Representation Theory and Number Theory Seminar, 2018-2019

Spring 2019: Friday, 3:30-4:30 PM, LCB 215

Monday January 14, 3:00-3:50pm, LCB 215: (note unusual day and time)
Speaker: Sean Howe, Stanford University
Title: A unipotent circle action on p-adic modular forms
Abstract: The horizontal translation action of the real line on the complex upper half plane descends to an action of the circle group S^1 on the "unstable locus", or image of Im \tau > 1, in the complex modular curve. In this talk, we explain an analogous action of the open p-adic unit disk centered at 1 on the Katz moduli space, a p-adic analytic covering space of the unstable locus on the p-adic modular curve whose ring of functions can be constructed by p-adically interpolating the coefficients of classical modular forms. The analogy is richer than one might first expect, and leads to new perspectives on classical notions in the p-adic theory of modular curves and modular forms such as Dwork's equation \tau=\log q and Hida's space of ordinary p-adic modular forms, with implications for the p-adic representation theory of GL_2 and p-adic Galois groups.

February 22:
Speaker: Kyo Nishiyama, Aoyama Gakuin and MIT

March 8:
Speaker: Wei Ho, University of Michigan

March 22:
Speaker: Eric Sommers, University of Massachusetts

April 5:
Speaker: Philip Kutzko, University of Iowa

April 12:
Speaker: Yunqing Tang, Princeton University

Fall 2018: Friday, 3:30-4:30 PM, LCB 215

August 24:
Title: Organizational meeting, 4:10-4:30

August 31:

September 7:
Speaker: Tony Feng, Stanford University
Title: The Artin-Tate pairing on the Brauer group of a surface
Abstract: There is a canonical pairing on the Brauer group of a surface over a finite field, and an old conjecture of Tate's predicts that this pairing is alternating. In this talk I will present a resolution to Tate’s conjecture. The key new ingredient is a circle of ideas originating in algebraic topology, centered around the Steenrod operations. The talk will advertise these new tools (while assuming minimal background in algebraic topology).

September 14:
Speaker: Adam Brown, The University of utah
Title: Arakawa-Suzuki functors for Whittaker modules
Abstract: The category of Whittaker modules, as formulated by Milicic and Soergel, is a category of Lie algebra representations which generalizes other well-studied categories of representations, such as the Bernstein-Gelfand-Gelfand category O. In this talk we will construct a family of exact functors from the category of Whittaker modules to the category of finite-dimensional graded affine Hecke algebra modules, for type A_n. These algebraically defined functors provide us with a representation theoretic analogue of certain geometric relationships, observed independently by Zelevinsky and Lusztig, between the flag variety and the variety of graded nilpotent classes. Using this geometric perspective and the corresponding Kazhdan-Lusztig conjectures for each category, we will prove that these functors map simple modules to simple modules (or zero). Moreover, we will see that each simple module for the graded affine Hecke algebra can be realized as the image of a simple Whittaker module.

September 21:
Speaker: Gil Moss, The University of Utah
Title: Characterizing the mod-\ell local Langlands correspondence for GL(n) using nilpotent gamma factors.
Abstract: In the complex representation theory of GL_n of a p-adic field, Rankin-Selberg gamma factors uniquely determine the supercuspidal support of a generic representation. This is known as a "converse theorem." By using a modified gamma factor that accounts for nilpotent elements in the mod-\ell Bernstein center, we can establish a converse theorem in the mod-\ell setting, addressing a conjecture of Vigneras. With an appropriate modification of gamma factors on the Galois side, this would give a characterization of Vigneras' mod-\ell local Langlands correspondence.

September 28:
Speaker: Kei Yuen Chan, University of Georgia
Title: Ext-branching law for reductive p-adic groups
Abstract: The local Gan-Gross-Prasad conjecture predicts the branching law for some tempered representations of classical reductive groups over real and p-adic fields. Dipendra Prasad suggests homological variations of those branching problems and in particular conjectures that there is no higher extensions between a generic representation of GL(n+1,F) and a generic one of GL(n,F). In a joint work with Gordan Savin, we give a positive answer to the conjecture. I shall report the work and other relevant results in the talk.

October 5:
Speaker: Stefan Patrikis, The University of Utah
Title: Lifting Galois Representations
Abstract: This talk will be an amply-motivated introduction to some of the different phenomena one encounters when trying to lift mod p representations of the absolute Galois group of a number field to ("geometric") p-adic representations. I will emphasize examples rather than general theory. Time permitting, I'll describe some recent joint work with N. Fakhruddin and C. Khare.

October 12: No Seminar (fall break)

October 19:
Speaker: Sabine Lang, The University of Utah
Title: Restriction of the oscillator representation using dual pairs
Abstract: In branching problems, the Ext functors vanish if the restriction of a representation is projective. We will study two restrictions of the oscillator representation of a real symplectic group: the restriction to a smaller symplectic subgroup and the restriction to an orthogonal subgroup. We will present conditions on the relative sizes of these groups under which these restrictions are projective. The main tools used in our proof are the theory of dual pairs and the theta correspondence, first introduced by Howe.

October 26:

November 2:
Speaker: Christian Klevdal, The University of Utah
Title: Recognizing Galois representations of K3 surfaces
Abstract: This talk centers around the question of whether a `cohomological object' (e.g. a Hodge structure or Galois representation) appears in the cohomology of an algebraic variety. For varieties over the complex numbers, Riemann's theorem that weight one Hodge structures appear in the cohomology of abelian varieties and the surjectivity of the period map for K3 surfaces provide positive answers. Over number fields, this is a very hard question to answer. However using certain motivic conjectures, recent work of Patrikis, Voloch and Zarhin provides an analogue of Riemann's theorem for Galois representations. We will then discuss the analogue of surjectivity of the K3 period map for Galois representations.

November 9:

November 16:
Speaker: Claus Sorensen, University of California, San Diego
Title: The local Langlands correspondence on eigenvarieties
Abstract: In this talk we will consider eigenvarieties for definite unitary groups, which are rigid analytic spaces parametrizing systems of Hecke eigenvalues appearing in spaces of finite slope p-adic modular forms. These varieties carry a natural coherent sheaf whose fibers interpolate the local Langlands correspondence for GL(n) at places away from p. To show this we consider the action on the sheaf itself of certain Bernstein center elements which appeared in works of Chenevier, and more recently in Scholze's proof of local Langlands. This bears a strong resemblance to "local Langlands in families" as formulated by Emerton, Helm, and Moss. In order to control the generic constituents of the fibers we make critical use of a newly established genericity criterion of Chan and Savin. This is a report on joint work with Christian Johansson and James Newton.

November 23: No Seminar (Thanksgiving break)

Tuesday December 4 (note unusual time), 3:30-4:30pm LCB 222:
Speaker: Nivedita Bhaskhar, UCLA
Title: Reduced Whitehead groups of algebras over p-adic curves
Abstract: Any central simple algebra A over a field K is a form of a matrix algebra. Further A/K comes equipped with a reduced norm map which is obtained by twisting the determinant function. Every element in the commutator subgroup [A*, A*] has reduced norm 1 and hence lies in SL_1(A), the group of reduced norm one elements of A. Whether the reverse inclusion holds was formulated as a question in 1943 by Tannaka and Artin in terms of the triviality of the reduced Whitehead group SK_1(A) := SL_1(A)/[A*,A*].
Platonov negatively settled the Tannaka-Artin question by giving a counter example over a cohomological dimension (cd) 4 base field. In the same paper however, the triviality of SK_1(A) was shown for all algebras over cd at most 2 fields. In this talk, we investigate the situation for l-torsion algebras over a class of cd 3 fields of some arithmetic flavour, namely function fields of p-adic curves where l is any prime not equal to p. We partially answer a question of Suslin by proving the triviality of the reduced Whitehead group for these algebras. The proof relies on the techniques of patching as developed by Harbater-Hartmann-Krashen and exploits the arithmetic of these fields.

December 7:
Speaker: Moshe Adrian, Queens College, CUNY
Title: On the sections of the Weyl group
Abstract: Let G be a connected reductive group over an algebraically closed field and W the Weyl group of a split maximal torus. In 1966, Tits defined a representative n_\alpha, in G, for each simple reflection s_\alpha in the Weyl group. These representatives satisfy the braid relations, and therefore define a "cross section" of W. This section has been used extensively ever since, in many works. On the other hand, there exist sections of the Weyl group that are not Tits' section. For example, in some cases (for example type B_n adjoint and even simpler, GL_2), W embeds into G, but the Tits section does not realize such an embedding. More importantly for our purposes (in the theory of p-adic groups), Tits' section does not realize a splitting of the Kottwitz homomorphism. Motivated by this question about the Kottwitz homomorphism, we have computed and classified all sections of the Weyl group that satisfy the braid relations, in types A through G. It turns out that these sections form a partially ordered set, with a unique maximal element. We will describe this poset, which is interesting in and of its own right. The maximal element in this poset is the "most homomorphic" of all of the sections. This "maximal" section answers our question from p-adic groups: it splits the Kottwitz homomorphism. Moreover, essentially by construction, it always realizes an embedding of W in G, whenever such an embedding exists.