Representation Theory / Number Theory Seminar


Seminar meets Wednesdays, 4-5pm MT (unless specified otherwise) in LCB 222.
Zoom link will be available for online talks*.

*For security reasons, Zoom links will not be posted here. If you would like to attend a talk, but do not have the link, please contact Gil Moss, Peter Wear, or Petar Bakic (last name at math.utah.edu).

Click here for the tentative Spring 2022 schedule.


Fall 2021

Wednesday, September 8, 4:00-5:00pm

Speaker: Hang Xue, University of Arizona
Title: The local Gan-Gross-Prasad conjecture for real unitary groups
Abstract: A classical branching theorem of Weyl describes how an irreducible representation of compact U(n+1) decomposes when restricted to U(n). The local Gan-Gross-Prasad conjecture provides a conjectural extension to the setting of representations of noncompact unitary groups lying in a generic L-packet. We prove this conjecture. Previously Beuzart-Plessis proved the "multiplicity one in a Vogan packet" part of the conjecture for tempered L-packets using the local trace formula approach initiated by Waldspurger. Our proof uses theta lifts instead, and is independent of the trace formula argument.


Wednesday, September 22, 4:00-5:00pm

Speaker: Mishel Skenderi, University of Utah
Title: Inverting the Siegel Transform in the Geometry of Numbers
Abstract: We begin this talk by introducing the general notion (due to Helgason) of generalized Radon transforms for homogeneous spaces in duality, together with some motivating examples of such transforms (the classical Radon transform and the Funk transform) and a brief discussion of the types of problems about such transforms. The rest of the talk is devoted to the primitive Siegel transform of (sufficiently nice) functions f : R -> R which is a particular kind of generalized Radon transform. The Siegel transform \widehat{f} of such a function f is a pseudo-Eisenstein series on SL_n(R)/SL_n(Z), the space of full-rank lattices in R^n up to covolume. After briefly discussing the history of this transform in the geometry of numbers, we show how classical formulae for the mean (due to Siegel) and inner product (when n \geq 3 and due to Rogers) of such transforms may be used to easily prove whenever n \geq 3 the injectivity of this transform on even functions and an inversion formula. We then explain why these easy proofs of injectivity and inversion do not apply in the classical case of n=2.


Wednesday, September 29, 4:00-5:00pm

Speaker: Ed Karasiewicz, University of Utah
Title: The Gelfand-Graev representation for linear groups and their covers.
Abstract: We will discuss the Gelfand-Graev representation for linear reductive groups and their nonlinear covers. In the first part of the talk we describe the spherical part of the Gelfand-Graev representation for linear groups as a module over the spherical Hecke algebra. Then we use this description to explain why the Casselman-Shalika formula can be expressed as the character of a dual group. In the second part of the talk we turn to nonlinear covers, where the multiplicity one theorem for Whittaker models fails. Here we will describe some ongoing work to generalize the description of the Iwahori part of the Gelfand-Graev representation due to Chan-Savin from linear groups to their nonlinear covers. This is joint work with Nadya Gurevich and Fan Gao.


Wednesday, October 6, 4:00-5:00pm

Speaker: Gabriel Dorfsman-Hopkins, UC Berkeley
Title: Untilting Line Bundles on Perfectoid Spaces
Abstract: Let X be a perfectoid space with tilt X♭. We construct a canonical map θ:PicX♭ → limPicX where the (inverse) limit is taken over the p-power map, and show that θ is an isomorphism if R=Γ(X,OX) is a perfectoid ring. As a consequence we obtain a characterization of when the Picard groups of X and X♭ agree in terms of the p-divisibility of PicX. The main technical ingredient is the vanishing of higher derived limits of the unit group R*, whence the main result follows from the Grothendieck spectral sequence.


Wednesday, October 13

No seminar - Fall break!


Wednesday, October 27, 4:00-5:00pm (Zoom)

Speaker: Anna Romanov, University of New South Wales
Title: A Soergel bimodule approach to the character theory of real groups
Abstract: Admissible representations of real reductive groups are a key player in the world of unitary representation theory. The characters of irreducible admissible representations were described by Lustig-Vogan in the 80's in terms of a geometrically-defined module over the associated Hecke algebra. In this talk, I'll describe a categorification of a block of the LV module using Soergel bimodules. This is joint work with Scott Larson.


Wednesday, November 3, 4:00-5:00pm (Zoom)

Speaker: Jennifer Berg
Title: Frobenius descent on subvarieties of constant abelian varieties
Abstract: If a variety X over a global field K has a rational point, then it always has points over each of its completions. Conversely if the presence of local solutions guarantees the existence of a global solution in X(K) then X is said to satisfy the local-to-global principle. Often this is too optimistic. However, when it fails to hold, one can systematically impose conditions on the collection of local points to narrow down the subset that captures the rational points, should any exist. A typically fruitful approach is via the method of descent which makes use of arithmetic information on various covers of X.
In this talk, we'll focus on constant subvarieties X of an abelian variety A defined over the function field of a curve over a finite field. This allows us to consider covers of X that arise from isogenies on A, such as the Frobenius isogeny which gives rise to "Frobenius descent." We'll describe this descent and the information it captures about rational points for certain surfaces (and higher dimensional subvarieties) and give a geometric interpretation in terms of maps between varieties over the finite field. This is joint work in progress with Felipe Voloch.


Wednesday, November 10, 4:00-5:00pm (Zoom)

Speaker: Maria Fox, University of Oregon
Title: Supersingular Loci of (2,m-2) Unitary Shimura Varieties
Abstract: The supersingular locus of a Unitary (2,m-2) Shimura variety parametrizes supersingular abelian varieties of dimension m, with an action of a quadratic imaginary field meeting the "signature (2,m-2)" condition. In some cases, for example when m=3 or m=4, every irreducible component of the supersingular locus is isomorphic to a Deligne-Lusztig variety, and the intersection combinatorics are governed by a Bruhat-Tits building. We'll consider these cases for motivation, and then see how the structure of the supersingular locus becomes very different for m>4. (The new result in this talk is joint with Naoki Imai.)


Wednesday, November 17, 4:00-5:00pm

Speaker: Thomas Hales, University of Pittsburgh
Title: Partition functions, spherical Hecke algebras, and the Satake transform
Abstract: This talk will describe a collection of partition functions that include Kostant's q-partition functions and the Langlands L-function of a spherical representation. These partition functions give explicit combinatorial formulas for such things as branching rules, the inverse of the weight multiplicity matrix, the inverse Satake transform, and Macdonald's formula. This work is motivated by the Fundamental Lemma (conjectured by Langlands and Shelstad and proved by Ngo) that arises in connection with the stable trace formula. This is joint work with Bill Casselman and Jorge Cely.


Wednesday, November 24

No seminar - Thanksgiving week!


Wednesday, December 1, 4:00-5:00pm (Zoom)

Speaker: Rachel Pries, Colorado State University
Title: Curves of genus 4 with infinitely many primes of basic reduction
Abstract: Elkies proved that an elliptic curve over the rationals has infinitely many primes of supersingular reduction. We generalize this result for curves of genus 4 that have an order 5 automorphism. This is joint work with Li, Mantovan, and Tang.


Wednesday, December 8, 4:00-5:00pm

Speaker: Chengyu Du, University of Utah
Title: Signatures of fintie-dimensional representations of real reductive Lie groups
Abstract: Let G be a real reductive Lie group and V be an irreducible finite-dimensional representation of G. Suppose V admits a G-invariant hermitian form. When it exists, it is unique up to a scalar. We use a twisted Dirac index to compute the signature (p,q) of the hermitian form, recovering a result of Kanilov-Vogan-Xu. For example, when G is compact, the form always exists and (up to a scalar) is positive definite, so the signature is just (dim(V),0). But for G noncompact and dim(V) greater than 1, the form is never definite. In this case the signature is a refinement of the dimension. In fact, the formula we prove can be thought of as a variant of the Weyl Dimension Formula. Motivated by the unitary representation theory of p-adic groups, we will also briefly explain how we generalize this method to the context of modules over the graded affine Hecke algebra to prove new results about signatures.



Spring 2021

Wednesday, January 20, 3:00-4:00pm

Speaker: Joseph Hundley, SUNY Buffalo
Title: Functorial Descent in the Exceptional Groups
Abstract: In this talk, I will discuss the method of functorial descent. This method, which was discovered by Ginzbug, Rallis and Soudry, uses Fourier coefficients of residues of Eisenstein series to attack the problem of characterizing the image of Langlands functorial liftings. We'll discuss the general structure, the results of Ginzburg, Rallis and Soudry in the classical groups, some recent attempts to extend the same ideas to the exceptional group, and challenges and new phenomena which emerge in these attempts. The new work discussed will mainly be joint with Baiying Liu. Time permitting I may also comment on unpublished work of Ginzburg and joint work with Ginzburg.


Wednesday, January 20, 3:00-4:00pm

Speaker: Joseph Hundley, SUNY Buffalo
Title: Functorial Descent in the Exceptional Groups
Abstract: In this talk, I will discuss the method of functorial descent. This method, which was discovered by Ginzbug, Rallis and Soudry, uses Fourier coefficients of residues of Eisenstein series to attack the problem of characterizing the image of Langlands functorial liftings. We'll discuss the general structure, the results of Ginzburg, Rallis and Soudry in the classical groups, some recent attempts to extend the same ideas to the exceptional group, and challenges and new phenomena which emerge in these attempts. The new work discussed will mainly be joint with Baiying Liu. Time permitting I may also comment on unpublished work of Ginzburg and joint work with Ginzburg.


Wednesday, January 27, 3:00-4:00pm

Speaker: Lucas Mason-Brown, Oxford
Title: What is a unipotent representation?
Abstract: The concept of a unipotent representation has its origins in the representation theory of finite Chevalley groups. Let G(Fq) be the group of Fq-rational points of a connected reductive algebraic group G. In 1984, Lusztig completed the classification of irreducible representations of G(Fq). He showed:
1) All irreducible representations of G(Fq) can be constructed from a finite set of building blocks -- called `unipotent representations.'
2) Unipotent representations can be classified by certain geometric parameters related to nilpotent orbits for a complex group associated to G(Fq).
Now, replace Fq with C, the field of complex numbers, and replace G(Fq) with G(C). There is a striking analogy between the finite-dimensional representation theory of G(Fq) and the unitary representation theory of G(C). This analogy suggests that all unitary representations of G(C) can be constructed from a finite set of building blocks -- called `unipotent representations' -- and that these building blocks are classified by geometric parameters related to nilpotent orbits. In this talk I will propose a definition of unipotent representations, generalizing the Barbasch-Vogan notion of `special unipotent'. The definition I propose is geometric and case-free. After giving some examples, I will state a geometric classification of unipotent representations, generalizing the well-known result of Barbasch-Vogan for special unipotents.
This talk is based on forthcoming joint work with Ivan Loseu and Dmitryo Matvieievskyi.


Wednesday, February 17, 4:00-5:00pm

Speaker: Hiraku Atobe, Hokkaido University
Title: The Zelevinsky-Aubert duality for classical groups
Abstract: In 1980, Zelevinsky studied representation theory for p-adic general linear groups. He gave an involution on the set of irreducible representations, which exchanges the trivial representation with the Steinberg representation. Aubert extended this involution to p-adic reductive groups, which is now called the Zelevinsky-Aubert duality. It is expected that this duality preserves unitarity. In this talk, we explain an algorithm to compute the Zelevinsky-Aubert duality for odd special orthogonal groups or symplectic groups. This is a joint work with Alberto Minguez in University of Vienna.


Wednesday, March 3, 3:00-4:00pm

Speaker: Allechar Serrano López, University of Utah
Title: Counting elliptic curves with prescribed torsion over imaginary quadratic fields
Abstract: A generalization of Mazur's theorem states that there are 26 possibilities for the torsion subgroup of an elliptic curve over a quadratic extension of $\mathbb{Q}$. If $G$ is one of these groups, we count the number of elliptic curves of bounded naive height whose torsion subgroup is isomorphic to $G$ in the case of imaginary quadratic fields.


Wednesday, March 10, 3:00-4:00pm

Speaker: Justin Trias, University of East Anglia
Title: Towards an integral local theta correspondence: universal Weil module and first conjectures
Abstract: The theta correspondence is an important and somewhat mysterious tool in number theory, with arithmetic applications ranging from special values of L-functions, epsilon factors, to the local Langlands correspondence. The local variant of the theta correspondence is described as a bijection between prescribed sets of irreducible smooth complex representations of groups G_1 and G_2, where (G_1,G_2) is a reductive dual pair in a symplectic p-adic group. The basic setup in the theory (Stone-von Neumann theorem, the metaplectic group and the Weil representation) can be extended beyond complex representations to representations with coefficients in any algebraically closed field R as long as the characteristic of R does not divide p. However, the correspondence defined in this way may no longer be a bijection depending on the characteristic of R compared to the pro-orders of the pair (G_1,G_2). In the recent years, there has been a growing interest in studying representations with coefficients in as general a ring as possible. In this talk, I will explain how the basic setup makes sense over an A-algebra B, where A is the ring obtained from the integers by inverting p and adding enough p-power roots of unity. Eventually, I will discuss some conjectures towards an integral local theta correspondence. In particular, one expects that the failure of this correspondence for fields having bad characteristic does appear in terms of some torsion submodule in integral isotypic families of the Weil representation with coefficients in B.


Wednesday, March 17, 3:00-4:00pm

Speaker: Ila Varma, University of Toronto
Title: Malle's Conjecture for octic $D_4$-fields.
Abstract: We consider the family of normal octic fields with Galois group $D_4$, ordered by their discriminant. In forthcoming joint work with Arul Shankar, we verify the strong form of Malle's conjecture for this family of number fields, obtaining the order of growth as well as the constant of proportionality. In this talk, we will discuss and review the combination of techniques from analytic number theory and geometry-of-numbers methods used to prove this and related results.


Wednesday, March 24, 3:00-4:00pm

Speaker: Baiying Liu, Purdue University
Title: On recent progress on Jiang's conjecture on wave front sets of representations in Arthur packets.
Abstract: In this talk, I will introduce some recent progress on Jiang's conjecture on wave front sets of representations in Arthur packets. Jiang's conjecture is a natural generalization of Shahidi's conjecture on tempered L-packets. It shows that there is a strong connection between the structure of Arthur parameters and the wave front sets of representations in the corresponding Arthur packets. This includes some work joint with Dihua Jiang, and joint with Freydoon Shahidi.


Wednesday, March 31, 3:00-4:00pm

Speaker: Michael Griffin, BYU
Title: Moonshine
Abstract: In the 1970's, during efforts to completely classify the finite simple groups, several striking apparent coincidences emerged connecting the then-conjectural “Monster group” to the theory of modular functions. Conway and Norton turned these observed 'coincidences' into a precise conjecture known as “Monstrous Moonshine.” Borcherds proved the conjecture in 1992, embedding Monstrous Moonshine in a deeper theory of vertex operator algebras which have important physical interpretations. Fifteen years after Borcherds' proof, Witten conjectured an important role of Monstrous Moonshine in his search for a theory of pure quantum gravity in three dimensions. Under Witten's theory, the irreducible components of the Monster module represent energy states of black holes. The distribution of these energy states can be found using tools from number theory. Moonshine-phenomena have also been observed for other groups besides the Monster. These include the Umbral Moonshine conjectures of Cheng, Duncan, and Harvey which arise from the symmetry groups of each of the 24-dimensional Niemeier lattices. Recently, Moonshine for other sporadic simple groups have been shown to connect arithmetic properties of certain elliptic curves to the class numbers of certain imaginary quadratic fields.


Thursday, April 8, time TBD (Colloquium talk)

Speaker: Aaron Pollack, UCSD
Title: TBA
Abstract: TBA


Wednesday, April 14, 3:00-4:00pm

Speaker: Martin Weissman, UC Santa Cruz
Title: The compact induction theorem for rank-one p-adic groups
Abstract: A folklore conjecture predicts that when G is a p-adic group, every irreducible supercuspidal representation of G is induced from a compact-mod-center open subgroup. This was proven for GL(n) by Bushnell and Kutzko. For other groups, e.g., classical groups, tame groups, etc., the conjecture is proven for sufficiently large p thanks to hard work by many people. In this talk, I will describe a recent proof of the conjecture which applies to all groups G of relative rank one, with no assumptions about p. The method is to use the work of Schneider and Stuhler to connect supercuspidal representations to sheaves on the Bruhat-Tits tree of G, and "refine" these sheaves until the induction theorem becomes obvious.