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Lattices, finite quadratic modules, and Weil representations

Published in Courses
Wednesday, 09-25-2019

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.

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