# 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.

Wednesday February 20, 4:00-5:00pm, LCB 215: (note unusual day and time)
Title: Hypergeometric decomposition of symmetric K3 quartic pencils
Abstract: In this talk, we will show the hypergeometric functions associated to five one-parameter deformations of Delsarte K3 quartic hypersurfaces in projective space. We compute all of their Picard–Fuchs differential equations; we count points using Gauss sums and rewrite this in terms of finite field hypergeometric sums; then we match up each differential equation to a factor of the zeta function, and we write this in terms of global L-functions. This computation gives a complete, explicit description of the motives for these pencils in terms of hypergeometric motives. This is joint work with Charles F Doran, Tyler L Kelly, Steven Sperber, John Voight, and Ursula Whitcher.

February 22:
Speaker: Kyo Nishiyama, Aoyama Gakuin and MIT
Title: Steinberg theory for symmetric pairs and generalized Robinson-Schensted correspondence for partial permutations
Abstract: Let G be a connected reductive group and B its Borel subgroup. Steinberg considered a double flag variety X = G/B \times G/B with diagonal G action, and got a correspondence between the Weyl group W and nilpotent orbits in Lie(G) in terms of moment maps. In type A, this correspondence amounts to the famous RS correspondence between permutations and pairs of standard Young tableaux. In this talk, we generalize it to get an RS correspondence for partial permutations. We also interprete it by using double flag varieties for symmetruc pairs, and get a further generalization to obtain a correspondence between partial permutations and signed Young diagrams parametrizing nilpotent orbits for a symmetric pair. This reveals an interesting interchange between geometry and combinatorics. The talk is based on the on-going joint work with Lucas Fresse at IECL (France).

March 8:
Speaker: Wei Ho, University of Michigan
Title: Integral points on elliptic curves
Abstract: Elliptic curves are fundamental and well-studied objects in arithmetic geometry. However, much is still not known about many basic properties, such as the number of rational points on a "random" elliptic curve. We will discuss some conjectures and theorems about this "arithmetic statistics" problem, and then show how they can be applied to answer a related question about the number of integral points on elliptic curves over Q. In particular, we show that the second moment (and the average) for the number of integral points on elliptic curves over Q is bounded (joint work with Levent Alpoge).

March 22:
Speaker: Eric Sommers, University of Massachusetts
Title: Generalized Bott-Samelson resolutions for Schubert varieties
Abstract: A Schubert variety is the closure of a B-orbit on the flag variety G/B of a reductive algebraic group G and is indexed by an element w of the Weyl group of G. In general these varieties are singular and some of the structure of their singularities can be understood via the Bott-Samelson resolutions and the resulting interplay between the intersection cohomology of the Schubert variety and the ordinary cohomology of the fibers of these resolutions. Bott-Samelson resolutions arise from any factorization of w into simple reflections and they are iterated P^1 bundles over P^1. This talk focuses on joint work with Jennifer Koonz, on generalized Bott-Samelson resolutions, which are iterated G/Q-bundles (for different parabolic subgroups Q of G). These resolutions also provide information about the singularities of Schubert varieties. Special cases have appeared in work of Wolper, Ryan, Polo, and Zelevinsky in the case where G is the general linear group. They also appear in other types in the work of Vanchinathan-Sankaran and Billey-Postnikov. After introducing these resolutions and some of their properties, we will suggest a possible best generalized resolution for a given Schubert variety. For these best resolutions, there appears to be a connection with work of Williamson on torsion in intersection cohomology. Finally, we'll discuss a computer program that calculates local intersection cohomology (i.e., Kazhdan-Lustztig polynomials) using these resolutions.

April 19:
Speaker: Yunqing Tang, Princeton University
Title: Reductions of abelian surfaces over global function fields
Abstract: For a non-isotrivial ordinary abelian surface $A$ over a global function field with everywhere good reduction, under mild assumptions, we prove that there are infinitely many places modulo which $A$ is geometrically isogenous to the product of two elliptic curves. This result can be viewed as a generalization of a theorem of Chai and Oort. This is joint work with Davesh Maulik and Ananth Shankar.

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

August 24:
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Title: Organizational meeting, 4:10-4:30
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August 31:
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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)
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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:
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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:
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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)
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Tuesday December 4 (note unusual time), 3:30-4:30pm LCB 222: