Archive: Number Theory Seminar HS 11
Time: Friday at 14.15
Place: HWZ (HG G43)
Autumn Semester 2011
| Date |
Speaker |
Title |
Time |
Location |
| 23-sep-2011 (fri) |
Jens Funke
|
Spectacle cycles and half-integral weight modular forms
|
14:15-15:15 |
HG G 43 |
| Abstract: |
The classical Shintani lift is the adjoint of the Shimura correspondence. It realizes periods of even weight cusp forms as Fourier coefficients of a half-integral modular form. In this talk we revisit the Shintani lift from a (co)homological perspective. In particular, we extend the lift to Eisenstein series and give a geometric interpretation of this extension. This is joint work with John Millson. |
| Speakers: |
Prof. Dr. Jens Funke
(University of Durham)
Invited by: Ö. Imamoglu
|
|
| 30-sep-2011 (fri) |
Paul-Olivier Dehaye
|
The lower order terms conjecture for moments of the Riemann zeta function
|
14:15-15:15 |
HG G 43 |
| Abstract: |
In 2006, Conrey Farmer Keating Rubinstein and Snaith published a conjecture concerning moments of the Riemann zeta function on the critical line. Theirs supersedes the so-called random matrix conjecture (due to Keating and Snaith), which only gives the first order term. The conjecture follows from a "recipe" involving a small set of allowed manipulations of Dirichlet polynomials, but computations tend to be very complicated (even to show agreement with less precise conjectures). In my talk, I will present new tools to implement this recipe, based on symmetric function theory. This will allow me to present a more explicit version of the final conjecture, with potential applications. |
| Speakers: |
Dr. Paul-Olivier Dehaye
(ETH Zürich, Switzerland)
E-Mail:
|
|
| 7-oct-2011 (fri) |
Alexander Gamburd
|
Generalization of Selberg's 3/16 Theorem and Affine Sieve
|
14:15-15:15 |
HG G 43 |
| Abstract: |
A celebrated theorem of Selberg states that for congruence subgroups of the modular group there are no exceptional eigenvalues below 3/16. We will present joint work with Bourgain and Sarnak, in which we establish a generalization of Selberg's theorem for "thin" (infinite index) "congruence" subgroups of SL(2, Z). Time permitting, applications to affine sieve will also be discussed. |
| Speakers: |
Prof. Dr. Alexander Gamburd
(CUNY Graduate Center)
Invited by: E. Kowalski, M. Einsiedler
|
|
| 14-oct-2011 (fri) |
Francesco Veneziano
|
Bounds on the number of points "with small denominator" on algebraic curves
|
14:15-15:15 |
HG G 43 |
| Speakers: |
Dr. Francesco Veneziano
(University of Basel)
Invited by: C. Fuchs
|
|
| 21-oct-2011 (fri) |
Lenny Taelman
|
Herbrand-Ribet in positive characteristic
|
14:15-15:15 |
HG G 43 |
| Speakers: |
Dr. Lenny Taelman
(University of Leiden)
Invited by: R. Pink
|
|
| 28-oct-2011 (fri) |
|
No seminar. Heinz Hopf Lectures 2011 (25-oct and 27-oct-2011 at 17:15 in HG G 60; symposium on 26-oct-2011)
|
|
HG G 43 |
|
|
| 4-nov-2011 (fri) |
Djordje Milicevic
|
Large values of eigenfunctions and arithmetic hyperbolic 3-manifolds of Maclachlan-Reid type
|
14:15-15:15 |
HG G 43 |
| Abstract: |
One aspect of quantum chaos are extreme values of high-energy eigenfunctions, which, on Riemannian manifolds of negative curvature, are not well understood and can depend heavily on the global geometry of the manifold.
The principal result to be presented shows that there is a distinguished class of arithmetic hyperbolic 3-manifolds on which a sequence of L^2-normalized high-energy Hecke-Maass eigenforms achieve values as large as a power of the Laplacian eigenvalue. Power growth, first observed in this context by Rudnick and Sarnak, is (by far) not expected generically and stands in stark contrast with the statistical models suggested by the so-called random wave conjecture on eigenfunctions of classically ergodic systems.
The arithmetic hyperbolic 3-manifolds on which the exceptional behavior is exhibited can be characterized in terms of their invariant quaternion algebras and are, up to commensurability, precisely those containing immersed totally geodesic surfaces, as described by Maclachlan and Reid. We will explain how number theory allows us to see some of the global geometry of the manifold and discuss the question of identifying the Hecke-Maass eigenforms achieving power growth through representation theory. |
| Speakers: |
Dr. Djordje Milicevic
(MPI Bonn)
Invited by: Ö. Imamoglu
|
|
| 11-nov-2011 (fri) |
Noah Giansiracusa
|
GIT compactifications of M_(0,n) and flips
|
14:15-15:15 |
HG G 43 |
| Abstract: |
Since the early days of geometric invariant theory (GIT) it has been recognized that natural compactifications of the moduli space M_(0,n) of n distinct points on the line are obtained as GIT quotients P1^n//Aut(P1). These were famously used by Deligne and Mostow to investigate hypergeometric functions and prove arithmetic results. In this talk, I will discuss a generalization of these quotients where one considers n-pointed rational normal curves and their degenerations as parameterized by a Chow variety. One new feature of these GIT quotients is the existence of flips occurring through variation of GIT as in the work of Thaddeus and Dolgachev-Hu. |
| Speakers: |
Dr. Noah Giansiracusa
(University of Zurich)
Invited by: C. Fuchs
|
|
| 18-nov-2011 (fri) |
Hung Manh Bui
|
Zeros of the Riemann zeta-function
|
14:15-15:15 |
HG G 43 |
| Abstract: |
We discuss the method of Levinson and joint work with Conrey and Young, in which we show that more than 41% of the zeros of the Riemann zeta-function are on the critical line. |
| Speakers: |
Dr. Hung Manh Bui
(University of Zurich)
Invited by: E. Kowalski
|
|
| 25-nov-2011 (fri) |
David Masser
|
Pell's Equation over Polynomial Rings
|
14:15-15:15 |
HG G 43 |
| Abstract: |
It is classical that for any positive integer d not a perfect square there are positive integers x and y (y not 0) with x^2-dy^2=1. The analogous assertion for D,X,Y not 0 in C[t] with X^2-DY^2=1 clearly requires that the degree of D be even, and it is easy to see that it holds for all quadratic D. However for quartic D the problem changes character; this can be seen from that fact that the set of complex l such that X,Y exist for D=t^4+t+l is infinite but "scarce"; for example it contains at most finitely many rationals. For sextic D things change again, and we will sketch a proof that there are at most finitely many complex l such that X,Y exist for D=t^6+t+l. This follows from recent work with Umberto Zannier about unlikely intersections for semiabelian schemes. |
| Speakers: |
Prof. Dr. David Masser
(University of Basel)
Invited by: Ö. Imamoglu
|
|
| 2-dec-2011 (fri) |
Philipp Habegger
|
Small Heights and Non-Abelian Extensions
|
14:15-15:15 |
HG G 43 |
| Abstract: |
The absolute logarithmic Weil height can attain arbitrarily small, positive values on the field of algebraic numbers. In a finite field extension of the rational numbers there is a gap principle: the height is either zero or at least some positive constant depending only on the field. Regarding infinite algebraic extensions of Q, Amoroso and Dvornicich proved that the field F, the largest extension of Q with abelian Galois group, satisfies the same height-gap principle. Let E be an elliptic curve defined over the rationals without potential complex multiplication. The group of all torsion points of K generates an infinite and non-abelian Galois extension of Q. In the talk I will first introduce height functions, discuss some historical aspects, and then sketch a proof that K cannot contain points of small positive height. The Néron-Tate height on E(K) satisfies a similar gap principle. Both proofs use the theory of Lubin-Tate modules as well as archimedean and non-archimedean equidistribution results by Bilu, Chambert-Loir and others. |
| Speakers: |
Prof. Dr. Philipp Habegger
(University of Frankfurt)
Invited by: C. Fuchs, E. Kowalski
|
|
| 16-dec-2011 (fri) |
James Cogdell
|
The local Langlands correspondence for GL(n) and the exterior
|
14:15-15:15 |
HG G 43 |
| Abstract: |
The local Langlands correspondence is a correspondence between n-dimensional representations of the Weil-Deligne group of a local field F and the irreducible admissible representations of GL(n,F); it is a step towards a non-abelian local class field theory. As established by Harris-Taylor and Henniart, this correspondence preserves L- and epsilon-factors of pairs. But as envisioned by Langlands, this should preserve all L-functions and their epsilon factors. Henniart has shown that the correspondence preserves the exterior and symmetric square L-functions. We (with Shahidi and Tsai) can now show it preserves the epsilon factors as well. The proof is an application of local/global techniques and the stability of these local factors under highly ramified twists. |
| Speakers: |
Prof. Dr. James Cogdell
(Ohio State University)
Invited by: M. Einsiedler
|
|
The seminar is organized by G. Wüstholz, R. Pink, Ö. Imamoglu, E. Kowalski and C. Fuchs. If you have any questions send an email to one of the organizers.
