*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 Petar Bakic (last name at math.utah.edu).
Speaker: Mathilde Gerbelli-Gauthier, McGill University Title: Fourier Interpolation and the Weil Representation Abstract: In 2017, Radchenko-Viazovska proved a remarkable interpolation result for even Schwartz functions on the real line: such a function is entirely determined by its values and those of its Fourier transform at square roots of integers. We give a new proof of this result, exploiting the fact that Schwartz functions are the underlying vector space of the Weil representation W. This allows us to deduce the interpolation result from the computation of the cohomology of a certain congruence subgroup of SL2(Z) with values in W. This is joint work with Akshay Venkatesh.
Speaker: Lucas Mason-Brown, University of Oxford Title: Arthur packets and generalized endoscopy for real reductive groups Abstract: Let G and H be real reductive groups. To any L-homomorphism e : H^L → G^L one can associate a map e_* from virtual representations of H to virtual representations of G. This map was predicted by Langlands and defined (in the real case) by Adams, Barbasch, and Vogan. Without further restrictions on e, this map can be very poorly behaved. A special case in which e_* exhibits especially nice behavior is the case when H is an endoscopic group. In this talk, I will introduce a more general class of groups which exhibit similar behavior. I will explain how this generalized version of endoscopic lifting can be used to prove the unitarity of all Arthur packets. This is based on joint work with Jeffrey Adams and David Vogan.
Speaker: Justin Trias, Imperial College London / University of East Anglia Title: Theta correspondence in families for type II dual pairs Abstract: The classical local theta correspondence for p-adic reductive dual pairs defines a bijection between prescribed subsets of irreducible smooth complex representations coming from two groups (H,H'), forming a dual pair in a symplectic group. Motivated by new perspectives in the local Langlands correspondence for modular representations, Alberto Minguez extended the theta correspondence for type II dual pairs (i.e. when (H,H') is made of general linear groups) to the setting of representations with coefficients in algebraically closed fields of characteristic l as long as the characteristic l does not divide the pro-orders of H and H'. More recently, the work of Emerton and Helm extended the local Langlands correspondence to families of representations, that is over coefficient rings, with compatibility to both classical and modular local Langlands for general linear groups. We explain how to build a theta correspondence in families, i.e. with coefficients in rings like Z[1/p], for type II dual pairs that is compatible with reduction to residue fields of the base coefficient ring, where central to this approach is the integral Bernstein centre. We translate some weaker properties of the classical correspondence, such as compatibility with supercuspidal support, as a ring morphism between the integral Bernstein centres of H and H' and interpret it for the Weil representation. This ring morphism between the Bernstein centres brings a richer structure than a simple compatibility of supercuspidal supports and allows to ask new geometric questions for the theta correspondence: we prove that this map is surjective i.e. it is a closed immersion between the associated affine schemes. In particular our result implies a theta correspondence between characters of the Bernstein centre over any coefficient field of characteristic not p. This is joint work with Gil Moss.
Speaker: Yotam Svoray, University of Utah Title: On the fundamental groups of the surface parametrizing cuboids Abstract: Motivated by Chabauty-Kim theory, we prove that the surface parametrizing cuboids and its resolution of singularities have a trivial fundamental group. We also compute the fundamental group of the surface minus the divisor of degenerate cuboids, which turns out to be non-abelian. In addition, we conclude by studying the analogous fundamental groups of the surface of face cuboids.
Speaker: Nathan Geer, Utah State University Title: Some algebra behind non-semisimple TQFTs Abstract: In this talk I will give an introductory lecture on constructing Topological Quantum Field Theories (TQFTs) from non-semisimple categories. The main goal of the talk is to give a hint of what is needed to extend the Turaev-Viro and Crane-Yetter TQFTs from the useful setting of semisimple categories to the non-semisimple world. I will do this from an algebraic and categorical point of view. In particular, I will discuss what kind of structures are needed in non-semisimple categories to give rise to (2+1)-TQFTs. Then I will remark that any spherical tensor category (in the sense of Etingof, Douglas et al.) has such structures. This work is joint with Francesco Costantino, Benjamin Haïoun, Bertrand Patureau-Mirand and Alexis Virelizier and based on arXiv:2302.04509 and arXiv:2306.03225.
No seminar - spring break!
Speaker: Jared Weinstein, Boston University Title: Integral Ax-Sen-Tate theory Abstract: Let K be a local field of mixed characteristic, let G be the absolute Galois group of K, and let C be the completion of an algebraic closure of K. The Ax-Sen-Tate theorem states that the field of G-invariant elements in C is K itself: H^0(G,C)=K. Tate also proved statements about higher cohomology (with continuous cocycles): H^1(G,C)=K and H^i(G,C)=0 for i>1. Let O_C be the ring of integers in C. Our main theorem is that the torsion subgroup of H^i(G,O_C) is killed by a constant which only depends on the residue characteristic p (in fact p^6 suffices). This is a part of a project with coauthors Tobias Barthel, Tomer Schlank, and Nathaniel Stapleton.
Speaker: Peter Trapa, University of Utah Title: Matching real and p-adic Kazhdan-Lusztig polynomials Abstract: Let G be a complex reductive algebraic group, and let s denote a semisimple element in its Lie algebra. The centralizer L of s in G acts with finitely many orbits on the eigenspaces of ad(s). The singularities of these orbit closures (encoded in p-adic Kazhdan-Lusztig polynomials, i.e. local intersection cohomology Poincare polynomials) determines the multiplicities of irreducible subquotients in standard induced modules for unramified, and more generally unipotent, representations of p-adic forms of the dual group of G. Meanwhile, if K is a symmetric subgroup of G, the singularities of K orbit closures in the flag variety of G, encoded in "real" Kazhdan-Lusztig-Vogan polynomials, determines the multiplicities of irreducible subquotients in standard modules for real forms of the dual group of G. Our main result describes a natural hypothesis, which is always satisfied for the classical subgroups of GL(n), that implies that the p-adic polynomials are a subset of the real polynomials. This, in turn, matches certain multiplicities in standard modules for real and p-adic groups. This is joint work with Leticia Barchini.
Speaker: Peter Trapa, University of Utah Title: Matching real and p-adic Kazhdan-Lusztig polynomials, part 2 Abstract: see above
Speaker: Daniel Gulotta, University of Utah Title: Some progress toward the Kottwitz conjecture for local shtuka spaces Abstract: Langlands functoriality predicts that if two reductive groups over a p-adic field are inner forms of each other, then the representation theory of the groups should be related. One can compare representations of the groups in the following ways: - One can compare the trace distributions of the representations. - One can find a "local shtuka space" that has an action of both groups, and look at how the groups act on its cohomology. Fargues-Scholze have constructed local shtuka spaces in great generality and proved that their cohomology is not too large. I will explain some progress toward relating their construction to trace distributions. This talk is based partly on joint work with David Hansen and Jared Weinstein.
No seminar - fall break!
Speaker: Allechar Serrano López, Montana State Title: Counting number fields Abstract: A guiding question in arithmetic statistics is: Given a degree $n$ and a Galois group $G$ in $S_n$, how does the count of number fields of degree $n$ whose normal closure has Galois group $G$ grow as their discriminants tend to infinity? In this talk, I will give an overview of the history and development of number field asymptotics and we will obtain a count for dihedral quartic extensions over a fixed number field.
Speaker: Moshe Adrian, Queens College, CUNY Title: Converse problems over local fields: an overview Abstract: Let G be a connected reductive group over a local field. Attached to a representation of G are conjectural families of invariants called gamma factors. A local converse problem seeks to determine which families of these invariants uniquely determine a representation of G. In this talk, we will present the current state of affairs regarding local converse problems and theorems. We will review older results, present recent results, and discuss upcoming results.
Speaker: Loren Spice, Texas Christian University Title: Quasisemisimple actions on reductive groups and buildings (joint with Adler and Lansky) Abstract: In the early 21st century, Prasad and Yu proved that, for a field of characteristic exponent p ≥ 1, the identity component of the group of fixed points of a finite group \Gamma of prime-to-p order acting on a reductive group \tilde G is itself a reductive group; and that, for a local field of residual characteristic p, the building of the resulting fixed-point group is the fixed-point set for the action of \Gamma on the building of \tilde G. Applications by Adler and Lansky to lifting require analogous statements for quasisemisimple actions, which are those for which there are a Borel subgroup \tilde B of \tilde G, and a maximal torus in \tilde B, that are preserved by the action of \Gamma. The fixed-point group remains reductive in this case, at least after smoothing. Unfortunately, the directly analogous statement about buildings is too strong; but, fortunately, the directly analogous statement about buildings is too strong, so that we can explore the interesting ways that the building of the fixed-point group can fail to fill out the full fixed-point set in the building of \tilde G, and conditions that recover the exact analogue of the Prasad-Yu statement.
No seminar - Thanksgiving week!
Speaker: Jialiang Zou, University of Michigan Title: Theta correspondence, Hecke algebra and Springer correspondence Abstract: In this talk, we consider the finite field theta correspondence between principal 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 Zhiwei Yun, we geometrized the whole picture. Consequently, we obtained a relation between the Springer correspondence and theta correspondence.
Speaker: Sean Howe, University of Utah Title: Mini-course on the commutative algebra of modularity. Part 1: Galois representations and Hecke algebras Abstract: The goal of this course is to advertise some deep interactions between commutative algebra, number theory, and representation theory, especially to graduate students in these different areas. Roughly, the first two talks will introduce the Galois representations on Hecke algebras attached to modular forms and the basic deformation theory of Galois representations. They will also give some context describing how these objects arise in the Wiles/Taylor-Wiles proof of Fermat's Last Theorem via the Shimura-Taniyama-Weil conjecture and in more general modularity theorems. The third talk will discuss Wiles' numerical criterion for a map of rings to be an isomorphism of complete intersections, and the third talk will discuss a recent generalization of the numerical criterion due to Iyengar, Khare, and Manning.
Speaker: Sean Howe, University of Utah Title: Mini-course on the commutative algebra of modularity. Part 2: Deforming Galois representations Abstract: see above
Speaker: Qixian Zhao, University of Utah Title: The non-integral Kazhdan-Lusztig problem for Whittaker modules Abstract: Whittaker modules are Lie algebra representations motivated by the study of Whittaker functionals in group representations. Although Whittaker functionals considered in the group context are usually non-degenerate, degenerate Whittaker modules of complex semisimple Lie algebras have interesting connections with several important categories of Lie algebra representations such as the ordinary category O and its parabolic version. We are interested in the problem of finding composition factors of cyclic modules --modules generated by a single vector analogous to a Whittaker functional. This problem was resolved by Milicic-Soergel and Romanov for integral infinitesimal characters. In this talk, I will explain the background of the problem and how the integral results can be generalized to arbitrary infinitesimal characters. I will discuss possible implications of our result and techniques for Kazhdan-Lusztig problems in other categories of representations.
Speaker: Sean Howe, University of Utah Title: Mini-course on the commutative algebra of modularity. Part 3: Deforming Galois representations (continued) Abstract: see above
Speaker: Srikanth Iyengar, University of Utah Title: Mini-course on the commutative algebra of modularity. Part 4: The classical numerical criterion. Abstract: see above
Speaker: Srikanth Iyengar, University of Utah Title: Mini-course on the commutative algebra of modularity. Part 5: A generalization to positive defect. Abstract: see above
No seminar - Spring break!
Speaker: Naomi Sweeting, Harvard Title: Tate Classes and Endoscopy for GSp4 Abstract: Weissauer proved using the theory of endoscopy that the Galois representations associated to classical modular forms of weight two appear in the middle cohomology of both a modular curve and a Siegel modular threefold. Correspondingly, there are large families of Tate classes on the product of these two Shimura varieties, and it is natural to ask whether one can construct algebraic cycles giving rise to these Tate classes. It turns out that a natural algebraic cycle generates some, but not all, of the Tate classes: to be precise, it generates exactly the Tate classes which are associated to generic members of the endoscopic L-packets on GSp4. In the non-generic case, one can at least show that all the Tate classes arise from Hodge cycles. I'll explain these results and sketch their proofs, which rely on the theta correspondence.
Speaker: Manish Patnaik, University of Alberta Title: Whittaker functions on metaplectic covers and Quantum Groups at Roots of Unity Abstract: (Joint work with Valentin Buciumas). We describe the space of Whittaker functions on the metaplectic cover of a p-adic group G as a module over certain Hecke algebras of G. This object has been studied for some time, starting with the works of G. Savin and culminating in the recent work of E.Karasiewicz, F.Gao, and N.Gurevich. What we explain is how the Whittaker module can be expressed in terms of the quantum group at a root of unity attached to the Langlands dual group of G. To do this, we build on the ideas of Lascoux-Leclerc-Thibon and use certain metaplectic Demazure-Lusztig operators to develop a Gauss-sum twisted Kazhdan-Lusztig theory for the Whittaker spaces in question. As an application, we deduce a 'geometric' Casselman-Shalika formula for metaplectic covers, conjectured in a slightly related form by S.Lysenko, and which was one of the main motivations for this work.
Speaker: Andrew Kobin, Emory University Title: Categorifying zeta and L-functions Abstract: Zeta and L-functions are ubiquitous in modern number theory. While some work in the past has brought homotopical methods into the theory of zeta functions, there is in fact a lesser-known zeta function that is native to homotopy theory. Namely, every suitably finite decomposition space (aka 2-Segal space) admits an abstract zeta function as an element of its incidence algebra. In this talk, I will show how many 'classical' zeta functions in number theory and algebraic geometry can be realized in this homotopical framework. I will also discuss work in progress towards a categorification of motivic zeta and L-functions.
Speaker: Matteo Tamiozzo Title: The Hodge-Tate period map, semismallness and Jacquet-Langlands functoriality Abstract:I will introduce Scholze's Hodge-Tate period map and explain why it behaves similarly to semismall maps in complex geometry. I will then describe how the fibres of the Hodge-Tate period map for quaternionic Shimura varieties encode the Jacquet-Langlands correspondence. Finally, I will mention how these two properties can be used to study the cohomology with torsion coefficients of quaternionic Shimura varieties. This talk is partly based on joint work with Ana Caraiani.
Speaker: Ellen Eischen Title: Algebraic and p-adic aspects of L-functions, with a view toward Spin L-functions for GSp_6 Abstract: I will discuss recent developments and ongoing work for algebraic and p-adic aspects of L-functions. Interest in p-adic properties of values of L-functions originated with Kummer’s study of congruences between values of the Riemann zeta function at negative odd integers, as part of his attempt to understand class numbers of cyclotomic extensions. After presenting an approach to studying analogous congruences for more general classes of L-functions, I will conclude by introducing ongoing joint work of G. Rosso, S. Shah, and myself (concerning Spin L-functions for GSp 6). I will explain how this work fit into the context of earlier developments. All who are curious about this topic are welcome at this talk, even without prior experience with p-adic L-functions or Spin L-functions.
Speaker: Pol van Hoften, Stanford University Title: Hecke orbits on Shimura varieties of Hodge type. Abstract: Oort conjectured in 1995 that isogeny classes in the moduli space A_g of principally polarised abelian varieties in characteristic p are Zariski dense in the Newton strata containing them. There is a straightforward generalisation of this conjecture to the special fibres of Shimura varieties of Hodge type, and in this talk, I will present a proof of this conjecture. I will focus on the case of A_g since most of the new ideas can already be explained in this special case. This is joint work with Marco D'Addezio.
Speaker: Peter Wear, University of Utah Title: The conjugate uniformization via 1-motives Abstract: I will talk about a uniformization result for certain p-divisible groups and semi-abelian varieties. Much of the time will be spent talking about what uniformization means, what p-divisible groups are (and why you might care about them), and drawing intuition from the story over the complex numbers. This is joint work with Sean Howe and Jackson Morrow, and builds off of work of Iovita-Morrow-Zaharescu.
Speaker: Dougal Davis, University of Edinburgh Title: Mixed Hodge modules and real groups Abstract: In this talk, I will explain an ongoing program that aims to prove interesting facts about representations of real reductive groups using mixed Hodge modules on flag varieties. The theory of mixed Hodge modules provides refined structures, of deep geometric origin, on the twisted D-modules corresponding to Harish-Chandra modules under Beilinson-Bernstein localisation, which are expected to control various aspects of the representation theory. I will sketch some results and conjectures in this direction, including a key motivating conjecture of Schmid and Vilonen, and original results concerning both the Hodge modules of interest and links between Hodge theory and representation theory. Time permitting, I will explain how our results can be used to give a new proof of the key signature character formula in Adams, van Leeuwen, Trapa and Vogan's algorithm for the unitary dual. This is joint work with Kari Vilonen.
Speaker: Gal Porat, University of Chicago Title: Overconvergence of étale (φ,Γ)-modules in families Abstract: In recent years, there has been growing interest in realizing the collection of Langlands parameters in various settings as a moduli space with a geometric structure. In particular, in the p-adic Langlands program, this space should come in two different forms of moduli spaces of (φ,Γ)-modules: there is the "Banach" stack (also called the Emerton-Gee stack), and the "analytic" stack. In this talk, I will present a proof of a recent conjecture of Emerton, Gee and Hellmann concerning the overconvergence of étale (φ,Γ)-modules in families, which gives a link between the two different moduli spaces.
No seminar - Fall break!
Speaker: Jackson Morrow, UC Berkeley Title: Boundedness of hyperbolic varieties Abstract: Let $C_1$, $C_2$ be smooth projective curves over an algebraically closed field $K$ of characteristic zero. What is the behavior of the set of non-constant maps $C_1 \to C_2$? Is it infinite, finite, or empty? It turns out that the answer to this question is determined by an invariant of curves called the genus. In particular, if $C_2$ has genus $g(C_2)\geq 2$ (i.e., $C_2$ is hyperbolic), then there are only finitely many non-constant morphisms $C_1 \to C_2$ where $C_1$ is any curve, and moreover, the degree of any map $C_1 \to C_2$ is bounded linearly in $g(C_1)$ by the Riemann--Hurwitz formula. In this talk, I will explain the above story and discuss a higher dimensional generalization of this result. To this end, I will describe the conjectures of Demailly and Lang which predict a relationship between the geometry of varieties, topological properties of Hom-schemes, and the behavior of rational points on varieties. To conclude, I will sketch a proof of a variant of these conjectures, which roughly says that if $X/K$ is a hyperbolic variety, then for every smooth projective curve $C/K$ of genus $g(C)\geq 0$, the degree of any map $C\to X$ is bounded uniformly in $g(C)$.
Speaker: Daniel Gulotta, MSRI & Boston University
Title: Vanishing of the cohomology of Shimura varieties at unipotent level
Abstract: The Langlands correspondence relates automorphic forms and Galois
representations -- for example, the modular form η(z)^2 η(11z)^2 and
the Tate module of the elliptic curve y^2 + y = x^3 - x^2 - 10x - 20 are
related in the sense that they have the same L-function. The p-adic
Langlands program aims to interpolate the Langlands correspondence in
p-adic families. In this setting, the role of automorphic forms is
played by the completed cohomology groups defined by Emerton.
Calegari and Emerton have conjectured that the completed cohomology
vanishes above a certain degree, often denoted q_0. In the case of
Shimura varieties of Hodge type, Scholze has proved the conjecture for
compactly supported completed cohomology. We give a strengthening of
Scholze's result under the additional assumption that the group becomes
split over Q_p. More specifically, we show that the compactly supported
cohomology vanishes not just at full infinite level at p, but also at
unipotent level at p.
We also give an application of the above result to Galois
representations. For any totally real or CM field F, Scholze has
constructed Galois representations associated with torsion classes in
the cohomology of locally symmetric spaces for GL_n(F). We show that,
if F splits completely at the relevant prime, then the nilpotent ideal
appearing in the construction can be eliminated.
This talk is based on joint work with Ana Caraiani and Christian
Johansson and on joint work with Ana Caraiani, Chi-Yun Hsu, Christian
Johansson, Lucia Mocz, Emanuel Reinecke, and Sheng-Chi Shih.
Speaker: Raúl Gómez, Universidad Autónoma de Nuevo León
Title: A Stone-von Neumann equivalence of categories for smooth representations of the Heisenberg group.
Abstract: The classical Stone-von Neuman theorem relates the irreducible unitary representations of the Heisenberg group $H_{n}$ to non-trivial unitary characters of its center $Z$, and plays a crucial role in the construction of the oscillator representation for the metaplectic group.
In this talk we will discuss how we can extend these ideas to non-unitary and non-irreducible representations,
thereby obtaining an equivalence of categories between certain representations of $Z$ and those of $H_{n}$ .
This is joint work with Dmitry Gourevitch and Siddhartha Sahi.
Speaker: Spencer Leslie, Duke Title: Endoscopy for symmetric varieties Abstract: Relative trace formulas are central tools in the study of relative functoriality. In many cases of interest, basic stability problems have not been addressed. In this talk, I discuss a theory of endoscopy in the context of symmetric varieties with the global goal of stabilizing the associated relative trace formula. I outline how, using the dual group of the symmetric variety, one can give a good notion of endoscopic symmetric variety and conjecture a matching of relative orbital integrals in order to stabilize the relative trace formula. Time permitting, I will explain my proof of these conjectures in the case of unitary Friedberg-Jacquet periods.
Speaker: Anna Romanov, University of New South Wales Title: Costandard Whittaker modules and contravariant pairings Abstract: I'll discuss recent work with Adam Brown (IST Austria) in which we propose a new definition of costandard Whittaker modules for a complex semisimple Lie algebra using contravariant pairings between Whittaker modules and Verma modules. With these costandard objects, blocks of Milicic-Soergel's Whittaker category have the structure of highest weight categories. This allows us to establish a BGG reciprocity theorem for Whittaker modules. Our costandard objects also give an algebraic characterization of the global sections of costandard twisted Harish-Chandra sheaves on the flag variety.
Speaker: Rebecca Bellovin, University of Glasgow Title: Modularity of trianguline Galois representations Abstract: The Fontaine-Mazur conjecture (proved by Kisin and Emerton) says that (under certain technical hypotheses) a Galois representation \rho:Gal_Q\rightarrow GL_2(\overline{Q_p)$ is modular if it is unramified outside finitely many places and de Rham at p. I will discuss an analogous modularity result for Galois representations \rho:Gal_Q\rightarrow GL_2(L) which are unramified away from p and trianguline at p, when L is instead a non-archimedean local field of characteristic p. More precisely, I will show that such Galois representations are attached to points on the extended eigencurve.q
Speaker: Jeremy Booher, University of Canterbury Title: G-Valued Crystalline Deformation Rings in the Fontaine-Laffaille Range Abstract: I will speak about Galois representations valued in groups besides GL_n and why one should care about them. In particular, I will discuss a new approach to proving that crystalline deformation rings in the Fontaine-Laffaille range are formally smooth. This is joint work with Brandon Levin.
Speaker: Heidi Goodson, Brooklyn College, CUNY Title: Sato-Tate Groups in Dimension Greater than 3 Abstract: The focus of this talk is on Sato-Tate groups of abelian varieties -- compact groups predicted to determine the limiting distributions of local zeta functions. In recent years, complete classifications of Sato-Tate groups in dimensions 1, 2, and 3 have been given, but there are obstacles to providing classifications in higher dimensions. In this talk, I will describe my recent work on families of higher dimensional Jacobian varieties. This work is partly joint with Melissa Emory.
Speaker: Juan Esteban Rodriguez Camargo Title: Locally analytic completed cohomology of Shimura varieties Abstract: In this talk I explain how the work of Lue Pan on the locally analytic completed cohomology of the modular curve extends to arbitrary Shimura varieties. As a first application, one can deduce some vanishing for the rational completed cohomology proving a rational version of the Calegari-Emerton conjectures.
Speaker: Christian Klevdal, UNIST Title: Strong independence of \ell for Shimura varieties Abstract: Work of Deligne on the Weil conjectures shows that for Galois representations of number fields appearing in the \ell-adic cohomology of algebraic varieties, the characteristic polynomial of Frobenius elements are rational and independent of \ell. Recent work of Kisin-Zhou has shown a stronger independence of \ell result for Galois representations coming from abelian varieties. I will discuss their work, and ongoing work (joint with Stefan Patrikis) to prove strong independence of \ell for Galois representations coming from Shimura varieties not considered by Kisin-Zhou (e.g. Shimura varieties for exceptional groups). The main difference in our approach compared to that of Kisin-Zhou is an application of Margulis' superrigidity theorem in place of Serre-Tate theory.
No seminar - Spring break!
Speaker: Wanlin Li, CRM Montreal Title: Ceresa Cycle and Hyperelliptic Curves Abstract: The Ceresa cycle is an algebraic cycle obtained from curves that is algebraically trivial for hyperelliptic curves and non-trivial for a very general non-hyperelliptic curve. Via cycle class maps, the Ceresa cycle gives rise to various cohomology classes. In this talk, we will discuss the relation between the vanishing of these Ceresa classes and the curve being hyperelliptic. The talk is based on joint work with Bisogno-Litt-Srinivasan, Corey-Ellenberg and ongoing work with Corey.
Speaker: Robin Zhang, Columbia University Title: Modular Gelfand pairs and multiplicity-free triples Abstract: The classical theory of Gelfand pairs and its generalizations over the complex numbers has many applications to number theory and automorphic forms, such as the uniqueness of Whittaker models and the non-vanishing of the central value of a triple product L-function. With an eye towards similar applications in the modular setting, this talk presents an extension of the classical theory to representations over algebraically closed fields with arbitrary characteristic.
Speaker: Bryden Cais, University of Arizona Title: Iwasawa theory of class group schemes Abstract: Iwasawa theory is the study of the growth of arithmetic invariants in p-adic Lie towers of global fields. Beginning with Iwasawa's seminal work in which he proved that the p-primary part of the class group in Z_p-extensions of number fields grows with striking and unexpected regularity, Iwasawa theory has become a central strand of modern number theory and arithmetic geometry. While the theory has traditionally focused on towers of number fields, the function field setting has been studied by Crew, Katz, Mazur-Wiles, and others, and has important applications to the theory of p-adic modular forms. This talk will introduce an exciting new kind of p-adic Iwasawa theory for towers of function fields over finite fields of characteristic p.
Speaker: Alexis Aumonier Title: An h-principle for complements of discriminants Abstract: In classical algebraic geometry, discriminants appear naturally in various moduli spaces as the loci parametrising degenerate objects. The motivating example for this talk is the locus of singular sections of a line bundle on a smooth projective complex variety, the complement of which is a moduli space of smooth hypersurfaces. I will present an approach to studying the homology of such moduli spaces of non-singular algebraic sections via algebro-topological tools. The main idea is to prove an "h-principle" which translates the problem into a purely homotopical one. I shall explain how to talk effectively about singular sections of vector bundles and what an h-principle is. To demonstrate the usefulness of homotopical methods, and using a bit of rational homotopy theory, we will prove together a homological stability result for moduli spaces of smooth hypersurfaces of increasing degree.
Speaker: Bianca Thompson Title: Periodic points in towers of finite fields Abstract: Periodic points are points that live in a cycle upon iteration of a function. Suppose we fix a rational map over F_p, we can then ask what proportion of the points are periodic? We can create extensions of F_p by looking at F_{p^n} and ask the same question. As we allow n to go to infinity, does this density converge? This turns out to be hard to answer in general because iteration over rational maps don't tend to have a lot of structure, but for some special families of maps that have a group structure upon iteration can be explored. This talk will explore density results for towers of finite fields.
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.
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.
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.
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.
No seminar - Fall break!
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.
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.
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.)
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.
No seminar - Thanksgiving week!
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Speaker: Aaron Pollack, UCSD Title: TBA Abstract: TBA
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.