Heisenberg group

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Do curso por National Research University Higher School of Economics

Jacobi modular forms: 30 ans après

11 ratings

National Research University Higher School of Economics

Jacobi modular forms: 30 ans après

11 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)

Na lição

Jacobi modular group and the second definition of Jacobi forms. Special values of Jacobi modular forms

This module is devoted to the second definition of Jacobi forms. In this module we also consider special values of Jacobi forms. Also there is a peer review in the end of this module.

Heisenberg group13:38

The action of Jacobi modular group14:08

The action of Jacobi modular group (part 2)10:58

The second definition of Jacobi forms15:17

Special values of Jacobi forms11:14

The first theorem20:50

Conheça os instrutores

Valery Gritsenko

Laboratory of Algebraic Geometry and its Applications

[MUSIC]

Hi there, participants.

We start this lecture.

And the main aim of this lecture is to get

the second definition of Jacobi modular form.

So we continue this chapter.

More exactly, I would like to discuss now Jacobi modular groups and its subgroup.

We realize Jacobi modular group is a parabolic

subgroup of the Siegel modular group.

So by definition, the Jacobi

modular group is a group of

all four by four matrices of

the following type, a,1,

d,1 on the diagonal then b,

c, 0, 0,0, 1 the last line 0,

1, 0, the second column.

And we have some other elements in all other places.

We can see that all elements of this

type in the modular group Sp2(Z).

Please check that we really have a subgroup of the Siegel modular group.

As usual, we can define this metric,

four by four metrics, as two by two metrics

Of the following blocks A, B, C, D.

Then this is an element in

the symplectic group.

Therefore, a transposed.

I put the index of

the position to

the left on the lattice AD- CB = E2.

Using this property of the symplectic metrics, we get

Then the two by metrics

constructed by A, B, C and

D belong to SL2 (Z).

Please check this.

Moreover for any,

Element of the usual modular group, we can construct

Its embedding in the Jacobi modular group.

This will be the matrix of the following form,

a,1, d,1 on the main diagonal.

A, b, c, d,

is the corresponding

places of our group, and

0 in all other places.

We get an element on

the Jacobi modular group.

Therefore, we can construct

the group SL2(Z) as a subgroup

of Jacobi modular group.

Therefore, for any g,

the Jacobi modular group,

there is an element

in SL2(Z).

Such that M to the power of minus one g,

is equal to the following,

more simple matrix.

We'll have unit matrix

in the SL2 part of the Jacobi group.

And we have some other elements in all other place.

But using the same property for

this matrix, we see that ATD

= E2 for this lattice.

Therefore, we have the following

parameterization of the latest

in the consideration.

So, if we denote this element by q,

then here we get -q.

Moreover, here we have 0, 0, 0,

p, p, r, 0, 0, 0, 0.

In fact, we have here a symmetric

matrix because these element belong to SP2.

This gives us the second [SOUND] subgroup of Jacobi modular group.

The so called Heisenberg subgroup or

the [INAUDIBLE] subgroup of the Jacobi modular group.

This subgroup contains all 11 of these [INAUDIBLE].

[SOUND] We'll

denote this element

as follows.

The column, p, q, where p, q,

two parameters of this lattice and.

Where p, q, r [INAUDIBLE].

This is the unipotent subgroup of gamma J.

Moreover, the identity

which we proved,

shows that the Jacobi

modular group is the product

of 2 in subgroup.

The first subgroup is the image of the modular group in gamma.

The second set subgroup is [INAUDIBLE].

This is the first result about the algebraic structure of

the Jacobi modular group.

Now, I would like to analyze the property of the Heisenberg group.

First of all, we can

calculate the product

of two elements of

the Heisenberg group.

The result is the following,

p + p1, q + q1, r + r1 plus,

and this is very important,

is a determinant of the two by two

matrix, p,q, p1, q1.

In particular, we see that the group H(Z) is not commutative.

[SOUND]

Why?

Because after the permutation of the first and

the second elements, we have to permute this element.

We have to permute the column of this matrix.

And then we get minus before the determinant.

So this additional part is very, very important.

But if we calculate the inverse element, we'll have no problem at all.

Because this determinate

will be equal to 0,

-p, -q, -r.

So you see that this nearly additive group.

Then, we can calculate the commutator.

Or the element h and h1.

We get, h times h1 times h to the power -1,

h1 to the power -1.

You can check that we have the first column is 0, 0.

But here instead of r,

we get two times the determinate,

q, p, q, p1, q1.

In particular, we calculated the to the Heisenberg.

If this group is generated or

this group contains all element of the following form.

[MUSIC]

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