Charlotte Chan

RTG Number Theory Seminar, Fall 2023

Organizers: Kartik Prasanna, Tasho Kaletha, Charlotte Chan

Winter 2024

This semester, RTG Number Theory seminar will take place on Mondays 3:00-4:00p in EH 3088.

We will have local speakers give 60-minute research talks followed by discussion. Speakers will be encouraged to share a problem or problems at the end of their talk that can serve as a starting point for discussions and collaborations among participants.

Please email Charlotte if you would like to be on our mailing list.

1/22 Planning meeting
1/29 On Kloosterman sums and representation theory [speaker: Elad Zelingher]

Abstract: I will give a quick overview of twisted Kloosterman sums and their associated sheaves defined by Deligne--Katz. Then I will talk about matrix Kloosterman sums defined by Erdélyi--Tóth and about my recent work that reduces these to classical Kloosterman sums and Hall--Littlewood polynomials. In both parts of the talk, I will explain a relation between exterior/symmetric powers of the Kloosterman sheaf and a special value of a Bessel function of a representation of GLn(Fq).
2/5 Functoriality and double covers [speaker: Tasho Kaletha]

Abstract: A key player in the Langlands program is the L-group ^LG of a connected reductive group G over a local or global field. It underpins both pillars of the program: the reciprocity conjecture, which relates automorphic representations of G to Galois representations valued in ^LG, and the functoriality conjecture, which relates automorphic representations of two different groups whose L-groups are related. We will discuss how considerations of functoriality lead to an extension of the concept of L-group to certain non-linear covering groups, and how these can be organized using a new kind of fundamental group associated to G. We will focus on the case of a local field.
2/12 Theta correspondence and Arthur packets [speaker: Alex Hazeltine]

Abstract: The local theta correspondence does not preserve L-packets. As a remedy, Adams conjectured that instead of L-packets, the theta correspondence preserves Arthur packets. Moeglin verified Adams' conjecture when the theta correspondence has sufficiently large rank. Moeglin also showed that Adams' conjecture fails in low rank. Bakic and Hanzer showed that the failure can be managed: namely if Adams conjecture holds at some rank then it holds in any higher rank. In this talk, we discuss how to understand the failure of Adams' conjecture in low rank and how to remedy it.
2/19 Polynomial factorization modulo many primes [speaker: Dan Altman]

Abstract: The fastest known deterministic algorithms for factorising polynomials in F_p[x] have a worst-case runtime that is exponential in log p. We will discuss a new deterministic algorithm which factorises an integer polynomial modulo many primes simultaneously with amortised runtime that is polynomial in log p. Based on joint work with David Harvey.
3/4 Modular degrees of elliptic curves over function fields [speaker: Lukas Scheiwiller]

Abstract: The degree of the modular parametrization of an elliptic curve is an important invariant that is related to congruences of modular forms. In the case of an elliptic curve E over Q, the degrees of parameterizations of E by different Shimura curves are related to each other through Tamagawa numbers. We will discuss the analogous situation for elliptic curves over function fields, which admit different parameterizations by moduli of Drinfeld-Stuhler-modules.
3/11 Families of CM forms [speaker: Yu-Sheng Lee]
3/18 Affine Soergel bimodules [speaker: Calvin Yost-Wolff]
3/25 p-adic interpolation of Bessel periods for Siegel modular forms [speaker: Alex Bauman]
4/1 mod p sheaves on mixed-characteristic affine Grassmannians [speaker: Robert Cass]
4/8 BZSV duality and Lagrangian subvarieties of hyperspherical varieties [speaker: Jialiang Zou]
4/15 Zero cycles on products of elliptic curves [speaker: Kartik Prasanna]
4/22 Poincaré schemes [speaker: Viktor Burghardt]

Fall 2023

This semester, RTG Number Theory seminar will take place on Tuesdays 2:30-3:45p in EH 4096.

Please email Charlotte if you would like to be on our mailing list.

9/12 Planning meeting [Kartik Prasanna and Charlotte Chan]

9/26 Structures on ABV-packets for p-adic groups [speaker: Alex Hazeltine]

Abstract: In 1989, Arthur proposed a definition of local Arthur packets which arose from global considerations. A couple years later, ABV-packets were defined by purely local methods for connected reductive real groups and Vogan conjectured that they generalize local Arthur packets. More recently, Cunningham et al. defined ABV-packets for p-adic groups which also conjecturally generalize local Arthur packets. In this talk, we consider some intersection problems for ABV-packets of p-adic groups motivated by local Arthur packets.

10/3 Internal structure of L-packets for p-adic groups [speaker: Razan Taha]

Abstract: The local Langlands correspondence predicts that there is a finite-to-one map between the set of irreducible, admissible representations of a group G and the set of L-parameters of G. While the local Langlands correspondence has been established in many cases, the internal structure of L-packets is not explicitly known except in a few special cases. In this talk, we will describe one approach to study the internal structure. This approach is based on the case of SL(2,Qp) whose L-packets are parametrized by irreducible representations of subgroups of the Klein 4-group.

10/10 Artin Representations and GSp(4) [speaker: Alex Bauman]

Abstract: We review the Artin conjecture for GL(2), and investigate approaches for GSp(4), including the real-analytic theory of Kim-Yamauchi and higher Hida theory, due in this case to Pilloni and Boxer-Pilloni.

10/24 Additive Combinatorics, Uniformity, and Patterns in Primes [speaker: Henry Talbott]

Abstract: Additive combinatorics provides a powerful framework for understanding additive structure in finite-rank abelian groups. I'll give an overview of a few of the main techniques in this field, focusing on the structure/randomness dichotomy pioneered by Szemeredi and Gowers. Along the way, I’ll also introduce a close connection between additive combinatorics and ergodic theory, and explain how the transference principle developed by Green and Tao allows additive combinatorics to be applied to questions of additive structure in the set of prime numbers.

10/31 Speh representations and the Godement--Jacquet zeta integral [speaker: Elad Zelingher]

11/7 Connected components in the moduli space of L-parameters [speaker: Sean Cotner]

Abstract: There have been several recent approaches to defining a moduli space of Langlands parameters (or L-parameters) over the integers, in order to obtain refined versions of the local Langlands conjecture "at all primes away from p at once". I will discuss the approach of Dat-Helm-Kurinczuk-Moss, including the basic results and the example of split GL_2. There is a conjectural description of the connected components of this moduli space, of which I will outline a proof if time permits.

11/14 A family of period integral related to triple product L-functions [speaker: Pam Gu]

Abstract: Conjectures of Braverman-Kazhdan, Lafforgue, Ng\^{o} and Sakellaridis suggest that every affine spherical variety admits a generalized Poisson summation formula. We refer to this conjecture as the Poisson summation conjecture. The Poisson summation conjecture implies the functional equation and meromorphic continuation for fairly general Langlands $L$-functions, which by the converse theorem, implies Langlands functoriality in great generality. In collaboration with Jayce Getz, Chun-Hsien Hsu and Spencer Leslie, we constructed a family of period integrals using certain spherical varieties related to Braverman-Kazhdan spaces, which are holomorphic multiples of the triple product $L$-function in a domain that nontrivially intersects the critical strip. If time permits, I'll also talk about some current progress towards the analytic properties by introducing a family of Whittaker inductions to the picture.

11/21 Theta correspondence, Hecke algebra and Springer correspondence [speaker: Jialiang Zou]

Abstract: In this talk, we consider the finite field theta correspondence between principle series. Joint with Jiajun Ma and Congling Qiu, we explicitly describe this correspondence by analyzing the relevant Hecke algebra bimodules and applying a deformation argument. Joint with Jiajun Ma and Congling Qiu, Zhiwei Yun, we geometrized the whole picture. Consequently, we obtained a relation between the Springer correspondence and theta correspondence.

11/28 Motivic sheaves on affine flag varieties [speaker: Robert Cass]

Abstract: In this talk I will discuss a categorification of integral generic Hecke algebras using motivic sheaves on affine flag varieties. This eliminates the dependence on the chosen cohomology theory, such as l-adic or Betti cohomology, in previous works. At the unramified level this amounts to a motivic version of the geometric Satake equivalence. On the other extreme, at the Iwahori level we construct a motivic version of Gaitsgory's central functor. This is joint work with T. van den Hove and J. Scholbach.

12/5 [speaker: Amrita Acharyya]

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RTG Number Theory Seminar, Winter 2023: Introduction to Shimura varieties

This is a learning seminar on Shimura varieties organized by Tasho Kaletha, Kartik Prasanna, and Charlotte Chan. We meet 3:00-4:15p on Mondays in EH 4088. Talks should run around 60 minutes, with the remaining time left for questions, comments, discussion, and chatting. (Note that GLNT will meet 4:30p-5:30p on Mondays, also in EH 4088.)

Syllabus/Outline: PDF to appear here

Lecture notes of all talks [by Guanjie Huang]

Schedule:

1/9 Overview and introduction + planning meeting [speaker: Tasho Kaletha]

1/30 Global Shimura varieties [speaker: Calvin Yost-Wolff] [notes by Guanjie Huang and Calvin Yost-Wolff]
2/6 Cohomology of global Shimura varieties and the global correspondence [speaker: Alex Bauman]
2/13 Integral models of Shimura varieties [speaker: Patrick Daniels]
2/20 The Langlands-Rapoport conjecture [speaker: Tasho Kaletha]
3/13 The Langlands-Kottwitz method [speaker: Alexander Bertoloni Meli]
3/20 The Kottwitz set B(G) and the refined local and global conjectures [speaker: Andy Gordon]
3/27 Rapoport-Zink spaces [speaker: Kartik Prasanna]
4/3 Local Shimura varieties and the Kottwitz conjecture [speaker: Guanjie Huang]
4/10 The Fargues-Fontaine curve and vector bundles [speaker: Serin Hong]
4/17 Local Shtuka spaces [speaker: Robert Cass]
4/24 Integral models of local Shimura varieties [speaker: Patrick Daniels]

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RTG Number Theory Seminar, Fall 2022: the Sakellaridis--Venkatesh conjectures

This is a learning seminar on the relative Langlands program organized by Tasho Kaletha, Kartik Prasanna, and Charlotte Chan. The focus of the seminar will be on the Sakellaridis--Venkatesh conjectures. We meet 3:00-4:15p on Mondays in EH 4088. Talks should run around 60 minutes, with the remaining time left for questions, comments, discussion, and chatting. (Note that GLNT will meet 4:30p-5:30p on Mondays, also in EH 4088.)

Literature:

[SK] Sakellaridis and Venkatesh, Periods and harmonic analysis on spherical varieties
[L] Luna, Variétés sphériques de type A
[BP] Bravi and Pezzini, Primitive wonderful varieties
[LV] Luna and Vust, Plongements d'espaces homogènes
[K] Knop, The Luna-Vust theory of spherical embeddings
[KS] Knop and Schalke, The dual group of a spherical variety
[S] Sakellaridis, Spherical functions on spherical varieties
[GG] Gan and Gomez, A conjecture of Sakellaridis--Venkatesh on the unitary spectrum of spherical varieties

Lecture notes for all talks [by Guanjie Huang]

Schedule:

9/12 Overview on the absolute and relative Langlands programs + planning meeting [speaker: Tasho Kaletha]

9/19 Structure theory of homogeneous spherical varieties [L, BP] [speaker: Robert Cass]

Abstract: We present Luna's classification of homogeneous spherical varieties for a reductive group G, i.e., spherical varieties of the form G/H. The classification problem reduces to understanding those subgroups H which are "spherically closed." In this case, G/H has a unique wonderful compactification, and the problem is to classify wonderful varieties by combinatorial data. Luna achieved this classification for groups of type A, and much later Bravi and Pezzini showed Luna's classification works for general groups. Along the way we will give examples of spherical and wonderful varieties, and we will encounter the important notion of spherical roots of G.

9/26 Luna--Vust theory: structure theory of spherical embeddings (non-homogeneous spherical varieties) [LV, K] [speaker: Calvin Yost-Wolff]

Abstract: The classifications of spherical embeddings characterizes spherical varieties with dense G orbit G/H according to certain combinatorial data called colored fans. I will present the classification of spherical embeddings along with some examples. Then I will show how the colored fan relates to the geometry of the spherical variety along with how morphisms of spherical embeddings correspond to maps of colored fans.

10/3 The dual group of a spherical variety (after Knop-Schalke) [KS] [speaker: Charlotte Chan]

Abstract: We explain Knop--Schalke's construction of the dual group of any G-variety X. Their approach is based on first associating to X a "weak spherical datum", which is a weakening of Luna's homogeneous spherical datum. To this weak spherical datum, one associates two dual groups: one using weak spherical roots (this is the dual group of X) and one using associated roots (this is a subgroup of the dual of G). The construction of the Sakellaridis--Venkatesh's desired distinguished morphism is then obtained by analyzing centralizers of maps between these two dual groups. We mention some functoriality properties of Knop--Schalke's construction, which will likely come up later in Sakellaridis--Venkatesh's conjectures on "boundary degenerations" of spherical varieties.

10/10 Local geometry and asymptotics [SV, §4,5] [speaker: Alex Bauman]

Abstract: We construct certain "boundary degenerations" X_\Theta of a G-spherical variety X associated to a set \Theta of simple spherical roots of X, which are spherical varieties which look similar to X in some ways, but which are simpler and have extra automorphisms. We follow Sakellaridis-Venkatesh's study of the local geometry and asymptotics of X over a local field. First, they construct a bijection, for each compact open subgroup J, between J orbits on X and on X_\Theta near infinity. Then, they construct a G-equivariant map between the smooth functions on these spaces. The purpose of these maps are to break up the space of functions on X in terms of the boundary degenerations, which are simpler.

10/24 Bernstein morphisms, scattering theory, and the Plancharel formula: an overview [SV, §9-15] [speaker: Guanjie Huang]

Abstract: The study of the Plancherel decomposition of L^2 space of a spherical variety X is the core part of Sakellaridis-Venkatesh paper. In this talk, we will see how this can be reduced to the discrete spectrum of its boundary degenerations. In particular, we will construct the Bernstein morphisms which span L^2(X) from the discrete spectra of its boundary degenerations, and introduce the scattering theory to describe the possible overlap. If time permits, we will talk about how to write down explicit formulas for these morphisms, and how they lead to an explicit Plancherel decomposition of L^2(X).

10/31 The Sakellaridis--Venkatesh conjectures [SV, §16-18] [speaker: Kartik Prasanna]

Abstract: The goal of the S-V conjectures is to understand the relation between automorphic periods and L-values, as well as questions about distinction, both locally and globally. The motivating theorem is that of Tunnell-Saito-Waldspurger (T-S-W) for GL_2 and its inner forms, later generalized by the Gan-Gross-Prasad (G-G-P) and Ichino-Ikeda (I-I) conjectures. The SV conjectures are a further vast generalization of this circle of ideas to the setting of spherical varieties, though not yet formulated at the same level of precision. I will start by recalling the work of T-S-W, G-G-P and I-I to put things in context, then explain how the S-V conjectures generalize all of this.

11/7 Spherical functions and L-functions [S] [speaker: Jialiang Zou]

Abstract: In this talk, we study the unramified spectrum of a homogeneous spherical variety X . We will discuss Sakellaridis’s work on computing the eigenfunctions on spherical varieties under the action of the spherical Hecke algebra, which generalise the classical Casselman Shalika type formula. We will also discuss a variant of this formula, which involves the dual group of the spherical variety X and certain quotient of L-functions. As an application, we will present the unramified Plancherel formula for X.

11/14 Local relative character for some strongly tempered spherical varieties [speaker: Chen Wan]

Abstract: In this talk, I will explain how to compute the local relative character for strongly tempered spherical varieties in the unramified case. I will first explain the general strategy of the computation, then I will give some specific examples. This is a joint work with Lei Zhang.

11/21 Overview of the theta correspondence [speaker: Lukas Scheiwiller]

11/28 The work of Gan--Gomez [speaker: Guanjie Huang]

Abstract: Sakellaridis-Venkatesh conjectured a Plancherel decomposition of the local L2 spectrum of a spherical variety coming from the distinguished morphism between the dual groups. In this talk we will introduce how Gan and Gomez verified this conjecture for spherical varieties of low ranks using theta correspondence. If time permits, we will also introduce local relative character identity and its relation with factorization of global period.

12/5 Introduction to the relative trace formula [speaker: Elad Zelingher]

Abstract: I will discuss the motivation for the relative trace formula and explain the ideas of using it in the proof of Ichino--Ikeda conjecture for unitary groups.

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RTG Representation Theory Seminar, 2021-2022: Everything SL2

This is a learning seminar on topics in representation theory organized by Karol Koziol and Charlotte Chan. We meet 4:00-5:15p on Mondays in EH 4088. All talks should be focused on SL2 or GL2, in varying fonts. Talks should run around 60 minutes, with the remaining time left for questions, comments, discussions, and chatting.

Please email Karol Koziol (kkoziol [at] umich [dot] edu) to get on the mailing list.

Fall 2021

9/13 Planning meeting
9/20 Highest weight theory [speaker: Alex Bauman] [notes]

Abstract: In this talk, we introduce the Lie algebra sl_2(C) and find all its finite dimensional irreducible representations. We then show that every finite dimensional representation is a direct sum of irreducible representations. Finally, we introduce highest weight modules in general for sl_2(C) and discuss the Verma modules.
9/27 Borel--Weil--Bott over C [speaker: Calvin Yost-Wolff] [notes by Guanjie Huang]

Abstract: The Borel-Weil-Bott theorem describes the cohomology of line bundles on flag varieties as certain representations. In particular, the Borel-Weil-Bott theorem gives a geometric construction of the finite dimensional irreducible representations for reductive groups. In this talk, I will explicitly compute these representations for SL_2(C). I will then motivate our previous computations with induced representations and Serre duality, leading to the Borel-Weil-Bott theorem for SL_2(C). Lastly, I will use the Atiyah-Bott fixed point formula to deduce the Weyl character formula from our geometric representations.
10/4 Deligne--Lusztig varieties [speaker: Andy Gordon] [notes by Andy Gordon]

Abstract: In this talk we will construct the irreducible characteristic 0 representations of $SL_2(\mathbb{F}_q)$ for $q$ a prime power. This will be done via introducing the Drinfeld curve, a smooth affine variety whose cohomology groups contain the irreducible representations.
10/11 Weil representations and irreducible representations of SL2(Fq) [speaker: Calvin Yost-Wolff] [notes by Guanjie Huang]

Abstract: The Weil representation gives an algebraic method to classify the irreducible representations of symplectic groups Sp(2n,Fq). In this talk, we will work out this method on Sp(2,Fq) = SL(2,Fq), and show how in this case, it associates to each irreducible representation of SL(2,Fq) a maximal torus. We will first construct the Heisenberg group corresponding to a symplectic vector space and classify its irreducible representations via the finite Stone-Von-Neumann theorem. We then use an action of Sp(4,Fq) on the irreducible representations to construct the Weil representation of Sp(4,Fq). Finally, we will decompose the Weil representation into irreducible representations of Sp(2,Fq) via Howe duality.
10/25 Quantum sl2 and knots, I [speaker: Ilia Nekrasov]
11/1 Quantum sl2 and knots, II [speaker: Linh Truong]

Abstract: I will give an introduction to knots, the Jones polynomial, ribbon categories, and quantum knot invariants. We will focus on the quantum sl_2 knot invariant, which categorifies the Jones polynomial.
11/15 Representations of SL2(p-adic) [speaker: Guanjie Huang] [notes by Guanjie Huang]

Abstract: The supercuspidal representations of a p-adic group are those which cannot be obtained by parabolic induction. The usual way to construct a supercuspidal representation is compact induction. In this talk, we will construct the supercuspidal representations of SL(2,F) for a p-adic field F following Yu's construction. We will first explain how to obtain the so-called cuspidal G-data by studying the structure of SL(2,F). Then we will use these data to construct supercuspidal representations of SL(2,F) of zero and positive depth.
11/29 Representations of SL2(R) [speaker: Havi Ellers, notes by Havi Ellers]

Abstract: There are two parts to this talk, which are for the most part disjoint from each other. In the first part we introduce a diagramatic representation of indecomposable, quasi-simple, h-multiplicity free sl(2)-modules. In the second we discuss the vanishing of matrix coefficients for certain representations of SL2(R).
12/6 Dynamics and SL2(R) [speaker: Carsten Peterson]

Abstract: PSL(2, R) acts simply transitively on the unit tangent bundle of the hyperbolic plane. Under this identification, we can identify the left action of PSL(2, R) on itself as Mobius transformations. As for the right action, the diagonal subgroup corresponds to the geodesic flow, and the upper triangular subgroup corresponds to the horocycle flow. These actions descend to Gamma\PSL(2, R) which we may identify with the unit tangent bundle of a hyperbolic surface. When the hyperbolic surface has finite volume, since PSL(2, R) preserves Haar measure, we get that PSL(2, R) acts by measure-preserving transformations, and hence we get a unitary representation. Ergodicity of the geodesic flow corresponds to this representation, when restricted to the diagonal subgroup, only having one copy of the trivial representation, and mixing of the geodesic flow corresponds to vanishing of matrix coefficients at infinity in the orthogonal complement of the constant functions. Both of these hold true on hyperbolic surfaces of finite volume because of the Howe-Moore theorem. Non-trivial irreducible unitary representations of PSL(2, R) can be classified as either principal series, complementary series, or discrete series. SO(2) invariant functions on Gamma\PSL(2, R) correspond to functions on the hyperbolic surface. The Casimir operator restricted to such functions acts the same as the Laplacian on the hyperbolic surface. When the hyperbolic surface is compact, L^2(Gamma\PSL(2, R)) decomposes as a direct sum of irreducible representations. The complementary series in this decomposition are parametrized by eigenvalues of the Laplacian in (0, 1/4), and the principal series are parametrized by the eigenvalues in [1/4, infty) (and the trivial representation corresponds to the eigenvalue 0).

Winter 2022

1/24 Planning meeting
1/31 Trace formula for SL2 and its stabilization [speaker: Tasho Kaletha] ** 2 hour talk: 3p--5p ** **On Zoom: https://umich.zoom.us/j/96092436197, log-in required**
2/7 The Steinberg representation of G [speaker: Karthik Ganapathy]

Abstract: An algebraic group G is geometrically reductive if for every nonzero G-invariant vector v in a G-representation V, there exists a non-constant homogeneous G-invariant polynomial function on V which does not vanish on v. We will see Haboush's proof that every reductive group is geometrically reductive. The main character of this proof is the Steinberg representation; the bulk of the talk will be about this representation.
2/14 Characteristic p representations of SL2Fp [speaker: Nate Harman]
2/21 Modular supercuspidal representations of SL2Qp [speaker: Karol Koziol]
3/7 Tunnell--Saito: local epsilon factors and representations of GL2 [speaker: Alex Bauman]
3/14 Henniart's characterization of supercuspidal representations of GL2(non-arch) [speaker: Charlotte Chan]

Abstract: In the 1990s, Henniart proved that the (complex) supercuspidal representations of GL2(non-arch local field) can be distinguished by their character on a special set of regular semisimple elements which he called "very regular." Although the character formula of these representations is quite complicated in general, the character formula on these very regular elements is very simple: up to a sign, it is equal to the average over the W-orbit of an admissible character. This has seen many applications over the years, as it allows one to recognize a supercuspidal representation from a very small amount of simple data.
3/21 Representations of parahoric subgroups [speaker: Guanjie Huang]

Abstract: In this talk I will introduce Lusztig's cohomological construction of representations of reductive groups over finite rings, and compute the representations one can obtain in this way of SL2 in equal characteristic and rank 2. Lastly, I will introduce how this construction can be generalized to parahoric subgroups.
3/28 Satake isomorphism for SL2 [speaker: Carsten Peterson]

Abstract: The Harish-Chandra isomorphism allows one to understand the structure of the center of the universal enveloping algebra of a semisimple Lie algebra. It is also useful in understanding the structure of the algebra of invariant differential operators on a symmetric space. The Satake isomorphism is somewhat of a p-adic analogue which allows one to understand the structure of the spherical Hecke algebra of a semisimple p-adic Lie group. Special attention will be paid to how these isomorphisms work for SL_2.
4/4 Geometric Satake for SL2 [speaker: Andy Gordon] [notes by Andy Gordon]
4/11 Character sheaves for SL2 [speaker: Calvin Yost-Wolff]
4/18 Finite subgroups of SL2C [speaker: Henry Talbott] [slides by Henry Talbott]