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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.

### Speakers

#### Valentin Blomer

Universität Göttingen

#### Francis Brown

All Souls College (Oxford)

#### Roelof Bruggeman

Universiteit Utrecht

#### Henri Cohen

Université Bordeaux I

#### Henri Darmon

McGill University

ICTP

UCLA

#### Stavros Garoufalidis

Georgia Institute of Technology

#### Dorian Goldfeld

Columbia University (New York)

Yale University

Osaka University

ETH Zürich

#### Masanobu Kaneko

Kyushu University

#### Maxim Kontsevich

Institut des Hautes Scientifiques

MIT

#### Philippe Michel

Ecole polytechnique federale de Lausanne

#### Martin Möller

Universität Frankfurt

ICTP

#### Maryna Viazovska

Humboldt Universität Berlin

### Live stream

A live stream of most talks of the conference is available here.

Please note: You need a video player that supports rtsp like VLC to play the stream.

We plan to make some videos available later at this site.

### Conference schedule (May 22-26)

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
11:00-11:50

#### Roelof Bruggeman: Holomorphic modular forms and cocycles

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.
14:00-14:50

#### Stavros Garoufalidis: A meromorphic extension of the 3D index

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.
15:00-15:50

#### Francis Brown: A class of non-holomorphic modular forms

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.
09:00-09:50

#### Masanobu Kaneko: Fourier coefficients and singular moduli of modular functions

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
10:30-11:20

#### Fernando Rodriguez Villegas: Motivic supercongruences

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.
11:30-12:20

#### Martin Möller: Modular forms defining gothic cathedrals

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.
14:00-14:50

15:00-15:50

#### Dorian Goldfeld: Super-positivity for L-functions associated to modular forms

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

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.
12:00-12:30

14:00-14:50

#### Valentin Blomer: Spectral reciprocity and non-vanishing of L-functions

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)
15:00-15:50

#### Philippe Michel: Sums of Kloosterman sums and sums of L-functions

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.
09:30-10:20

10:30-11:00
Tea Break
11:00-11:50

#### Henri Darmon: Modular cocycles and class field theory

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.
09:30-10:20

#### John Lewis: Cotangent sums and Dirichlet L functions

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
11:00-11:50

#### Tomoyoshi Ibukiyama: Siegel modular forms and Ihara lifts to algebraic modular forms

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.
14:00-14:50

#### Maryna Viazovska: Modular forms and sphere packing

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.

### Live stream

A live stream of the talks of the summer school is available here.

Please note: You need a video player that supports rtsp like VLC to play the stream.

We plan to make some videos available later at this site.

### Summer School Speakers and Topics

#### Martin Möller

Universität Frankfurt

ICTP

#### Don Zagier

Max Planck Institute for Mathematics

08:30-09:00
Registration
09:00-10:40

10:40-11:10
Tea break
11:10-12:50

14:20-16:00

09:00-10:40

10:40-11:10
Tea break
11:10-12:50

14:20-16:00

09:00-10:40

10:40-11:10
Tea break
11:10-12:50

14:20-16:00

09:00-10:40

10:40-11:10
Tea break
11:10-12:50

14:20-16:00

09:00-10:40

10:40-11:10
Tea break
11:10-12:50

14:15-15:00

15:00-15:30
Tea break
15:30-16:15

### Registration

If you would like to register (for the summer school and/or the conference), please complete the registration form.

If you need financial support, you can add your request on the registration webform with a brief motivation. Requests for financial support will have to be made by December 31, 2016.

## Questions / Hotel reservations

For general questions, help with hotel reservations and questions regarding the conference, please write to MFeverywhere@mpim-bonn.mpg.de For question concerning the summer school you can contact us at MFsummer@mpim-bonn.mpg.de

If you would like to reserve a hotel on your own: Here is a list of hotels.

## Location

Travel directions

### Photos

...will be posted later.