Jacobi theta-series and Jacobi forms in many variables

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From the course by National Research University Higher School of Economics

Jacobi modular forms: 30 ans après

12 ratings

National Research University Higher School of Economics

Jacobi modular forms: 30 ans après

12 ratings

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)

From the lesson

Jacobi forms in many variables and the Eichler-Zagier Jacobi forms

This module is devoted to Jacobi forms in many variables. In this module we also define classical Eichler-Zagier Jacobi forms in terms of Jacobi forms in many variables. Also there is a peer review in the end of this module.

Jacobi theta-series and Jacobi forms in many variables15:03

Jacobi theta-series and Jacobi forms in many variables (part 2)15:57

Hyperbolic reformulation14:18

Examples of Jacobi forms in many variables14:55

Examples of Jacobi forms in many variables (part 2)16:20

Examples of Jacobi forms in many variables (part 3)16:29

Meet the Instructors

Valery Gritsenko

Laboratory of Algebraic Geometry and its Applications

[MUSIC]

Hi there, participants of my course.

Today, we are studying the lecture number seven.

The title of this lecture, Jacobi theta series.

And Jacobi modular form in several variables.

Jacobi forms in many variables.

This is very important for our cause.

Today we extend our subject to the case of many billion variables.

Jacobi theta series.

Was the hero of the last two lectures.

This is a very important function for our study.

I remind you the Fourier expansion of the definition of this function.

The connector character, summation of all integers.

Q to the power of 8R to the power N over two.

Or for the current lecture I will

use more explicit notations.

Let me change this term by e

to the power by i n square over

4 tau plus n zeta where tau

is in the usual upper plane and

zed is an arbitrary complex number.

Using this function we construct very many

interesting Jacobi forms I recall what function would be constructed.

First of all, the which

Jacobi form of weight minus two and

index one is defined as a potion of the square

of Jacobi Theta series over

the power six of the Dedekind function.

This function generates

The space of the weak of Jacobi form

of weight minus 2 and index 1.

Then, we defined the first

Jacobi cast form.

This function

generates the space of

Jacobi cast form,

Of weight 10 and index 1.

This is the first Jacobi cast form.

Moreover, we constructed

the first Jacobi 4 of odd weight.

[SOUND] This is theta to the power 21,

times Jacobi 30 series into that.

The eighth power of the Jacobi theta series

Is a Jacobi modular form, non-cusp form of weight 4 and index 4.

Now I would like to put the following question.

What is about the following product?

Let me introduce the following notation to make our formula shorter.

I would like to show you only the second argument of the Jacobi series.

Then I would like to consider the following function.

Theta in (z1) times theta (z2) so

on times theta in (z8).

So I would like to analyze the function

in eight Abelian were able.

This is caligraphoc is that,

this is the vector of frank eight.

So what function do we have?

This is the first example of Jacobi modular forms in many variables.

Now I would like to understand the property of this action.

[SOUND] First of all, any Jacobi form is a holomorphic

function satisfying two functional equations,

the modular equation and the Abelian equation,

or quasiperiodic condition.

Now [COUGH] let us analyze the modular behavior of this new function.

We have a new Theta in eight Abelian variables.

Now I would like to take the modular transformation

from the special linear group sl2z.

So here we use another variable

the vector z in eight coordinate.

What do we get?

The original Jacobi theta series is a modular form

Jacobi modular 4 of weight one half and

index one half with respect to the multiplied system

of the Theta function to the power 3 times the character of the heights in the group.

In the new function, we have eight factors.

It means that the product will

have the weight, 4.

8 x 1/2 is 4.

So the first factor,

(c tau + d) to the power 4.

Then we have no character or multiplier

factor because we have 8 factors.

And for each factor their multiplier

system is V eta to the power cube,

then Z 8 power of this multiplies system, is true.

So have no character, then we have E,

Ti because theta has index one half and

C then the sum of the squares of our variable

Over C plus D.

And then Theta

function, In eight variables.

So now let's analyze this formula.

The sum of the square, which we have here,

we can write as the scalar square of

the, Vector z.

So this is the modular behavior of our function.

Now I would like to analyze Its periodic property.

So, we change z by

lambda tau + mu

where lambda and

mu are vectors.

There's a eight vectors with eight integral cordon.

So using the function equation or

the usual that the series, I guess the full increase.

Minus one to the power is the sum

of Lambda e plus mu e [SOUND] e,

from 1 till 8.

This is the character of the Heisenberg group.

Times e to the power minus pi.

And now I can use the scalar product.

It will be Lambda.

Scalar product, it means lambda 1 to the square plus lambda 2 to the square and so

on, tau + 2 (the scalar product of lambda and

z) means lambda 1 z1 plus lambda 2 z2 and so on.

Times Theta 2 z.

This is a functional equation of our function.

Now I would like to have a function with triple character.

In what case, we have no plus minus one in this functional equation.

So, I would like to have one here.

To have one, we have to take Lambda and mu in the lattice D8.

By the definition, the lattice D8,

this is the lattice of all integral vectors.

With even sum of the coordinate.

Sublatens of index two

is an incladen latest of frank 8.

So, in this case if lambda and mu have these properties.

It means then, the sum of the coordinate lambda e and,

the sum of the coordinate of mu j is equal to 0, modular 2.

Under this condition, we have always the factor one in this equation.

We get the first example

of the so-called

Jacobi Form for

the latest D8 or

a Jacobi form was 8 billion

variable that that is a vector.

It was 8 complex coordinate.

And this is I repeat this is a Jacobi form with respect to the latest D8.

Now I would like to give you the general definition

of Jacobi modular form in many variables.

[MUSIC]

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