Representation Theory and Number Theory Seminar, 2019-2020

Spring 2020: Monday, 3:00-4:00 PM, LCB 222

January 6:
Speaker: Peter Wear, UC San Diego
Title: Perfectoid covers of abelian varieties and the weight-monodromy conjecture
Abstract: The theory of perfectoid spaces was initially developed by Scholze to prove new cases of the weight-monodromy conjecture. He constructed perfectoid covers of toric varieties that allowed him to translate results from characteristic p to characteristic 0. We will give an overview of Scholze’s method, then explain how to use an analogous construction for abelian varieties to prove the weight-monodromy conjecture for complete intersections in abelian varieties.

January 27:
Speaker: Adam Brown, IST Austria
Title: Contravariant forms on Whittaker modules
Abstract: Contravariant forms are symmetric bilinear forms on modules over a semisimple complex Lie algebra which satisfy an invariance property with respect to the action of the Lie algebra. Certain well-studied modules, such as highest weight modules, admit a unique contravariant form up to scaling. In this talk I will outline joint work with Anna Romanov, classifying contravariant forms on irreducible Whittaker modules. Specifically, we will discuss the non-uniqueness of contravariant forms on irreducible Whittaker modules and the resulting implications for potential algebraic notions of duality in the category of Whittaker modules.

February 3:
Speaker: Mathilde Gerbelli-Gauthier, University of Chicago
Title: Cohomology of Arithmetic Groups and Endoscopy
Abstract: How fast do Betti numbers grow in a congruence tower of compact arithmetic manifolds? The dimension of the middle degree of cohomology is proportional to the volume of the manifold, but away from the middle the growth is known to be sub-linear in the volume. I will explain how automorphic representations and the phenomenon of endoscopy provide a framework to understand and quantify this slow growth. Specifically, I will discuss how to obtain explicit bounds in the case of unitary groups using Arthur’s stable trace formula. This is joint work in progress with Simon Marshall.

February 10:
Speaker: Patrick Daniels, University of Maryland
Title: A Tannakian framework for G-displays and Rapoport-Zink spaces
Abstract: Formal p-divisible groups over a p-adic ring are equivalent to linear algebraic objects called displays. In this talk, we present a Tannakian framework for group-theoretic analogs of displays, which correspond to formal p-divisible groups with additional structures. We use these G-displays to define a Rapoport-Zink functor which generalizes the purely group-theoretic one of Bueltel and Pappas, and we show that this functor recovers the classical one of Rapoport and Zink in the unramified EL-type situation. Representability of this functor in general would provide integral models for local Shimura varieties.

February 24 3-4pm LCB 323: (Note unusual room)
Speaker: Kevin Childers, University of Utah
Title: The density of automorphic points in Galois deformation spaces
Abstract: Serre's conjecture (a theorem of Khare and Wintenberger) states that an absolutely irreducible 2-dimensional Galois representation of the rational numbers valued in a finite field has a modular lift if it is "odd," in the sense that the determinant of the image of complex conjugation is negative one. In fact, modular lifts are dense in the deformation space of such a representation. In this talk we will explore the density of modular (or automorphic) points in deformation spaces of other Galois representations.

Friday February 28 3:30-4:30 LCB 215: (Note unusual day, time, and room)
Speaker: Bianca Viray, University of Washington
Title: Quadratic points on intersections of quadrics
Abstract: The theorems of Springer and of Amer and Brumer together imply that an intersection of 2 quadrics has a k-rational point if and only if it has a point over an odd degree extension. In other words, an intersection of quadrics has a k-rational point if and only if there is a 0-cycle of degree 1. In joint work with B. Creutz, we consider whether anything stronger can be said in the case when there is no 0-cycle of degree 1, in particular what conditions imply that there is a 0-cycle of degree 2.

April 6, 3:30-4:30pm: (Remote Seminar)
Speaker: Jeffrey Manning, UCLA
Title: The Wiles defect for Hecke algebras that are not complete intersections
Abstract: In his work on modularity theorems, Wiles proved a numerical criterion for a map of rings R->T to be an isomorphism of complete intersections. He used this to show that certain deformation rings and Hecke algebras associated to a mod p Galois representation at non-minimal level were isomorphic and complete intersections, provided the same was true at minimal level. In addition to proving modularity theorems, this numerical criterion also implies a connection between the order of a certain Selmer group and a special value of an L-function. In this talk I will consider the case of a Hecke algebra acting on the cohomology a Shimura curve associated to a quaternion algebra. In this case, one has an analogous map of ring R->T which is known to be an isomorphism, but in many cases the rings R and T fail to be complete intersections. This means that Wiles' numerical criterion will fail to hold. I will describe a method for precisely computing the extent to which the numerical criterion fails (i.e. the 'Wiles defect"), which will turn out to be determined entirely by local information at the primes dividing the discriminant of the quaternion algebra. This is joint work with Gebhard Bockle and Chandrashekhar Khare.

Fall 2019: Monday, 3:00-4:00 PM, LCB 215

August 29:
Title: Brief Organizational Meeting

August 26:
Speaker: Stephen Miller, Rutgers
Title: An update on the sphere packing problem
Abstract: I will discuss recent work on the optimal arrangement of points in euclidean space. In addition to the solution to the sphere packing problem in dimensions 8 and 24 from 2016, the "Universal Optimality" conjecture has now been established in these dimensions as well. This shows that E8 and the Leech lattice minimize energy for any completely monotonic function of distance-squared. Previously there were no proofs in any dimension > 1 that a particular configuration minimizes energy for any nonzero example of such a potential. Beyond giving a new proof of the sphere packing results, Universal Optimality also gives information about long-range interactions. The techniques reduce the problem to a question in single-variable calculus, which is ultimately solved using modular forms and methods introduced by Viazovska. This will be a colloquium-style talk. (Joint work with Henry Cohn, Abhinav Kumar, Danylo Radchenko, and Maryna Viazovska)

September 9:
Speaker: Gil Moss, Utah
Title: The Whittaker model of Serre's universal unramified module
Abstract: Let F be a nonarchimedean local field with residue field of order q, and let l be a prime different from p. The mod-l representation theory of the F-points of reductive groups can exhibit very different behavior than the complex theory. We will discuss the interplay between Whittaker models and systems of spherical Hecke eigenvalues for representations of GL_n(F). The results can be applied toward an open conjecture in the theory of mod-l automorphic forms.

September 16:
Speaker: Ronno Das, University of Chicago
Title: Points and lines on cubic surfaces
Abstract: The Cayley-Salmon theorem states that every smooth cubic surface in CP^3 has exactly 27 lines. Their proof is that marking a line on each cubic surface produces a 27-sheeted cover of the moduli space M of smooth cubic surfaces. Similarly, marking a point produces a 'universal family' of cubic surfaces over M. One difficulty in understanding these spaces is that they are complements of incredibly singular hypersurfaces. In this talk I will explain how to compute the rational cohomology of these spaces. I'll also explain how these purely topological theorems have (via the machinery of the Weil Conjectures) purely arithmetic consequences: the average smooth cubic surface over a finite field F_q contains 1 line and q^2 + q + 1 points.

September 23:
Speaker: Stefan Patrikis, Utah
Title: Lifting irreducible Galois representations
Abstract: This will be a largely expository talk on "odd" Galois representations. I will begin with motivation from the theory of automorphic forms and representation theory of semisimple Lie groups, and then, with an emphasis on examples, I will discuss some recent results with Fakhruddin and Khare on lifting odd irreducible mod p Galois representations to geometric p-adic representations.

September 30:
Speaker: Sean Howe, Utah
Title: A(nother) conjecture about zeta functions, or, "it's zeta functions all the way down."
Abstract: We conjecture a unification of arithmetic and motivic/topological statistics over finite fields through a natural analytic topology on the ring of zeta functions. A key step will be to explain exactly what it means to evaluate the zeta function of a zeta function at a zeta function. This is joint work with Margaret Bilu.

October 21:
Speaker: Gordan Savin, Utah
Title: Exceptional Siegel-Weil formula
Abstract: Joint work with W.T. Gan

November 4:
Speaker: Petar Bakic, Utah
Title: The local theta correspondence
Abstract: The global theta correspondence is a standard tool in the study of automorphic forms. To understand it, one would like to have a description of its local variant, i.e. the local theta (aka Howe) correspondence. In the first part of the talk, I will go over the basic setup and the most important general results concerning theta correspondence. After that, I will discuss the recent work with M. Hanzer in which we give an explicit description of the local theta correspondence for dual pairs of Type I.

November 18:
Speaker: Andrea Dotto, University of Chicago
Title: Functoriality for Serre weights
Abstract: By work of Gee--Geraghty and myself, one can transfer Serre weights from the maximal compact subgroup of an inner form D* of GL(n) to a maximal compact subgroup of GL(n). Because of the congruence properties of the Jacquet--Langlands correspondence this transfer is compatible with the Breuil--Mezard formalism, which allows one to extend the Serre weight conjectures to D* (at least for a tame and generic residual representation). This talk aims to explain all of the above and to discuss a possible generalization to inner forms of unramified groups.

November 22, 3:30-4:30pm, LCB 225:
Speaker: Zheng Liu, UC Santa Barbara
Title: Doubling archimedean zeta integrals for symplectic and unitary groups
Abstract: In order to verify the compatibility between the conjecture of Coates--Perrin-Riou and the interpolation results of the p-adic L-functions constructed by using the doubling method, a doubling archimedean zeta integral needs to be calculated for holomorphic discrete series. When the holomorphic discrete series is of scalar weight, it has been done by Bocherer--Schmidt and Shimura. I will explain a way to compute this archimedean zeta integral for general vector weights by using the theory of theta correspondence.

November 25:
Speaker: Allen Moy, Hong Kong University of Science and Technology
Title: Decompositions of Euler-Poincaré presentations and resolutions
Abstract: Work of Bezrukavnikov-Kazhdan-Varshavsky uses an equivariant system of trivial idempotents of Moy-Prasad groups to obtain an Euler-Poincaré presentation of the r-depth Bernstein projector. Bestvina-Savin, generalizing earlier work of Schneider-Sthuler, showed this system of Moy-Prasad groups, allows a resolution of a smooth representation generated by its depth r-vectors. We report on work in progress with Gordan Savin establishing a direct sum decomposition of the equivariant system and therefore of the Euler-Poincaré presentation and the resolution.