Rutgers University at Newark
Algebra & Number Theory Seminar
Academic year 2011-2012

The seminar is meeting in Smith Hall, Room 204 (mathematics seminar room).
The seminar is organized by Zhengyu Mao and Yiannis Sakellaridis.

Format: This is mostly a learning seminar, with occasional invited talks. During the fall of 2011 we will be studying the theta correspondence and its applications to automorphic forms.

Click on the name of the speaker or scroll down for the abstract (if available).

Spring Semester

Thursday 2 February, 1:00-2:20pm
A result of Gan-Gomez on the unitary spectrum of some spherical varieties.
Yiannis Sakellaridis
Thursday 9 February, 10:00-11:20am
A result of Gan-Gomez on the unitary spectrum of some spherical varieties.
Yiannis Sakellaridis
Thursday 16 February, 1:00-2:20pm
Introduction to the conjectures of Stark
Robert Sczech
Thursday 23 February, 1:00-2:20pm
Introduction to the conjectures of Stark
Robert Sczech
Thursday 8 March, 1:00-2:20pm
Introduction to the conjectures of Stark
Robert Sczech
Thursday 29 March, 1:00-2:20pm
Introduction to the conjectures of Stark
Robert Sczech
Monday 16 April, 2:30-3:50pm
Euler Systems from CM cycles for Unitary Shimura Varieties and the Gross-Prasad Conjectures
Dimitar Jetchev (EPF Lausanne)
Sunday 22 April, 2:00-3:20pm
Products of distinct Whittaker coefficients on the metaplectic group and the relative trace formula.
Cesar Valverde
Monday 30 April, 2:30-3:50pm
Transfer relations in essentially tame local Langlands correspondence.
Geo Kam-Fai Tam (Toronto)
Thursday 3 May, 3:00-5:00pm
Non-standard comparison of relative trace formulas.
Yiannis Sakellaridis

Fall Semester

Wednesday 28 September, 4:00-5:15pm
Toric periods on GL(2): the result of Waldspurger.
Zhengyu Mao
Wednesday 5 October, 2:30-3:45pm
Introduction to the Weil representation.
Cesar Valverde
Wednesday 2 November, 3:45-5:00pm
Reductive dual pairs.
Zhengyu Mao
Wednesday 30 November, 10:00-11:20am
Towards the proof of Howe duality I. (Notes)
Yiannis Sakellaridis
Wednesday 7 December, 10:00-11:20am
Towards the proof of Howe duality II.
Yiannis Sakellaridis

Last Year's seminar
Back to the Department of Mathematics & Computer Science.

Abstracts

Feb. 2 and 9
Title: A result of Gan-Gomez on the unitary spectrum of some spherical varieties.
Speaker: Yiannis Sakellaridis
Abstract: A conjecture of Venkatesh and myself states that for a spherical variety X over a local field the space L^2(X) should have a Plancherel decomposition in terms of certain "X-distinguished" Arthur parameters. I will present a recent paper of Wee Teck Gan and Raul Gomez, where they use the theta correspondence to prove this conjecture for most varieties of rank one.

Feb. 16 and 23, Mar. 8 and 29
Title: Introduction to the conjectures of Stark.
Speaker: Robert Sczech
Abstract: A fundamental problem of algebraic number theory is the problem of constructing all abelian extensions of a given algebraic number field. That problem was solved by Kronecker ("Jugendtraum") in the special case of the rational number field and the case of an imaginary quadratic field. The conjectures of Stark (from the 1970's) represent a partial answer to that problem for an infinite class of base fields. In my talk I will report on the most accessible cases of that conjecture starting with the case of the rational number field. The talk will be elementary and therefore accessible to interested graduate students.

Apr. 16
Title: Euler Systems from CM cycles for Unitary Shimura Varieties and the Gross—Prasad Conjectures
Speaker: Dimitar Jetchev (EPF Lausanne)
Abstract: Euler systems have been invented by Kolyvagin as a tool to algebraically model L-functions and have been successfully used in proving various deep results towards the Birch and Swinnerton-Dyer conjecture, the Iwasawa main conjecture and various modularity theorems. Currently, there are only few constructions of Euler systems known in the literature: the Euler systems of cyclotomic units, Stickelberger elements, elliptic units, Siegel units (Kato's construction) and Heegner points (Kolyvagin's construction). It is an open question to understand more conceptually the construction of Euler systems and to place it in a more general-representation theoretic context. In this talk, we discuss a novel, higher-dimensional construction of an Euler system from CM 1-cycles on certain Shimura varieties for the group U(2,1)xU(1,1) via the Gross—Prasad restriction problem for the Gelfand pair U(1,1) embedded diagonally in U(2,1)xU(1,1). This construction can be used to prove new results towards a generalization of the Birch and Swinnerton-Dyer conjecture closely related to a recent Gross—Zagier type formula studied by Zhang—Zhang—Yuan in the same case. In addition, it indicates a general strategy for constructing Euler systems out of restriction problems for automorphic representations in the context of general recent conjectures by Gan—Gross—Prasad. Part of this project is joint work in progress with Yiannis Sakellaridis.

Apr. 22
Title: Products of distinct Whittaker coefficients on the metaplectic group and the relative trace formula.
Speaker: Cesar Valverde (Rutgers-Newark)
Abstract: I will talk about a relative trace formula between the metaplectic and the general linear group. As a consequence, one expects a relation between a product of distinct Whittaker coefficients on the metaplectic group and a non-split period on the general linear group.

Apr. 30
Title: Transfer relations in essentially tame local Langlands correspondence.
Speaker: Geo Kam-Fai Tam (Toronto)
Abstract: We first describe the essentially tame local Langlands correspondence of GL_n constructed by Bushnell and Henniart. Then we relate their results to endoscopic relations of Langlands and Shelstad by comparing twisted characters of representations. Finally we interpret the essentially tame correspondence using admissible embeddings of L-groups constructed by Langlands and Shelstad.

May 3
Title: Non-standard comparison of relative trace formulas.
Speaker: Yiannis Sakellaridis
Abstract: The standard paradigm of endoscopy involves an orbit-by-orbit comparison of the geometric sides of two trace formulas. I will present a different way to compare trace formulas, using certain integral transforms between the pertinent spaces of orbital integrals. More precisely, I will compare the Kuznetsov trace formula to Jacquet's relative trace formula for torus periods on GL(2), obtaining a new proof of a well-known result of Waldspurger (also proven by Jacquet). The global argument involves a Poisson summation formula for functions defined on a meager subset of the adeles!