The first Jacobi cusp form of weight 2

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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]

So we proved the theorem about theta quarks.

Let me write down this formulation once more.

So the theta-quarks

are products of three

theta functions divided by one eta function.

So this is a theta block times of types three theta,

means the product of zeta over one eta,

and you see in the beginning of our lecture

we put this question, how many Jacobi forms we

can construct using only theta and eta.

But why we analyze only product?

We did it in the first part of this lecture.

But at the end we find this construction.

We can see they're not only products, but also quotients.

And this is not the only possible quotient.

This is theta block of type theta.

Theta ab.

The product of three theta blocks of this type,

the product of three theta quarks, gives us Jacobi form of weight three.

Can we generalize this construction?

Yes, this is really possible.

For example, I can give you an explicit formula for

the Jacobi form of weight 2, and index 37.

This is the first

cusp form of

weight two.

See please the book of Eichler and

Zagier about Jacobi modular form.

In this book is the author found the first several

Fourier coefficient of this Jacobi form.

But now I can give you the explicit formula for this function.

Theta 5 times theta 4 times theta

3 to the squared times theta 2 to the cube

times theta to the cube over eta to the power of 6.

So here we have 10

theta over 6 eta.

Certainly, this is Jacobi form of weight two and some index,

and this is holomorphic Jacobi form.

And in our case, in our particular case, this is a cusp form.

So now we can calculate as many Fourier coefficients as we like,

because using for example you can get all Fourier or

possible Fourier coefficients of this function.

So you see that this idea of theta block is very, very fruitful.

So, this is my formula from 2002.

But now, certainly, we can give you better explanation why theta-quark, and

why these theta blocks of times 10 over 6 are holomorphic Jacobi forms.

So we, at the moment, have at least three proofs of

this fact is our joined project with Nils Skuroppa and

Zegier, which I hope will finish maybe the next year.

But, I can give you one more result, maybe it's the best possible and

very useful theta blocks which we have at the moment.

This is the theta blocks of type 34 theta over 30 eta.

Certainly, this, again, in Jacobi form of weight 2 and some index.

But you see, then the structure of this theta block is much more complicated.

At least in the block, you'll see in my joint paper with Cris Poor and David Yuen.

[SOUND] Maybe this result from

we found these examples two years ago.

And I cannot tell you that this subject is totally finished,

because in the moment we don't have the full classification of theta block.

We have some algorithms, how to check that this theta blocks are holomorphic modular

form, but we don't have the full theory of them and

you see that it might be that all Jacobi forms

we can get using some constructions of this type.

For weight two, you see, J was finished by J2 because

according to the classical result of Nils Skoruppa, this is his,

the theorem, very important theorem from his PhD thesis from '95.

I think he, no '95, '85.

I'm sorry.

He proved that there is no, Nils-Peter Skoruppa,

then there are no Jacobi form of weight one.

All the spaces are free of them.

So, the way two is minimal possible, and at the moment I cannot tell you when we

really know the structure of Jacobi form two, in terms of theta blocks.

But this object you can continue, and

now I propose you to develop this subject

using not only this type of theta eta

but [SOUND] another type.

So one more example.

Let us consider the following function, which I denote by theta 3 over 2,

Theta times theta in

2 Zed over theta Zed.

This is the holomorphic Jacobi form of

weight one half, index 4 minus 1/2,

3/2 with multiplied system v eta times VH.

And to check then this is holomorphic Jacobi form,

I can propose you to prove that we have the following

Fourier expansion for this function.

Now have connect a symbol of level 12 Q.

This is the same character which we use in

the Fourier expansion of the [INAUDIBLE] function or

[INAUDIBLE] Kronecker symbol from the Euler formula.

Here we have n to the square over 24 times r to the power n over 2.

Please prove this Fourier expansion.

And you can do it using the Fourier expansion for theta series and for

the function.

And now you see that in the theory of theta blocks,

we can add also theta quotient because we still have holomorphic Jacobi form.

Now I can put the following question for you.

So I started this lecture by the following

table ,so please construct a similar table for

the theta products with theta three half.

Then you have to consider this relation not modular 8, but

modular 24, so try to generalize, to generalize

This table to the product with theta three half.

Certainly the table will be much longer,

but you can formulate very many interesting questions.

So today, we're doing mathematics.

[SOUND]

[MUSIC]

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