## The University of Toronto Number Theory/Representation Theory Seminar  2007-08: Wednesdays 2:10-3:00PM, Bahen Centre 6183 , University of Toronto

This seminar is organized by Jim Arthur, Henry Kim and Stephen Kudla. If you would like to speak at the seminar, please email them (arthur, henrykim, skudla). For inquiries regarding this web page, please email Jonathan (jkorman).

## Fall 2007 Term Schedule

 DATE and TIME SPEAKER TITLE Oct. 3,  Wednesday 2:10--3:00PM John Friedlander     U of T Hyperbolic Prime Number Theorem. Oct. 10,  Wednesday 2:10--3:00PM Patrick Ingram     U of T Diophantine approximation in arithmetic dynamics. Oct. 17,  Wednesday 2:10--3:00PM Roman Holowinsky Sieve Method for Shifted Sums of Hecke Eigenvalues. Oct. 24,  Wednesday 2:10--3:00PM Fiona Murnaghan     U of T Spherical characters of distinguished representations. Oct. 31,  Wednesday 2:10--3:00PM Gerald Gotsbacher     U of T Arithmetic groups and their cohomology: the cocompact case. Nov. 7,  Wednesday 2:10--3:00PM Paul Mezo     Carleton University A tMorten1966wisted Paley-Wiener theorem for real reductive groups. Nov. 14,  Wednesday 2:10--3:00PM Jeff Adams     University of Maryland Algorithms for Representation Theory. Nov. 21,  Wednesday 2:10--3:00PM Brooke Feigon     U of T Averages of central L-values of Hilbert modular forms. Nov. 28,  Wednesday 2:10--3:00PM Chao Li     U of T A local twisted trace formula. Dec. 5,  Wednesday 2:10--3:00PM Valentin Blomer     U of T Summing Hecke eigenvalues over quadratic polynomials.

## Winter 2008 Term Schedule

 Jan 16,  Wednesday 2:10--3:00PM Stephen Kudla     U of T Arithmetic cycles for Shimura varieties attached to unitary groups (joint work with M. Rapoport). Jan 23,  Wednesday 2:10--3:00PM Ben Howard     Boston College Intersection theory on Shimura surfaces. Jan 30,  Wednesday 2:10--3:00PM David Whitehouse     MIT On a result of Waldspurger in higher rank. Feb 13,  Wednesday 2:10--3:00PM Jim Arthur     Toronto Recent history of the trace formula. Feb 27,  Wednesday 2:10--3:00PM John Friedlander     Toronto Selberg og Soldmetoden: En Positiv Tilnaerming (Selberg and the Sieve: A Positive Approach). Mar 12,  Wednesday 2:10--3:00PM Gerald Gotsbacher     Toronto Arithmetic groups and their cohomology: Eisenstein series. Mar 26,  Wednesday 2:10--3:00PM Chao Li     Toronto Some twisted orthogonality relations. Apr 16,  Wednesday 2:10--3:00PM Sukumar Adhikari     Allahabad Some classical zero-sum problems in combinatorial number theory.

## Summary

Oct. 3
Title: Hyperbolic Prime Number Theorem.
Speaker: John Friedlander (University of Toronto)
Abstract: In joint work with H. Iwaniec we count the number of quadruples $$(x_1,x_2,x_3,x_4) \in \zb^4, \, {\rm{for}\,\, {which}}\,\, p =x_1^2+x_2^2+x_3^2+x_4^2\leq X$$ is a prime number and satisfying the determinant condition: $x_1x_4-x_2x_3=1$. By means of the sieve, one shows easily the upper bound $S(X)\ll X/\log X$. Under a hypothesis about prime numbers, which is stronger than the Bombieri-Vinogradov theorem but is weaker than the Elliott-Halberstam conjecture, we prove that this order is correct: $S(X)\gg X/\log X$.

Oct. 10
Title: Diophantine approximation in arithmetic dynamics.
Speaker: Patrick Ingram (University of Toronto)
Abstract: In 1978, Lang conjectured a lower bound on the canonical height of a non-torsion point on an elliptic curve which depends on various data related to the curve. This conjecture remains open (although there are several partial results). Much more recently, Silverman posited a more general conjecture about lower bounds on canonical heights of `arithmetic dynamical systems' (that is, systems defined by a morphism mapping a variety to itself), which at least roughly reduces to Lang's Conjecture when the underlying system is that defined by the multiplication-by-n map on an elliptic curve (n>1). We'll discuss the first result towards Silverman's conjecture in which the underlying structure is not that of an abelian variety. Some other problems related to heights in arithmetic dynamics will come up along the way.

Oct. 17
Title: Sieve Method for Shifted Sums of Hecke Eigenvalues.
Speaker: Roman Holowinsky
Abstract: Naturally connected with the Quantum Unique Ergodicity(QUE) conjecture are shifted sums of Hecke eigenvalues. We'll discuss a sieve technique for analyzing such sums and demonstrate how the method may be applied to shifted sums of general multiplicative functions. We'll also discuss the remaining obstacles in application to QUE.

Selberg og Soldmetoden: En Positiv Tilnaerming
(Selberg and the Sieve: A Positive Approach). Oct. 24
Title: Spherical characters of distinguished representations.
Speaker: Fiona Murnaghan (University of Toronto)
Abstract: Let H be the fixed points of an involution of a connected reductive p-adic group G. Spherical characters are H-biinvariant distributions on G that are attached to H-distinguished representations of G. They play an important role in harmonic analysis of the reductive p-adic symmetric space G/H. We will recall the definitions of H-distinguished representations and spherical characters of H-distinguished representations. Then we will give a general description of some new integral formulas for spherical characters of distinguished supercuspidal representations. These formulas are analogous to well-known integral formulas for the ordinary characters of supercuspidal representations. We will also discuss results showing that some spherical characters vanish near the identity element.

Oct. 31
Title: Arithmetic groups and their cohomology: the cocompact case.
Speaker: Gerald Gotsbacher (University of Toronto)
Abstract: After introducing the notion of an arithmetic subgroup $\Gamma$ for a real linear Lie group $G$ I will focus on the discussion of \emph {compact} locally symmetric spaces and their cohomology. That is, on de Rham and relative Lie algebra cohomology of such. In particular, I will presentOn a result of Waldspurger in higher rank Matsushima's formula for the decomposition of the cohomology in terms of the spectrum of $\Gamma$, and talk about the problems resulting from it: to determine the cohomological representations of $G$ and their multiplicities. All of this shall be illustrated throughout by means of a guiding example. The talk is of an expository kind.

Nov. 7
Title: A twisted Paley-Wiener theorem for real reductive groups.
Speaker: Paul Mezo (Carleton University)
Abstract: The original Paley-Wiener theorem characterizes the Fourier transform of smooth compactly supported funIntersection theory on Shimura surfacesctions on the real numbers. This characterization was generalized to a representation theoretic context on real reductive groups by Clozel and DelormeMorten1966. We provide a characterization in the case that the representations are stable under a group automorphism. This is joint work with P. Delorme.

Nov. 14
Title: Algorithms for Representation Theory.
Speaker: Jeff Adams (University of Maryland)
Abstract:The first goal of the Atlas project is to implement the Langlands classification for real groups by computer. The problem of converting the Langlands classification to an explicit algorithm,On a result of Waldspurger in higher rank suitable for computer computation, is a highly nontrivial one; solving it has given us a better, and simpler, understanding of the mathematics. I will discuss this algorithm, and discuss its application to computation of unipotent representations, Arthur's conjectures, and the unitary dual. Arithmetic cycles for Shimura varieties attached to unitary groups

Nov. 21
Title: Averages of central L-values of Hilbert modular forms.
Speaker: Brooke Feigon (University of Toronto)
Abstract: We use the relative trace formula to obtain exact formulas for central values of certain twisted quadratic base change L-functions averaged over Hilbert modular forms of a fixed weight and level. We apply these formulas to the subconvexity problem for these L-functions. This talk is based on joint work with David Whitehouse.

Nov. 28
Title: A local twisted trace formula.
Speaker: Chao Li (University of Toronto)
Abstract: Around 1990, Arthur proved a local (ordinary) trace formula for real or p-adic connected reductive groups. The local trace formula is a powerful tool in local harmonic analysis of reductive groups. For non- connected reductive groups, Arthur suggested that there should be a local twisted trace formula to make the ordinary one into a special case. In this talk, I will describe a twisted vision of the local trace formula. It should have applications to the classifications of representations of classical groups.

Dec. 5
Title: Summing Hecke eigenvalues over quadratic polynomials.
Speaker: Valentin Blomer (University of Toronto)
Abstract: While most classical arithmetic functions are reasonably well understood on average, the situation becomes much harder if one considers sums over sparse sequences, for example values of a polynomial of degree exceeding 1 (think of summing the von Mangoldt function over numbers of the form $n^2+1$). Here we derive bounds with a power saving for sums of Hecke eigenvalues over quadratic polynomials $n^2+an+b$.

Jan 16
Title: Arithmetic cycles for Shimura varieties attached to unitary groups (joint work with M. Rapoport).
Speaker: Stephen Kudla (University of Toronto)
Abstract: We give a modular definition of the cycles in Shimura varieties attached to the unitary group U(1,n-1) that were originally considered in a Riemannian setting in work with John Millson. This definition allows us to extend the cycles to an integral model over Z_p for a good prime p. We describe the intersection a collection of such cycles in terms of their fundamental matrix. In the case in which the cycles intersect in a finite set of points, we determined the length of the corresponding Artin scheme (This should be the local contribution at p to the height pairing of the cycles.) and show that it is related to the p-part Morten1966of the derivative of a Fourier coefficient of an Eisenstein series on U(n,n). This talk will provide an overview and background.

Jan 23
Title: Intersection theory on Shimura surfaces.
Speaker: Ben Howard (Boston College)
Abstract: Kudla, Rapoport, and Yang and have proved the equality of two modular forms of weight 3/2. One is an Eisenstein series, and the other is a formal q-expansion which encodes the arithmetic intersection numbers of CM points on a Shimura curve. Using this equality of modular forms those authors deduce a formula relating the height of a CM point in a modular Jacobian to the central derivative of an L-series, much in the spirit of the Gross-Zagier theorem. I will discuss a similar result for Shimura surfaces, relating the intersection multiplicities of special cycles to the Fourier coefficients of a Hilbert modular form of parallel weight 3/2.

Jan 30
Title: On a result of Waldspurger in higher rank.
Speaker: David Whitehouse (MIT)
Abstract: An important result of Waldspurger relates certain central L-values of automorphic forms on GL(2) to period integrals over tori. Subsequently this result was reproved by Jacquet using the relative trace formula. We will explain some progress on extending Waldspurger's result to higher rank via a generalization of Jacquet's approach.

Feb 13
Title: Recent history of the trace formula.
Speaker: Jim Arthur (Toronto)
Abstract:This will be a geSome twisted orthogonality relationsneral talk (ie. for a general mathematical audience) on the history of the trace formula since Selberg. I shall try to discuss the underlying ideas, rather than the formula itself. I will also discuss basic applications, past, present and future.

Feb 27
Title: Selberg og Soldmetoden: En Positiv Tilnaerming
(Selberg and the Sieve: A Positive Approach).
Speaker: John Friedlander (Toronto)
Abstract: We survey the contributions of A. Selberg to sieve methods. This is the same talk which was given at the Selberg Memorial Conference, January 11--12, 2008 at the Institute for Advanced Study. It is intended to be accessible to a general mathematical audience. (The lecture will be given in English.)

Mar 12
Title: Arithmetic groups and their cohomology: Eisenstein series.
Speaker: Gerald Gotsbacher (Toronto)
Abstract: Let G/Q be a reductive algebraic group of positive semisimple Q-rank, and Gamma in G(Q) an arithmetic subgroup. One way to construct non-trivial classes in the group cohomology of Gamma; is by means of Eisenstein series. I shall outline the set up for this approach, explain the construction and illustrate the insight it provides in terms of a certain class of orthogonal groups.

Mar 26
Title:Some twisted orthogonality relations.
Speaker: Chao Li (Toronto)
Abstract: It is well-known that there are Schur's orthogonality relations for characters of compact groups. For non-compact groups, Harish-Chandra defined the characters of infinite dimensional representations via the language of distributions. And he conjectured that there should be a vanishing property for elliptic characters of a connected reductive group and it was proved by Kazhdan. In the beginning of 1990's, Arthur proved some more general and explicit orthogonality relations for elliptic characters of a connected reductive group. In this talk I will introduce some twisted orthogonality relations for non-connected reductive groups. These results should have applications in the classifications of representations of classical groups.

Apr 16
Title:Some classical zero-sum problems in combinatorial number theory.