University of Toronto
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Title: On Eisenstein series and the cohomology of arithmetic groups
Speaker: Joachim Schwermer (University of Vienna)
Abstract: The automorphic cohomology of a reductive k--group G, defined in terms of the automorphic spectrum of G, captures essential analytic aspects of the arithmetic subgroups of G and their cohomology. We discuss the actual construction of cohomology classes represented by residues or principal values of derivatives of Eisenstein series. We show that non--trivial Eisenstein cohomology classes can only arise if the point of evaluation features a half--integral property. This rises questions concerning the analytic behavior of certain automorphic L--functions at half--integral arguments. The case of the group G obtained from the general linear group GL2 over a generalized quaternion algebra defined over an algebraic number field will serve as a specific example.
Title: Quantum affine algebras, canonical bases and q-deformation of arithmetical functions
Speaker: Henry Kim (University of Toronto)
Abstract: We interpret Gindikin-Karpelevich formula and Casselman-Shalika formula as sums over Lusztig's canonical bases, generalizing the results of Bump-Nakasuji to arbitrary split reductive groups. We also obtain their affine analogues. Suggested by these formulas, we define natural q-deformation of arithmetical functions such as (multi-)partition function and Ramanujan tau-function, and prove various identities among them. In some examples, we recover classical identities by taking limits. We also consider q-deformation of Kostant's function. This is a joint work with Kyu-Hwan Lee.
Title: Arithmetic cycles on Picard modular varieties
Speaker: Stephen Kudla (University of Toronto)
Abstract: In this talk I will describe the connections between algebraic and arithmetic cycles in Picard modular varieties and the Fourier coefficients of certain hermitian modular forms, especially Eisenstein series. This is joint work with M. Rapoport.
Title: Frobenius rigidity
Speaker: Ying Zong (University of Toronto)
Abstract: We show that the sequence of iterated Frobenius endomorphisms of schemes of positive characteristics have a strong rigidity. An application to reduction of shimura varieties will be discussed.
Title: An Erdos-Kac theorem for multiplicative order
Speaker: Leo Goldmakher (University of Toronto)
Abstract: A famous theorem of Erdos and Kac asserts that the number of distinct prime factors of n behaves like a normally distributed random variable with mean and variance log log n. In this talk I will describe recent work on an analogue of this theorem, on the distribution of the multiplicative order of a fixed integer a (mod p) as p varies over primes. This is joint work with Greg Martin (UBC).
Title: Ngo's proof of the fundamental Lemma
Speaker: Joel Kamnitzer (University of Toronto)
Abstract: I will give an expository talk on Ngo's proof of the fundamental lemma, focusing on the geometric aspects of the proof. I will discuss affine Springer fibres, the Hitchin fibration, and Ngo's support theorem.
Title: The Siegel-Weil formula for classical groups
Speaker: Shunsuke Yamana (Osaka City University)
Abstract: The Siegel-Weil formula is an identity between a value of a certain Eisenstein series and an integral of a theta function. Such an identity was first proven by Siegel and then extended to classical dual reductive pairs by Weil under the assumption that the Eisenstein series is absolutely convergent. Later, Kudla and Rallis extended this formula for symplectic-orthogonal dual pairs in certain cases beyond the range of absolute convergence. In this talk I survey the work of Kudla and Rallis as well as my recent work in which the Siegel-Weil formula is extended to all quaternion dual pairs.
Title: The L^2 restriction of a GL(3) Maass form to GL(2)
Speaker: Xiaoqing Li (SUNY-Buffalo)
Abstract: In this talk, we sill study the L^2 restriction problem of a GL(3)Maass form to GL(2). By Parseval's formula, the problem becomes bounding averages of different families of GL(3)xGL(2) L-functions. Assuming the Lindelof hypothesis for these GL(3)xGL(2) L-functions as we usually do, one can prove the L^2 restriction is bounded by q^\vepsilon where q is the analytic conductor of the varying GL(3) Maass form. However, we will give an unconditional proof of this bound for selfdual GL(3) Maass forms. For nonselfdual GL(3) Maass forms, our bounds depend on the bounds for the first Fourier coefficients of the GL(3) Maass forms. This is a joint work with Matt Young.
Title: Pairs of polynomials over Q taking infinitely many common values
Speaker: Ben Weiss (University of Michigan)
Abstract: For two polynomials G(X), H(Y) with rational coefficients, when does G(X) = H(Y) have infinitely many solutions over the rationals? Such G and H have been classified in various special cases by previous mathematicians. I will discuss both the known results, the new results in my thesis, and related problems which have yet to be solved.
Title: The combinatorics of local models for Shimura varieties
Speaker: Brian Smithling (University of Toronto)
Abstract: I will discuss some combinatorial problems on the enumeration of certain Schubert varieties in affine flag varieties. These problems arise in the theory of local models for Shimura varieties, but they admit a purely group-theoretic formulation. Therefore essentially no background on Shimura varieties or their local models will be necessary.
Title: The Congruence Subgroup Kernel and the Fundamental Group of the Reductive Borel-Serre Compactification
Speaker: John Scherk (University of Toronto)
Abstract: Let G be simple, simply connected algebraic group defined over a number field k, and let S be a finite set of places which includes all the infinite ones. The congruence subgroup problem asks whether all S-arithmetic subgroups of G are congruence subgroups, or equivalently, whether the congruence subgroup kernel is trivial. The cohomology of compactifications of locally symmetric spaces of G has been studied extensively, largely because of its relation to the cohomology of (S-)arithmetic subgroups and to automorphic forms. It turns out that the fundamental group of the reductive Borel-Serre compactification is related to the congruence subgroup kernel. This involves describing the fundamental group in terms of "elementary matrices".
Title: Nekovar duality over p-adic Lie extensions of global fields
Speaker: Meng Fai Lim (University of Toronto)
Abstract: In his monograph, Nekovar gave formulations of analogues of Tate local duality and Poitou-Tate duality for finitely generated modules over a complete commutative local Noetherian ring R with finite residue field of characteristic a fixed prime p. In the usual formulation of Tate local duality and Poitou-Tate duality, one takes the Pontryagin dual which does not in general preserve the property of finite generation. Nekovar takes the dual with respect to a dualizing complex of Grothendieck so as to have a duality between bounded complexes of R-modules with finitely generated cohomology groups. In this talk, we give a generalization of this result to the setting of nonabelian p-adic Lie extensions. This is a joint work with Romyar Sharifi.
Title: Values of Green functions at CM points
Speaker: Stephen Kudla (University of Toronto)
Abstract: I will survey of various generalizations of the classic Gross-Zagier formula giving an explicit formula for the factorization of singular moduli. The Green functions on Shimura varieties of orthogonal type are constructed via the Borcherd's regularized theta lifts. Their special values at various types of CM points can be expressed in terms of derivatives of Fourier coefficients of incoherent Eisenstein series. In the case of `big' CM points, these results are joint work with Jan Bruinier and Tonghai Yang.
Title: Classification of Representations
Speaker: James Arthur (University of Toronto)
Abstract: Suppose that G is a quasisplit orthogonal or symplectic group over a field F of characteristic 0. We shall describe a classification of the irreducible representations of G(F) if F is local, and the automorphic representations in the discrete spectrum of G if F is global.
Title: Monodromy Exercise
Speaker: Ying Zong (University of Toronto)
Abstract: With Kumar Murty, we give a proof of a theorem of Serre on big monodromy of abelian varieties over number fields. We also prove a new theorem of big monodromy.
Title: Post-critically finite polynomials
Speaker: Patrick Ingram (University of Waterloo)
Abstract: In classical holomorphic dynamics, rational self-maps of the Riemann sphere whose critical points all have finite forward orbit under iteration are known as post-critically finite (PCF) maps. A deep result of Thurston shows that if one excludes examples arising from endomorphisms of elliptic curves, then PCF maps are in some sense sparse, living in a countable union of zero-dimensional subvarieties of the appropriate moduli space (a result offering dubious comfort to number theorists, who tend to work over countable fields). We show that if one restricts attention to polynomials, then the set of PCF points in moduli space is actually a set of algebraic points of bounded height. This allows us to give an elementary proof of the appropriate part of Thurston's result, but it also provides an effective means of listing all PCF polynomials of a given degree, with coefficients of bounded algebraic degree (up to the appropriate sense of equivalence).
Title: Nilpotence join theorem for Lie algebras
Speaker: Hamid Usefi (University of Toronto)
Abstract: I will prove that if L is a Lie algebra generated by subideals H and K and L/L' is finite-dimensional, then given positive integers a and b there exists a positive integer c such that \gamma_c(L) is contained in the sum of \gamma_a(H) and \gamma_b(K). One of the main ingredients is a similar result for the augmentation ideals. (Based on a joint work with Salvatore Siciliano.)
Title: Some results about Arthur's packets in the real case, and a topological application
Speaker: Laurent Clozel (Universite Paris-Sud 11)
Abstract: This is joint work with N. Bergeron. We prove (conditional on some expected results pertaining to the twisted trace formula) that certain hyperbolic 7-manifolds have trivial 1st Betti number. This depends on the understanding of the local "packets" defined by Arthur in his classification of automorphic forms on classical groups. In the lecture I will explain our partial results in this direction.
Apr. 6 (1st
Title: A geometric variant of Titchmarsh divisor problem
Speaker: Amir Akbary (University of Lethbridge)
Abstract: We formulate a geometric analogue of the Titchmarsh Divisor Problem in the context of elliptic curves and more generally for abelian varieties. For any abelian variety A defined over rationals, we study the asymptotic distribution of the primes which split completely in the division fields of A. This is a joint work with Dragos Ghioca (UBC).
Apr. 6 (2nd
Title: Zeros of a Family of Approximations of the Riemann Zeta-Function
Speaker: Steve Gonek (University of Rochester)
Abstract: Let U(s) = ∑ n ≤ N n-s be a partial sum of the Riemann zeta-function ζ(s), where s = σ + it is a complex variable. We set ζN(s) = U(s) + χ(s) U(1-s). If N = √|t| / 2π, the approximate functional equation for ζ(s) tells us that ζN(s) is an excellent approximation of ζ(s). Here we describe the distribution of zeros of ζN(s) when N is independent of t. For instance, we discuss "critical" strips in which most of the zeros lie (on and off the Riemann Hypothesis), the number of zeros of ζN(s) in rectangles, and lower bounds for the number of zeros of ζN(s) on the critical line Re s = ½. In particular, we show that if N ≥ 1 is fixed and T is sufficiently large, then 100% of the zeros with ordinates in (T,2T] lie on the critical line and are simple. This is joint work with Hugh Montgomery.