results are parallel to some results regarding special values of Whittaker functions for unramified representations.

According to the relative Langlands functoriality conjecture, an admissible morphism between the L-groups of spherical varieties should induce a functorial transfer of the corresponding local and global automorphic spectra. Via the relative trace formula approach, two fundamental problems on the geometric side are the smooth transfer and the fundamental lemma. For rank-one spherical varieties with morphism of L-groups being the identity map, Y. Sakellaridis has already established the local transfer, and we will talk about the fundamental lemma.

In this talk, we study murmurations in families of real Dirichlet characters weighted by a smooth compactly supported function. We show that the murmuration density interpolates the phase transition in the 1-level density of the associated symplectic family of L-functions.

Beyond Endoscopy, the proposal of Langlands to prove the principle of functoriality requires, as its first step, the removal, from the discrete part of the trace formula, of the traces of nontempered distributions.Later, it was proposed that the way to achieve this should be by an application of Poisson Summation. Altug carried it for the first time for $GL(2, \mathbb{Q})$. In the next several years, works followed in $GL(2)$ for different more general contexts. In this talk we will discuss this step for $GL(3, \mathbb{Q})$. This is joint work with Taiwang Deng.

We begin with a brief introduction to automorphic forms in the setting of loop groups. Our main focus is then on the Weil representation, the Siegel–Weil formula, and the theta lift for loop symplectic groups. As an illustration, we present examples of lifting modular forms to loop orthogonal groups.