Definition of Jacobi forms

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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

Jacobi modular forms: the first definition

This module is devoted to the first definition of Jacobi forms. In this module we also define Jacobi modular group. Also there is a peer review in the end of this module.

Definition of Jacobi forms12:22

Definition of Jacobi forms (part 2)13:04

Basic properties of Jacobi forms12:54

Jacobi modular group11:56

Symplectic group11:30

Jacobi modular group (part 2)13:32

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Valery Gritsenko

Laboratory of Algebraic Geometry and its Applications

[MUSIC]

Dear students, welcome to the second lecture of my course.

The title of this second section

is the First Definition of Jacobi Form.

Certainly, in the title, I would like to emphasize the word adjust.

Because there in this lecture,

you will see also the second definition of Jacobi form.

Let me explain to you the plain of this legend.

First of all, we consider the concept of Jacobi forms.

Then I defined the so called Jacobi module group.

We'll have two points of view on this group.

First of all this group is an extension of the usual modular group s to z,

which will be a group of this group.

But from another side,

itself will be parabolic sub group of the so

called Zed module group.

Or the integral group of genus two.

This group is very interesting for

many points of view, particularly this group,

is the semi direct product of the group s into z, and

the unit per 10 group, h which is the Heisenburg group.

Of the latest that it's better to say of the direct product of zed by.

So the concept of Jacobi modular group is very useful,

because at the end of this lecture,

we come to the second definition the second,

more factorial definition of Jacobi form

But let me start by the first definition.

[SOUND] In this definition.

Tau is an element of the upper half plane.

It means the imaginary power of tau is positive.

Then Zed is a complex number.

K is an integer.

And m is non-negative integer.

We need this data for

the definition of Jacobi form.

Consider a holomorphic

function, p into

two variables tau and z.

These holomorphic on the direct product of the upper half plane and

the field of complex standards.

This function satisfies

two functional equation.

The first equation, the so

called modular equation,

phii, a, tau plus b over c theta plus d,

zeta over c-theta plus d is

equal to c-theta plus d to the power k.

K, this is an integer.

Times e to the power

2 fi im c times z

to the square or

1 c tau plus d

times v tau zed.

M is the second integer, which we have fixed for the definition,

and we have this functional equation the modular equation.

For any metrics, A,

B, C, D in the modular

group added to zed.

The second functional equation is the so-called elliptic equation.

Now, We take an arbitrary,

Abelian translation of the variable zed.

Where lambda and nu two integers.

Then I get e to the power -2 t i

m lambda to the square tau plus

2 lambda zed phi tau zed, and

we have this elliptic equation for

arbitrary lambda and mu.

This is two functional equations.

The modular equation, which we have for

arbitrary element also modular group.

And the second elliptic equation,

which were have of all translations

of zed by this element.

So from this point of view the first module

equation described our function phi as

a modular form with respect to the variable,

This factor is the standard at a morphic factor,

which we have seen in the definition of fusion modular form,

was a modular group, but

in this definition, we have the second rather

complicated factor related to the number M.

In the definition of modular form, of Jacobi modular form.

K is the weight of this

Jacobi modular form,

and m is the index.

So you see index m in this rather complicated

exponential factor in the module equation,

and we see the index in the second equation as well.

In the second equation, we describe our modular form as a double periodic

Function in zed, it's not really double periodic,

but this is double periodic up to this Data factor.

But nevertheless, we can

consider the variable z as an element.

The complex field modular [INAUDIBLE] z,

[INAUDIBLE] plus z.

Which we just as well know,

I hope you know that this is so called elliptic curve.

Determined by the period tau.

This is not the full definition of Jacobian modular form.

In the moment, we consider only holomorphic function

on the product of the upper half plane ends the complex number.

A direct product of [INAUDIBLE] plane C, and

this function such as [INAUDIBLE] two functional equation.

We, to finish this definition we'll have to put a special

condition on the free expansion of this function

[MUSIC]

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