## Courses (6)

### Quantum Ergodicity and Subconvexity of L-functions

**Topic and importance**

The interplay between mathematical physics and automorphic forms led to the development of %the area of

Arithmetic Quantum Chaos. With pioneering work by Sarnak, Rudnick, and Luo, and culminating with work of Lindenstrauss and Soundararajan, it immediately became known that questions such as non-concentration of masses of eigenforms are intimately related to the problem of subconvexity of L-functions.

%This also blossomed in the last twenty years under Michel, Venkatesh, et al. to become one %of the centre points of the analytic theory of automorphic forms.

Currently an area of great activity, subconvexity results and their consequences have become central in the analytic study of automorphic forms; however, this is a difficult area for students

to penetrate. The purpose of this mini-course is to provide participants an entrance to this area by using a concrete, accessible set-up, where the solutions are known.

**Syllabus**

After introducing the notion of Quantum Ergodicity for chaotic systems, we look at the spectral decomposition of (arithmetic) hyperbolic surfaces, the (non-holomorphic) Eisenstein series, and Maass forms. We discuss the L-functions of Maass cusp forms and Rankin-Selberg convolutions. We explain the significance of subconvexity of L-functions to Quantum Unique Ergodicity (QUE) and prove the subconvex results needed for QUE of Eisenstein series: Weyl's bound for zeta on the critical line, estimates near the edge of the critical strip, subconvexity bounds for the L-functions of Maass cusp forms.

### Homogeneous Dynamics and Number Theory

**Topic and importance**

Homogeneous dynamics is the study of actions of subgroups of Lie groups on their quotients called homogeneous spaces. This has been a very active area of research for several decades. Developments in the area have also led to important breakthroughs in other areas, in particular number theory and mathematical physics. A few examples are the proof of the Oppenheim conjecture (Margulis, 1986), a proof of Quantum Unique Ergodicity (QUE) for particular arithmetic surfaces (Lindenstrauss, 2006), and a strong result towards the Littlewood conjecture (Einsiedler, Katok, Lindenstrauss, 2006). Another application is to describe the periodic Lorentz gas in the Boltzmann-Grad limit (Marklof, Strombergsson, 2011). Many of the results obtained in homogeneous dynamics are intrinsically ineffective, and there is currently much interest in the problem of proving effective versions of such results.

**Syllabus **

We will start by giving an introduction to the setting and problems in homogeneous dynamics, and we will discuss various examples in detail. We will also describe the basic set-up for how results from homogeneous dynamics are applied in the proof of the Oppenheim conjecture, the Littlewood conjecture and Arithmetic QUE. Finally, we will discuss a recent effective result on the Oppenheim conjecture by Lindenstrauss and Margulis, and recent applications of homogeneous dynamics in the study of the Boltzmann-Grad limit of the Lorentz gas for a periodic or quasicrystalline scatterer configuration.

### Lattices, finite quadratic modules, and Weil representations

**Topic and importance **

Since the development of the theory of Jacobi forms, the applications of vertex operator algebras in quantum field theory, and Borcherds' discovery of automorphic products, vector valued elliptic modular forms have gained increasing importance and interest. Key to developing a satisfactory theory of these modular forms are the finite dimensional representations of the elliptic modular group (or its unique twofold central extension) whose kernels are congruence subgroups.

The category of these representations is essentially equivalent to the catagory of Weil representations of the modular group. More recent publications by Gritsenko, Ibukiyama, Schheithauer, Skoruppa, Zagier et al. indicate that in fact the theory of Jacobi forms, Siegel modular forms of genus two and, more generally, the theory of orthogonal modular forms of low weight (more specifically, of singular and critical weight) is essentially equivalent to the study of invariants of Weil representations.

From a completely different point of view, and pursuing a different aim, the monograph of Nebe, Rains, Sloane (Self-Dual Codes and Invariant Theory) is centered around Weil representations and applications to the structure of rings of weight enumerators.

Recently, work of Skoruppa has shown there is a wonderful interplay between the theory of integral non-degenerate latices, finite quadratic modules, and the theory of Weil representations of the elliptic modular group. The goal of this mini-course is to introduce students to this theory and to enable them to apply its ideas and methods to solve questions in the theory of automorphic forms.

** Syllabus **

Lecture 1:

Finite quadratic modules, ABC and Jordan decomposition;

Lecture 2:

Witt groups, anisotropic modules:

Lecture 3:

the Grothendieck groups of finite quadratic modules, connection to the category of lattices and the associated Grothendieck groups, orthogonal groups of finite quadratic modules;

Lecture 4:

generators and relations for $SL(2,\Z)$ and its twofold central extension $Mp(2,\Z)$, Weil representations associated to $SL(2,\O)$ for general rings $\O$ , Weil representations associated to finite quadratic modules;

Lecture 5:

decomposition of Weil representations into irreducible parts, invariants of Weil representations;

Lecture 6:

Jacobi forms of singular and critical weight as spaces of invariants of Weil representations.

### Kronecker limit formalism and moonshine type groups

**Topic and importance:**

Kronecker’s limit formula refers to the evaluation of the first two terms in the Laurent expansion of the non-holomorphic Eisenstein series E(s,z) associated to the discrete group PSL(2,Z) at s=1. In this case, the first term is a pole with residue equal to 1/V, where V is the co-volume of PSL(2,Z), and the second term involves the weight twelve cusp form, suitably normalized, associated to PSL(2,Z).

More generally, one can consider generalizations of Kronecker’s limit formula to be any result which studies the Laurent expansion of any zeta or L function about its first pole. There are various examples where such considerations exist, including the study of spectral zeta functions (which yields determinants of the Laplacian and analytic torsion) and various types of Eisenstein series (which yields constructions of certain modular forms). Going further, the variation of the Kronecker limit function may be of additional interest, one example of which is the manifestation of Dedekind sums.

**Syllabus:**

In these talks the speakers will outline their point-of-view of Kronecker’s limit functions and Dedekind sums associated to various Eisenstein series for co-finite quotients of the hyperbolic upper half plane. Specific attention will be paid to the so-called “groups of moonshine type”, which we define as normalizer subgroups associated to Gamma_{0}(N). In lieu of a problem solving session, the speakers will offer a number of open problems.

### Explicit L-functions, automorphic forms and Galois representations

**Topic and importance:**

Automorphic forms and L-functions have played prominent roles invarious branches of mathematics and physics since the 20th century,mostly at a theoretical level. However, recent advances in explicitand computational methods have made a more hands-on approach possible,and with it a host of applications in explicit and algorithmic numbertheory have arisen.

**Syllabus:**

This course will survey some of these methods and applications, in twoparts. The first part will focus on the computational theory ofL-functions and applications of an analytic nature, such asHelfgott's proof of the ternary Goldbach conjecture. The second partwill focus on explicit equations for modular varieties, the arithmeticof Jacobians of curves, and applications of an algebraic nature, suchas Couveignes and Edixhoven's polynomial-time algorithm for computingthe Ramanujan \tau-function.

### Galois Representations and Diophantine Problems

**Topic and importance:**

Wiles' proof of Fermat's Last Theorem was one of the crowningachievements of 20th century mathematics, and has led to twomajor research programmes. The first aims to establish moregeneral and powerful modularity theorems for Galois representations,and has yielded the proof of modularity of elliptic curves over therationals by Wiles, Breuil, Conrad, Diamond and Taylor, the proof ofSerre's modularity conjecture by Khare and Wintenberger, and powerfulmodularity lifting theorems by Kisin, Barnet-Lamb, Gee, Geraghty andothers. The second programme aims to apply the proof strategy ofFermat's Last Theorem to solve Diophantine problems, especially casesof the notorious Beal conjecture (also known as the generalized Fermat conjecture).The original strategy of Hellegouarch, Frey, Serre andRibet has been refined and extended by many, most notably Bennett,Dahmen, Darmon and Kraus. The two programmes are still intricatelyconnected, as evident in the recent work of Freitas and Siksek on theFermat equation over totally real fields.

**Syllabus: **

The minicourse will use Diophantine equations as a vehicle tointroduce and motivate the study of Galois representations of ellipticcurves and of modular forms and their relationship (modularity). Thestudents will learn the basics of 2-dimensional Galoisrepresentations, understand the statements of modularity theorems(including the more recent modularity lifting theorems), and how toapply them to solve basic ternary Diophantine problems.