Max Planck Institute for Mathematics, Bonn

Summer School (May 15-19) Conference (May 22-26)Modular forms are omnipresent in mathematics. Recently they have even turned up in the study of black holes and string theory. Their presence usually indicates a deep underlying structure teeming with symmetry.

In the first week, there will be a school for advanced students which is immediately followed by a research conference in the second week. These activities will focus on the many facets of modular forms, especially those related to the work of Don Zagier, a giant in the theory of modular forms.

- Kathrin Bringmann, Universität zu Köln
- Michael Griffin, Universität zu Köln
- Maxim Kontsevich, IHES
- Pieter Moree, MPIM Bonn
- Ken Ono, Emory University
- Martin Raum, Chalmers Technical University

Universität Göttingen

All Souls College (Oxford)

Universiteit Utrecht

Université Bordeaux I

McGill University

ICTP

UCLA

Georgia Institute of Technology

Columbia University (New York)

Ecole polytechnique federale de Lausanne

Universität Frankfurt

Humboldt Universität Berlin

Some recordings of the conference talks are available on this site.

09:00-09:30

Registration

09:30-10:20

Maxim Kontsevich: Modular q-difference modules

Abstract: I will propose a definition of modular families of q-difference
modules via a q-version of Riemann-Hilbert correspondence.

10:30-11:00

Tea Break

Abstract: I'll speak about joint work with YoungJu Choie and Nikos Diamantis on
the cocycles that one can attach to holomorphic modular forms. Knopp
has shown that there is a generalization of the classical
Eichler-Shimura theory tocusp forms of real weight. We consider a map
to cohomology from the space of holomorphic functions with modular
transformation behavior (without any growth condition at the cusps).
For weights that are not integers at least two the results differ
considerably from the classical Eichler-Shimura theory, and are analogous to earlier results for Maass wave forms studied by Don
Zagier, John Lewis and me.

Abstract: The 3D-index of Dimofte-Gaiotto-Gukov is a collection of q-series with integer coefficients which is defined for 1-efficient ideal triangulations, and gives topological invariants of hyperbolic manifolds, in particular counts the number of genus 2 incompressible and Heegaard surfaces. We give an extension of the 3Dindex to a meromorphic function defined for all ideal triangulations, and invariant under all Pachner moves. Joint work with Rinat Kashaev.

Abstract: I will define an elementary theory of non-holomorphic modular forms and
describe some of its basic properties.
Within this family, there exists a class of functions which correspond
to certain mixed motives.
They are constructed out of single-valued iterated integrals of holomorphic
modular forms, and are closely related to a problem in string theory.

Abstract: The generating function of traces of singular moduli of the modular j-invariant
becomes a modular form of weight 3/2. This is Don's celebrated discovery, inspired
by a work of R. Borcherds. Using this modular form, one can obtain a formula for
the Fourier coefficients of the modular j-invariant in terms of singular moduli.
In this talk, I shall review these works, and introduce recent developments
regarding an application of the formula (due to R. Murty and K. Sampath)
as well as generalizations (due to T. Matsusaka).

10:00-10:30

Tea Break

Abstract: Certain congruences between truncated hypergeometric
polynomials and unit roots of their associated motives appear to hold to
higher powers of primes than expected. We will discuss how this
phenomenon, generally known as supercongruences, is tied to Hodge theory
and is more widespread than previously thought. This is joint work with
D. Roberts.

Abstract: Flat surfaces with the floorplan of gothic cathedrals define an exceptional series of
Teichmüller curves. We give an overview of the ways to define Teichmüller curves
using Hilbert modular forms of non-parallel weight and the flat surfaces invariants
that can be computed from this viewpoint.

Abstract: to be added.

Abstract: Zhiwei Yun and Wei Zhang introduced the notion of "super-positivity of self-dual L-functions" which specifies that all derivatives of the completed L-function (including Gamma factors and power of the conductor) at the central value s=1/2 should be non-negative. They proved that the Riemann hypothesis implies super-positivity for self dual cuspidal automorphic L-functions on GL(n).
This talk is based on recent joint work with Bingrong Huang where we prove, for the first time, that there are infinitely many L-functions associated to modular forms for SL(2,ℤ) each of which has the super-positivity property.

09:30-10:20

Özlem Imamoglu: The Kronecker limit formulas and geometric invariants for real quadratic fields, part I

Abstract: The Kronecker Limit formulas lead to some beautiful relations between quadratic fields, L functions, elliptic curves and, of course, modular forms! Associated to the ideal classes of a quadratic field are some well known geometric invariants attached to a modular curve: CM points in the imaginary quadratic case and closed geodesics in the real quadratic case.

10:30-11:00

Tea Break

11:00-11:50

William Duke: The Kronecker limit formulas and geometric invariants for real quadratic fields, part II

12:00-12:30

Speed talks

Abstracts: to be added.

Abstract: We develop a reciprocity formula for a spectral sum over central values
of L-functions on GL(4) x GL(2). As an application we show that for any
self-dual spherical cuspidal automorphic representation Pi on GL(4),
there exists a self-dual representation pi on GL(2) such that L(1/2, Pi
x pi) does not vanish. An important ingredient is a functional equation
of a certain double Dirichlet series involving Kloosterman sums and
GL(4) Hecke eigenvalues. (Joint work with X. Li and S. Miller)

Abstract: This talk is a review a recent series of works by V. Blomer, E.
Fouvry, E. Kowalski, myself, D. Milicevic, W. Sawin as well as R.
Zacharias.
We will describe various estimates on sums of Kloosterman sums (or more
generally trace functions)
proven using methods from $\ell$-adic cohomology and some of their
applications to the study of analytic properties
of character twists of L-functions on average over the family of
Dirichlet characters of some large prime modulus.

Abstract: to be added.

10:30-11:00

Tea Break

Abstract: The factorization of differences of singular moduli is described in a landmark 1985 article
of Gross and Zagier bearing the same title. I will describe a project which aims to uncover similar
phenomena in the setting of real quadratic fields. This is joint work with Jan Vonk.

12:00-12:50

Henri Cohen: Classical Modular forms in Pari/GP

Abstract: At present, computer packages for working with classical
modular forms are available in Magma and in Sage, both based on
modular or Manin symbols. I will describe a new and extensive package
available in Pari/GP based on trace formulas, including in particular
modular forms of weight 1.

Abstract: We define a sequence of finite sums of cotangents at arguments equal to selected rational multiples of π, with the selection made by a certain order-relation algorithm. We then show how these sums are related to information about the location of the zeros of the Dirichlet L function L(s, χ4).

10:30-11:00

Tea Break

Abstract: Around in 1963, Ihara gave a theory of lifts from elliptic modular forms to automorphic forms
of compact symplectic group of complex rank two. This can be regarded as a compact version
of Saito-Kurokawa lift and Yoshida lift, which were found much later. In this talk, we give a
precise conjectural global correspondence (of Langlands type) between Siegel modular forms
and automorphic forms on compact symplectic group, for discrete subgroups which are parahoric
subgroups locally, based on exact dimension formulas of automorphic forms and a lot of numerical
examples. As a byproduct, we propose a conjecture on precise images of the Ihara lift.

Abstract: In this talk we will give an application of modular forms to the sphere
packing problem. First, we will explain the Cohn-Elkies linear
programming bound for the best packing constant in Euclidean space.
Next, we will show that the linear programming problems corresponding to
dimensions 8 and 24 can be solved explicitly. As a byproduct of our
method, we will prove a new type of interpolation formula for the
Schwartz functions. Finally, using this approach we will present the
solution of the sphere packing problem in dimensions 8 and 24.

Organizers: Kathrin Bringmann, Stephan Ehlen,

Michael Griffin, Larry Rolen, Mike Woodbury

Some video recordings of the lectures are available on this site.

ICTP

Max Planck Institute for Mathematics

08:30-09:00

Registration

09:00-10:40

Martin Möller (1/5)

10:40-11:10

Tea break

11:10-12:50

Don Zagier (1/5)

14:20-16:00

Fernando Rodgriguez Villegas (1/5)

09:00-10:40

Martin Möller (2/5)

10:40-11:10

Tea break

11:10-12:50

Fernando Rodriguez Villegas (2/5)

14:20-16:00

Don Zagier (2/5)

09:00-10:40

Fernando Rodriguez Villegas (3/5)

10:40-11:10

Tea break

11:10-12:50

Martin Möller (3/5)

14:20-16:00

Don Zagier (3/5)

09:00-10:40

Fernando Rodriguez Villegas (4/5)

10:40-11:10

Tea break

11:10-12:50

Don Zagier (4/5)

14:20-16:00

Martin Möller (4/5)

09:00-10:40

Don Zagier (5/5)

10:40-11:10

Tea break

11:10-12:50

Martin Möller (5/5)

14:15-15:00

Fernando Rogriguez Villegas (5/5) - part 1

15:00-15:30

Tea break

15:30-16:15

Fernando Rodriguez Villegas (5/5) - part 2