Examples of Jacobi forms (part 2)

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來自 National Research University Higher School of Economics 的課程

Jacobi modular forms: 30 ans après

11 個評分

National Research University Higher School of Economics

Jacobi modular forms: 30 ans après

11 個評分

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)

從本節課中

The Jacobi theta-series as Jacobi modular form. The basic Jacobi modular forms

This module is devoted to detailed study of Jacobi theta-series. We will discuss abelian, modular and some other properties of this function. Also there is a peer review in the end of this module.

Examples of Jacobi forms (part 2)16:14

Properties of Jacobi theta-function9:31

Properties of Jacobi theta-function (part 2)13:36

Properties of Jacobi theta-function (part 3)15:26

Construction of Jacobi forms using theta-series15:42

Construction of Jacobi forms using theta-series (part 2)13:24

與講師見面

Valery Gritsenko

Laboratory of Algebraic Geometry and its Applications

[MUSIC]

Hi, we are studying our fifth lecture and

we continue the six chapter.

Jacobi theta series.

This is one of the main object of my course

and due to this function Eichler–Zagier

called the function we are talking about Jacobi modular forms.

But you cannot find the Jacobi theta series in the book of Eichler–Zagier but

for my course, this is really one of the main tool.

By definition.

This is the following series.

The [INAUDIBLE] symbol,

E to the power P-I N to the square

over 4, tao, +nz.

Summation is taken over all integers.

As usual, tao are able from the upper plane and z is a complex number.

4n is equal to + or -1.

If M is to + or- 1, modular 4.

And zero if M is even.

Let us study this function.

The first question is about the convergence.

This function.

Converge.

Absolutely.

And normally.

On any compact set.

Of H 1 cross C.

More exactly,

we can make the following estimation of the general term of this function.

Let's assume that imaginary part

of Z is smaller than c.

And imaginary part of tau which is a positive

number is strictly larger than epsilon.

So we estimate the general term of this series, it's absolute value.

Restrict e to the power,

Pi epsilon and

to the square over.

Which is

equal to e p.

N C -epsilon

n 0 over 4,

times e to

the power -p

epsilon n,

n -n0 over.

Now, we can fix n0 in

such a way that this

number is negative.

Then, this is strictly

smaller than E to the pi

to the power nine,

E to the power, n, n -n0 over.

So you see that this series converges very, very rapidly.

[NOISE] Now, it's easy to see that this

function is anti symmetric with respect to z.

Or it's better to say that this function is and z.

This is directly follows from the definition because our character,

the chronicle character, is in our function.

Therefore.

Our theta series vanishes in there.

Now, I would like to study its quasi periodicity.

More exactly, we prove that this

function satisfies the following

functional equation for any lambda in mu.

In that.

-1 to the power lambda + mu

times e to the power -pi i,

lambda to the square [INAUDIBLE]

+two lambda theta, z.

So you see, that here we have two factors.

The second factor is similar to the factor we saw in the definition of Jacobi form.

The only difference that in the definition of the Jacobi form of weight k and

index m, we had the factor 2 pi i m.

Therefore.

The Jacobi theta series suggests phi as

a correspondent elliptic equation for m = 1/2.

The first one is + -1.

It depends on lambda and mu.

Let's prove this equation.

So this is the elliptic equation of Jacobi theta series.

To prove it, [COUGH] we have to use the definition of Jacobi form.

This function is equal to the following sum.

E to the power T i n

square over 4 tau +.

N z plus lambda tau +.

Let's try to find a resumption in these series.

First of all.

M is always odd.

Because they're connected character -4 over n.

4 over n is 0 4 even n.

Therefore, e to the power

phi i n mu is = to -1 to the power mu.

We analyze this.

Now, I would like to write the rest.

We can write this expression in full length form.

First of all,

Tor times n^2 over 4 +.

N lambda and I would like to add lambda to the square.

Here.

We get the full square.

N over 2 = lambda to the square.

It would denote this as (N over 2) to the square.

Now.

We have to cancel the additional term lambda to the square.

+ z(n + 2 lambda)

because I would like

to have here N.

-2 lambda z.

So we were right.

The expression which we have on the exponent.

So let me continue summation.

So we get then our expression

star (*) = to the sum over -4 over n.

E to the power -p i capital

N to the square over 4

tau + capital Nz -, or

I can put simply here.

E to the power -pi i

lambda to the square

2 + 2 lambda z.

Wait capital N Capital N

is = +2 lambda,

but then you can check -4

Over small n is equal to -1 to

the power lambda the connector

symbol -4 over n +2 lambda.

If lambda is even this is evident.

But if lambda equal one modulo four,

we change one to three and three to one modulo four.

So this property is more or less evident.

So to finish we can change.

Our character.

The character -1 to the power lambda, -4 capital N.

And we get.

First of all, we have additional Sin

-1 to the power mu, which we have to,

certainly to add to this summation.

And as a result, we have (-1)

to the power Lambda +mu times theta

series into Z times the additional

vector for the.

The e to the power -Ti

( lambda squared tau + 2 lambda z).

Therefore, we proved that our Jacobi theta series is quasi periodic function.

[MUSIC]

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