The Weil representation

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Из курса от партнера National Research University Higher School of Economics

Jacobi modular forms: 30 ans après

11 оценки

National Research University Higher School of Economics

Jacobi modular forms: 30 ans après

11 оценки

This is a master course given in Moscow at the Laboratory of Algebraic Geometry of the National Research University Higher School of Economics by Valery Gritsenko, a professor of University Lille 1, France. Jacobi forms are holomorphic functions in two complex variables. They are modular in one variable and abelian (or double periodic) in another variable. The theory of Jacobi modular forms became an independent research subject after the famous book of Martin Eichler and Don Zagier “Jacobi modular forms” (Progress in Mathematics, vol. 55, 1985) which was cited more than a thousand times in research papers. This is due to many applications of Jacobi forms in arithmetic, topology, algebraic and differential geometry, mathematical and theoretical physics, in the theory of Lie algebras, etc. The list of mentioned subjects shows that my course might be useful for master and Ph.D. students working in different directions. Motivated undergraduate students can also study this subject. To follow the course one has to know only elementary basic facts from the theory of modular forms (for example, the paragraphs 1-4 of the chapter VII of Serre’s “A Course in Arithmetic” are enough). The main hero of the course is the Jacobi theta-series. Using it we will construct a lot of concrete examples of Jacobi forms in one or many abelian variables, in particular, Jacobi forms for root systems. For some of you, who will be successful with the theoretical exercises of the course, I am ready to formulate research problems for Master or Ph.D. thesis. (Ph.D. support might be available at CEMPI in Lille or at the Faculty of Mathematics of National Research University Higher School of Economics in Moscow)

Из урока

The Weil representation and vector valued modular forms. Jacobi forms of singular weight

This module is devoted to very useful notion of the Weil representation and vector-valued modular forms. In this module we also define Jacobi forms of singular weight. Also there is a peer review in the end of this module.

The Weil representation11:58

The Weil representation (part 2)15:49

Jacobi forms of singular weight14:03

Jacobi forms of singular weight (part 2)15:34

Jacobi forms of critical weight15:34

Jacobi forms of critical weight (part 2)12:14

Познакомьтесь с преподавателями

Valery Gritsenko

Laboratory of Algebraic Geometry and its Applications

[SOUND]

[MUSIC]

Hi there colleagues and dear participants of my course.

Today, we start the ninth section.

It's titled the Weil,

Representation.

This is closely related to the splitting

principle we analyzed in the last lecture.

Let phi be in Jacobi form of weight k with

a lattice L, a holomorphic Jacobi form.

According to the splitting principle,

we can represent our Jacobi form

as a product of a function,

Depending only on tau.

The summation of all elements in the dual group,

in the dual lattice, modulo Z lattice times,

The Jacobi theta series with characteristic h.

So we can write it as,

The vector-valued function,

phi(tau) times the vector-valued theta series L.

So phi L(tau), this is

the vector (fh(tau)),

the vector of lengths determinant of L.

The same for the vector of the theta series.

This is the splitting principle we analyzed, In the last lecture.

Moreover, we got an explicit formula for

the Fourier expansion of the function f(tau).

This is the sum of the Fourier

coefficients of the Jacobi form phi,

the following form (N + h to the square / 2,

h) e to the power 2 pi i capital N tau,

where capital N is a non-negative

rational number such that N is congruent

to -h to the square / 2 modulo Z.

And here, the capital N is non-negative because,

2N = the hyperbolic norm,

Of the index of the corresponding, Fourier coefficient.

And this, the hyperbolic norm of its index is non-negative.

This is because phi is holomorphic.

In this definition, we see that

we compare N with h squared / 2 modulo Z.

And this number, the square of h,

is well defined modulo 2Z, modulos are even numbers.

So the next small section is the discriminant,

Group, Of the lattice L.

My definition, as a group, as a discriminant group,

D(L), this is the quotient of the dual lattice modulo lattice.

The order of the group is

the determinant of L.

But the square of any vector

h in L star is well defined,

Modulo 2Z because if we compare this product,

we get h to the square + 2 (h,v) + v to the square.

This number is even integral.

And v to the square is even integral because the lattice L is even.

So we always work with even lattice.

Moreover, the usual

scalar product for

any h and g in L star ( h + v,

g + u) = (h,g) +

(v,g) + (h,u) +

(v,u) is congruent,

Of (h,g) modulo Z.

Therefore, on the discriminant group,

we have a symmetric by linear pairing,

With the values in the field of rational

numbers modulo the integral lattice.

And this pairing is simply induced by our quadratic formula.

Moreover, For

any h bar in D(L) in class,

its square with respect to this

pairing is well defined modulo 2Z.

So if we analyze the formula from the splitting principle,

now you see that this number is well defined,

Modulo Z.

And our formula in this splitting principle is really

determined only on the discriminant group.

So we can see them only determinant of L,

classes in this discriminant group.

[SOUND] Now I would like to analyze [SOUND] the modular

behavior of vector-valued theta series.

Modular properties,

Of our vector-valued theta series,

with determinant L components.

Let me start with one component.

I would like to calculate it for tau + 1.

By definition,

we'll have the theta series with

characteristic e to the power pi i,

then the scalar square of ((v + h,

v + h) tau + 2(v + h, z)),

the summation of all v in L.

And we analyze its behavior under this transformation.

So we add here + 1.

What do we get?

The square, Is well

defined modulo 2Z.

So we get our function up to the vector

e to the power pi i h to the square.

[SOUND]

[MUSIC]

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