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).

Last Term's Schedule

Fall 2006 Term Schedule

Winter 2007 Term Schedule

Summary

Oct. 4
Title: De Concini-Procesi compactifications of semi-simple groups, Jacquet functors and semi-regular bimodules over semi-simple Lie algebras.
Speaker: Sergey Arkhipov (University of Toronto)
Abstract: This is a joint work in progress with Mitya Donin. We recall the geometry of the De Concini-Procesi compactification of a semi-simple group G over complex numbers. We define the Jacquet functor on the category of Harish-Chandra bimodules over the corresponding Lie algebra g and its geometric realization due to Nadler and Vilonen. Finally we provide a localization for the semi-regular bimodule over g in terms of this geometric Jacquet functor.

Oct. 11
Title: Equidistribution of eigenvalues of Hecke operators.
Speaker: Kaneenika Sinha (University of Toronto)
Abstract: In 1997, Serre proved the following "vertical" variant of the Sato-Tate conjecture : if $p$ is a fixed prime , then the eigenvalues of the p-th Hecke operator acting on the spaces S(N,k) of cusp forms of weight k and level N are equidistributed with respect to a certain measure as we vary k and N . In this talk, we make Serre's equidistribution effective, that is, we find out explicit error terms and constants. This is done with a careful investigation of the Eichler-Selberg trace formula combined with some interesting trigonometric polynomials due to Selberg-Beurling-Vaaler. If time permits, we will also discuss some interesting consequences of making Serre's theorem effective. This is joint work with M. Ram Murty.

Oct. 18
Title: Trilinear forms and the central values of triple product L-functions.
Speaker: Atsushi Ichino (Osaka City University)
Abstract: Harris and Kudla proved Jacquet's conjecture relating the non-vanishing of the central value of a triple product L-function to the non-vanishing of a certain global invariant trilinear form. Such trilinear forms have been studied by many people and some explicit formulas have been established by Gross and Kudla, Boecherer and Schulze-Pillot, and Watson. In this talk, we discuss a generalization of their formulas.

Oct. 25
Title: Equidistribution of eigenvalues of Hecke operators--part 2.
Speaker: Kaneenika Sinha (University of Toronto)
Abstract: This is a continuation of a talk given in this seminar on the 11th of October. We will carefully investigate the Eichler-Selberg trace formula and some trigonometric polynomials due to Selberg-Beurling-Vaaler to find an effective version of Serre's theorem on equidistribution of Hecke eigenvalues.

Nov. 22
Title: Elliptic curve analogues of the twin prime conjecture
Speaker: Alina Carmen Cojocaru (University of Illinois at Chicago and Fields Institute)
Abstract: Let E be an elliptic curve over Q, and let E_p be its reduction modulo a prime p. In 1988, Neal Koblitz conjectured that there are infinitely many primes p such that the order of the group of points of E_p is also a prime. This conjecture may be viewed as a higher dimensional analogue of the classical twin prime conjecture. I will discuss results (due to A.Miri & K. Murty, J. Steuding & A. Weng, H. Iwaniec & J. Urroz, and myself). obtained on Koblitz's conjecture by exploiting the analogy with the twin prime conjecture. I will also discuss a recent result (due to A. Balog, C. David and myself) asserting that for most elliptic curves, Koblitz's conjecture holds.

Nov. 29
Title: CM values of Hilbert Modular Functions.
Speaker: Tong Hai Yang (University of Wisconsin)
Abstract: As a testing case for their famous Gross-Zagier formula, Gross and Zagier discovered a beautiful factorization formula for the singular moduli, more precisely, the norm of $j(z_1) -j(z_2)$, where $j$ is the usual  modular $j$-function, and $z_i$ are Heegner points of disc. $d_i$ with $(d_1, d_2)=1$. In this talk, we explain a generalization of this work to CM values of certain Hilbert modular functions (Borcherds products) at Cm points associated to non-biquadratic CM quartic fields. The work raises some interesting questions about arithmetic intersections on Hilbert modular surfaces. This is a joint work with Jan Bruinier.

Jan. 10
Title: Mean square of the sum of the Möbius function in short intervals.
Speaker: Peng Gao (University of Toronto)
Abstract: Concerning the mean square distribution of primes in short intervals, Selberg showed (assuming RH) that \begin{equation*} \int^X_{0} \Big ( \psi \left ( (1+\delta)x \right) - \psi (x)-\delta x \Big )^2 \frac {dx}{x^2} \ll \delta (\log X)^2 \end{equation*} uniformly for $1/X \leq \delta \leq 1/\log X$. A similar case for higher moments was recently studied by Montgomery and Soundararajan. As an analogue of Selberg's result, this talk is concerning the size of the mean square of $M(x+h)-M(x)$ for small $h$. I will explain my result that assuming RH, for $X \geq 2$ and $h \geq \log^A X$ (for some explicit constant $A$) that \begin{equation*} \int^{2X}_{X}|M(x+h)-M(x)|^2dx=o(Xh^2). \end{equation*}

Jan. 17
Title: Semi-regular module for an algebraic group as an exotic Hopf algebra.
Speaker: Sergey Arkhipov (University of Toronto)
Abstract: Given an algebraic group G with two subgroups B and N such that g=Lie(G) is the direct sum of b=Lie(B) and n=Lie(N) we consider the semi-regular module for the pair (G,B) (which is a g-bimodule) has an associative algebra structure in the category of bicoimodules over O(B) and an associative coalgebra structure in the category of bimodules over U(n). We introduce an exotic tensor product (called semi-invariants) in the category of g-bimodules such that the semiregular module becomes a bialgebra in this tensor structure. We introduce a certain exotic Lie algebra (closely related to the normal bundle to B in G) and show that the semiregular bimodule as a bialgebra should be considered as the universal envelope of this Lie algebra. We establish the connection between the category of modules over this Lie algebra and the one of B-integrable G-modules. We are especially interested in the case when G is the square of an algebraic group H and B is the diagonal subgroup.

Feb. 7
Title: Hyperbolic Lattice Point Count in Infinite Volume with Applications to Sieves.
Speaker: Alex Kontorovich (Columbia University)
Abstract: There are very few examples of thin sets known to contain primes. Some of the most famous are the Piatetski-Shapiro prime number theorem and Friedlander and Iwaniec's polynomial X^2+Y^4 (and subsequently, Heath-Brown's polynomial X^3+2 Y^3). Consider Fermat's original problem of primes in the sum of two squares, c^2+d^2, but take (c,d) to be the bottom rows of matrices in an infinite index non-elementary subgroup of SL(2,Z). The work of Bourgain, Gamburd, and Sarnak implies that this set contains infinitely many "R-almost primes" (integers with at most R factors), but their theorem is so general that it gives an unspecified R. We will first execute a hyperbolic lattice point count in infinite volume to show that this set is indeed thin. Then we will use knowledge of an infinite volume spectral gap (expander property) to count this set along arithmetic progressions. Finally, we will use a combinatorial sieve to show that the set contains infinitely many R-almost primes, where R decreases as the Hausdorff dimension of the limit set of the subgroup approaches 1.

March 7
Title: Towards a p-adic Langlands program.
Speaker: Matt Emerton (Northwestern University)
Abstract:The Langlands program sets up a dictionary (conjectural in general, but proved in many significant cases) relating (certain) representations of matrix groups over the adeles (of a global field F) to representations of the Galois group of F that are ``motivic'' (i.e. arise from geometry). It is known that motivic Galois representations are in general not isolated, but move in p-adic families. Although such families may pass through a dense collection of motivic points, a general point in the family will not be motivic. It is natural to ask whether there is a corresponding phenomenon on the other side of the Langlands correspondence -- that is, whether representations of adelic groups can be interpolated into p-adic families. In this talk I will discuss some recent progess (due to Berger, Breuil, Colmez, and myself) on this question which shows that in the simplest non-abelian case, namely the group GL_2 over Q, the Langlands correspondence does indeed admit such a p-adic interpolation. I will also discuss some applications of this p-adic Langlands correspondence to some questions arising from the classical Langlands program (in particular, to a conjecture of Fontaine-Mazur on the modularity of certain two- dimensional Galois representations).

March 14
Title: Hirzebruch-Zagier, intersection homology, and modular generating functions, I.
Speaker: Jayce Getz (University of Wisconsin and University of Toronto)
Abstract: In a famous Invent. Math. paper, Hirzebruch and Zagier proved that certain generating functions for intersection numbers of cycles on Hilbert modular surfaces are elliptic modular forms.  In this talk, we will first discuss this result and how it can be viewed as a geometric manifestation of quadratic base change for quadratic extensions of $\mathbb{Q}$.  We will then introduce intersection homology with a view towards incorporating Hirzebruch-Zagier into a higher-dimensional theory.

March 21
Title: Sum of squares and a conjecture of Zagier.
Speaker: Sanoli Gun (University of Toronto)
Abstract: Sum of squares and a conjecture of Zagier Abstract: We discuss about a subspace of the space of modular forms $M_{k+1/2}(4N)$ ($k$ integer,$N$ odd and squarefree integer) the existence of which was conjectured by Zagier and it's application to the representation of integers as sums of odd number of squares.

March 28
Title: Hirzebruch-Zagier, intersection homology, and modular generating functions, II.
Speaker: Jayce Getz (University of Wisconsin and University of Toronto)
Abstract: We will discuss joint work with Mark Goresky that provides a device for producing Hilbert modular generating functions out of intersection homology classes on Hilbert modular varieties of arbitrary dimension.  Depending on time and interest, we will discuss connections with the theory of quadratic base change for $\mathrm{GL}_2$ and the theory of distinguished representations.

April 4
Title: The Lang-Trotter conjecture.
Speaker: M. Ram Murty (Queen's University)
Abstract: In the 1970's, Lang and Trotter made several conjectures concerning the value distribution of Hecke eigenvalues attached to a fixed eigenform and prescribed weight. The distribution conjecture for weight 2 is strikingly different from the case k>2 with the former having some relationship to certain classical conjectures of Hardy and Littlewood concerning primes represented by a polynomial of degree 2. For weight k>3, Lang and Trotter predict that eigenvalues with a prescribed value are only finitely many. In the case the prescribed value is coprime to 2, we will prove this conjecture in the level 1 case as well as discuss what happens at higher levels. Finally, we invoke the abc conjecture and introduce a new and exotic Dirichlet series which gives us information on the number of solutions for a fixed value. This is joint work with V. Kumar Murty.

April 25
Title: Integer solutions to decomposable form inequalities.
Speaker: Min Ru (University of Houston)
Abstract: In this talk, I'll discuss the finiteness of the number of integer solutions to certain decomposable form equations and inequalities, using Diophantine approximation. If time permits, I'll talk about the corresponding results in Nevanlinna theory as well.