Schedule-at-a-Glance

30 PRIMA 2013 AbstractsThis talk is about a method, based on model theory, thatcan be used to get some uniform in p estimates for integralsover p-adic fields in the cases when it is hard to seedirectly how such integrals behave for different places p.One example of such a situation is orbital integrals onp-adic groups. Let G be a reductive p-adic group. It wasproved by Harish-Chandra that the orbital integrals, normalizedby the discriminant, are bounded (for a fixed testfunction). However, it is not easy to see how this boundbehaves if we let the p-adic field vary (for example, if thegroup G is defined over a number field F , and we considerthe family of groups G v = G(F v), as v runs over the set offinite places of F ), and how it varies for a family of testfunctions. Using a method based on model theory andmotivic integration, we prove that the bound on orbitalintegrals can be taken to be a fixed power (dependingon G) of the cardinality of the residue field. This statementhas an application to the recent work of S.-W. Shinand N. Templier on counting zeroes of L-functions. Thisproject is joint work with R. Cluckers and I. Halupczok.On the center of Lie algebrasMasoud KamgarpourUniversity of Queensland, Australiamasoud@uq.edu.auI will discuss the center of the universal enveloping algebraof four different kinds of Lie algebras:(1) Harish-Chandra’s description of the centre of a simplealgebra g.(2) Duflo-Kirillov’s description of the centre of an arbitraryfinite dimensional algebra.(3) Center of the algebra of jets g[[t]] and g((t)).(4) Feigin-Frenkel description of center of the vertex Liealgebra associated to g, via Langlands dual group.I will then define the notion of quasi-Verma modules forthe abovementioned vertex algebra, and stipulate how theFeigin-Frenkel Center should act on these modules.A homological study of Green polynomialsSyu KatoKyoto University, Japansyuchan@math.kyoto-u.ac.jpWe first explain how to categorify the Green/Kostka polynomialsof reductive groups. Then, we discuss how theyare related to the relevant orthogonality relation of somemodules of affine Hecke algebras and p-adic groups, andtheir consequences.On the structure of Selmer groupsMasato KuriharaKeio University, Japankurihara@math.keio.ac.jpAs a refinement of Iwasawa theory, which gives a refinedrelationship than the usual main conjecture betweenSelmer groups and p-adic L-functions, I describethe structure as an abelian group of the classical Selmergroup of an elliptic curve, using modular symbols. I alsotalk on Euler systems and Kolyvagin systems of Gausssum type, which give nontrivial elements in Galois cohomologygroups.On the first Betti number of hyperbolic arithmeticmanifoldsJianshu LiHong Kong University of Science and Technology, HongKong, Chinamatom@ust.hkIn the 1970’s Thurston conjectured that every hyperbolicmanifold admits a finite cover with non-zero first Bettinumber. In the arithmetic case, this property is relatedto the congruence subgroup problem for SO(n, 1). Forarbitrary dimension n there are two types of arithmetichyperbolic manifolds, which arise from either quadraticforms over totally real number fields, or skew-hermitianforms over certain quaternion algebras. In 1976 Millsonproved Thurston’s conjecture for the first type of arithmetichyperbolic manifolds using geometric methods. Forthe second type of manifolds, similar results were establishedin the 1990’s .In this talk we will briefly review this history, and discussan improved version of an old construction which leads tosome modest related results. We will also discuss somerelated open problems.On the elliptic part of the trace formula for˜Sp(2n)Wen-Wei LiChinese Academy of Sciences, Chinawwli@math.ac.cnIn this talk, I will introduce the stabilization of Arthur-Selberg trace formula for reductive groups and its applications,then proceed to its generalization to the metaplectictwofold cover ˜Sp(2n) of Sp(2n). Based on therecent progress on geometric transfer and the invarianttrace formula for metaplectic group, I will explain thestabilization of the elliptic terms in the trace formula of˜Sp(2n).Endoscopic classification of automorphic representationsof classical groupsChung Pang MokMcMaster University, Canadacpmok@math.mcmaster.caWe will give an exposition of the recent works on endoscopicclassification of automorphic representations forclassical groups. Some recent arithmetic applications ofthe endoscopic classification will be discussed.Twisted spectral transfer for real groupsPaul MezoCarleton University, Canadamezo@math.carleton.caKottwitz and Shelstad generalized the framework of endoscopyto include twisting by group automorphisms orcentral characters. This generalization contained conjecturalidentities between orbital integrals, constituting atransfer from functions on a group to functions on oneof its endoscopic groups. This geometric transfer has recentlybeen proven by Shelstad. Dual to geometric transferis spectral transfer, which is a collection identities betweencharacters of L-packets of a group and one of itsendoscopic groups. We show how some work of Bouaziz,Duflo and Shelstad may be adapted to the twisted endoscopyof real reductive groups in order to achieve spectraltransfer.Zelevinsky involution and l-adic cohomologyof Rapoport-Zink spacesYoichi MiedaKyoto University, Japanmieda@math.kyoto-u.ac.jp