Taylor expansion of Jacobi forms

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来自 National Research University Higher School of Economics 的课程

Jacobi modular forms: 30 ans après

10 个评分

National Research University Higher School of Economics

Jacobi modular forms: 30 ans après

10 个评分

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)

从本节课中

Zeros of Jacobi forms. The Jacobi theta-series, the Dedekind eta-function and the first examples of Jacobi modular forms

This module is devoted to zeros of Jacobi modular forms, their Taylor extensions and the first examples of Jacobi forms. Using classical Jacobi theta-series and Dedekind eta-function we construct a series of Jacobi forms. Also there is a peer review in the end of this module.

The zeros of elliptic functions13:30

The zeros of Jacobi forms (part 2)11:08

Taylor expansion of Jacobi forms12:47

Taylor expansion of Jacobi forms (part 2)13:27

Dimensions of some spaces of Jacobi forms10:31

Examples of Jacobi modular forms14:24

与讲师见面

Valery Gritsenko

Laboratory of Algebraic Geometry and its Applications

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Theorem.

The dimension of the space holomorphic Jacobi form of weight k and

the index m is a finite dimension.

Proof.

In the previous lecture we construct,

Special pullbacks of Jacobi modular form.

More exactly, for any irrational vector.

P minus Q in Q power of 2.

We construct and for any Jacobi form,

polymorphic Jacobi form of weight k and index m.

We construct the following, function.

By definition,

this is e to the power 2 pi i,

q to the square tau plus pq,

times phi (tau) q tau plus p.

This special value of the Jacobi

modular form in this rational

point is a modular form of weight k,

with respect to a congruence of group gamma x,

of the group with the character x,

gamma x is subgroup, SL2.

Now we can choose many rational plans.

For example, we can choose 2M point X1,

X2 and so on, then The map,

Pi tao Z to pi xi I

tao 2m Xi is not

equal to Xj EQ2

then this mat is

injective.

Why this is true?

Because we proved in

the corollary 3.

Then the elliptic function phi is determined

uniquely up to multiplicative comfort by zero.

Therefore, this function

is defined uniquely by these images.

Immediately, we get.

Then, as a dimension.

Of the space of holomorphic Jacobi form of weight k, in index m.

Smaller or equal than the sum of the dimension

of the corresponding spaces of modular forms.

And this sum is finite.

This theorem is proved.

I would like to tell you that this estimation Is not good at all.

Later we get better result about the dimension

of the space of polymorphic Jacobi form.

But, at the moment, this estimation gives us this very important result.

Which we claim In our first lecture.

Then for all k and m.

The space of the Jacobi modular form is a fine to be matched.

To get a better estimation, I would like to consider

Taylor expansion to Jacobi modular form.

It means that I would like to analyse Jacobi form

in the neighborhood of 0 of the complex plane.

Section 5.2.

Taylor Expansions.

Of Jacobi forms.

Let i

B in Jacobi form, what is corresponding Fourier expansion.

In the summation, 4nm- l to the square is greater or equal to 0

q = e to the power of 2 pi i tau,

r = e to the power 2 pi i z.

I would like to analyze the Taylor expansion in z.

Fd(tau) z to the power d,

d is greater or equal to d0 is greater or

equal to 0, where d0.

This is order of zero, of Jacobi form.

Analyzing the Taylor expansion,

I would like to get a better estimation on

the dimension of the space of Jacobi forms.

First of all, let me analyze F0.

By definition, this is a value on Jacobi form zero,

and we can find it's Fourier expansion.

Sum over N, great equal to zero,

then it's the sum over 8 charged at 4 and

NM minus L to the square is greater or equal to 0.

So this sum is finite.

N,l, q to the power n.

This is the modular form of weight K with the respect to SL2.

The function phi, the Jacobi modular form, is periodic.

Is equal to five tao vector.

Therefore, all Taylor coefficients are also periodic.

And we can define Fourier expansion of this function.

The following property is true.

If d is positive then we

don't have the constant

in the Fourier expansion.

N is strictly positive here.

Why do you have this property?

Let's analyze once more,

the Fourier expansion, of Jacobi form.

If n is equal to zero.

Then the corresponding folic

coefficient is zero for

all A, for all non-zero A.

Therefore, we can

get positive power.

Of D, Only

from purification

with M positive.

It means that, any power,

Z to the power D comes with some positive

power of Q So

this, Property is true.

But in general, this is not a modular form

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