Dimension of the space of Jacobi forms of odd weight

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From the course by 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)

From the lesson

Theta-blocks, theta-quarks and the first Jacobi cusp form of weight 2

This module is devoted to very important notion of theta-blocks and theta-quarks. In this module we also construct the first Jacobi form of weight 2. Also there is a peer review in the end of this module.

Theta-blocks14:49

Theta-blocks (part 2)13:59

Theta-quarks14:06

Dimension of the space of Jacobi forms of odd weight14:43

Theta-quarks (part 2)11:41

The proof of theorem about theta-quarks16:11

The first Jacobi cusp form of weight 210:48

Meet the Instructors

Valery Gritsenko

Laboratory of Algebraic Geometry and its Applications

[MUSIC]

I would like to add one exercise which is not so simple.

Exercise.

Question.

The function which we construct,

is it the first cusp form of weight 4 and index 7?

I can formulate it in the following way.

If you consider the space of Jacobi cusp form of weight, four,

in index seven, index six, this space is trivial or not?

Certainly the same equation we can put in our index five, four, three, and so on.

So this function, repeat, this function is a constructor.

This is a first cusp form

of weight for 4 or not.

The method which we discussed today is not only a series of examples.

Using this approach, we can prove some theorem.

Now, I would like to prove the following theorem.

As a dimension of the space of

Jacobi form of odd weight,

any odd weight, and

index 2 is equal to the dimension

of the space of usual modular

cusp form of weight 2 k + 2.

So I can formulate it as follows.

As linear space.

These spaces are isomorphic.

Moreover, in the proof of the theorem,

we can construct this isomorphism explicitly.

As a proof, I will use the idea of theta products or theta blocks as a proof.

First of all, I would like

to analyze the divisor of any

Jacobi form of odd weight.

The divisor of a holomorphic

Jacobi form of weight, 2k + 1 and

index M always contains four points,

modular, the standard latest,

zed tau plus zed 0 one-half tau over

2 tau plus 1 over 2 modular zed tau + zed.

I can write down the same fact

in the following simpler way.

The divisor of the corresponding Jacobi form

contains the latest one-half zed Tao plus zed.

Okay, we can prove this using the functional equation of Jacobi form.

First of all, the value of this Jacobi

form in 0 is a modular form of odd weight

with respect to the full modular group.

Certainly, the space is trivial.

This is the first 0 of the corresponding Jacobi form.

Now.

Let me calculate its value.

At point, Tau over 2 minus tau.

Using the elliptic equation

of our Jacobi form,

we get e to the power -2 pi i m.

Then, lambda to the square tau but

lambda is equal to -1 here, so, we have tau.

Minus, because lambda is equal to minus one two times zed.

But zed is tao over two fe to

the power two k plus one m

Tau, tau over two.

So, I used only the elliptic equation.

This term, corresponding factor,

certainly is one.

But, tau over two minus tau

Is minus tau over 2.

And this is -1 to the weight of our function

Any Jacobi form of odd weight is odd function.

So, this is minus one.

We have proved that this function

Vanishes and tau over 2.

Exactly the same proof

The same proof using the elliptic equation,

we can realize for any other points of order two,

so for one half and tao plus 2 over 2.

So we'll prove this very important property.

Then the divisor of any Jacobi form of odd

weight contains at least four points.

In particular, I can fix this [INAUDIBLE].

This fact.

This space of Jacobi form of

odd weight in index 1 is 3.

Because any Jacobi form Holomorphic or

weak, it doesn't matter.

Any function of index one has only two zeroes

in any fundamental domain of our latest.

But arbitrary Jacobi form of odd

weight has at least four zeros.

Now, I would like to analyze the Jacobi form of weight of index two, sorry.

To construct all Jacobi forms of index two and odd weight,

I would like to take the following theta block.

The Jacobi form, weight form, of weight -1 and index 2.

We consider it this function as the last lecture.

This is the product, the quotient

of theta function over eta q tau.

And this function has exactly, has

this divisor.

[INAUDIBLE] in four points of the fundamental

domain because the divisor of this function

is equal to the divisor of theta 2 zed.

This is equal exactly 1/2.

And now, let me analyze in arbitrary

Jacobi form of index two, and an odd weight.

The divisor of this function contains this divisor and exactly equal to it.

Because, we know, that the Jacobi form of weight two has exactly four divisors.

Follows when this function is

holomorphic and weak.

Holomorphic function in this weak Jacobi

form of weight 2 k plus 2 and index zero.

Why is Jacobi form is a weak form, because we

know the Fourier expansion of this function.

The Fourier expansion starts by r minus r to the power minus one plus q.

So this quotient contains only non negative

powers of q in this fourier expansion so

this function is weak but this space of weak

Jacobi form of weight 2k plus 2 is equal.

To the space of usual modular form for the full modular group.

So we see then our Jacobi form

which we try to analyze is equal.

Pi minus 1 2 times a modular form in

one variable of weight 2k + 2.

If, Pi is Eisenstein series,

Eisenstein series of correspondent weight.

And the Eisenstein series starts the Forier expansion of

Eisenstein series starts by 1.

Then, phi- 2 times Eisenstein series

is not holomorphic because it starts for

the expansion by q something.

And the index, the hyperbolic norm of the correspondent index.

Is equal to 4 times 0 times the integer

Jacobi form minus 1 to the square, is equal to -1.

So certainly, this Jacobi form is not holomorphic.

But if we multiply this function by cusp

form of [INAUDIBLE] 2k+2 cusp form.

Then we can take [INAUDIBLE]

delta function here as a vector.

And this function is our function, phi 11, 12, which is holomorphic.

Holomorphic L infinity.

So, we proved that

in this case,

we get a function in

J 2k+1, 2.

[MUSIC]

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