Motivations

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來自 National Research University Higher School of Economics 的課程

Jacobi modular forms: 30 ans après

12 個評分

National Research University Higher School of Economics

Jacobi modular forms: 30 ans après

12 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

Jacobi modular forms: motivations

This module is devoted to motivations to study Jacobi forms. We provide some first examples including theta-functions. Also there is a peer review in the end of this module.

Motivations12:16

Theta-function12:16

Modular and abelian transformations12:41

Pullbacks of theta-function12:00

Modular forms12:48

與講師見面

Valery Gritsenko

Laboratory of Algebraic Geometry and its Applications

[MUSIC]

Dear students, I am very glad that you decided to start my course,

so very welcome to my first lecture.

I am planning, first of all,

to give you some motivations for the theory of Jacobi form.

So the title of the first section, Motivations.

The plan of the first section is the following.

First of all, I would like to analyze

the arithmetic generating function.

Generating,

Functions was arithmetic problem of

the representation of numbers by what's rated for.

Then we consider partition function of the same problem.

After that, I would like to give you such definition of

the classical modular form in one variable.

And after that, we construct an elliptization

of the famous Ramanujan delta function.

So elliptization,

Of Ramanujan delta function delta and tau.

More exactly, I construct here an elliptization of some infinite product.

Let me start by the first definition.

This is the definition of the latest because I would like to construct

the generating function, Of a quadratic lattice.

So let, L be an integral,

Even positive definite,

Quadratic.

Lattice.

What is it?

First of all, Lattice,

this means that L is a free

Z module of rank n.

So n is isomorphic as integral model to Z to the power n.

Then, Integral quadratic.

It means then there is this

symmetric bilinear pairing,

defined under direct product

of L with integrally.

In another [INAUDIBLE] we have here an integral quadratic form.

Even, what is it?

It means that the no

of any vector in L is even,

for any v in L.

And the last property, which is more or less evident, positive definite,

Means that for any nonzero for

v in L, its scalar square

is strictly positive.

This is a object, which we would like to work with.

So let me repeat a, b, and even integral

positive definite quadratic lattice of rank n.

Here certainly, where u that n is strictly positive.

So now I would like to define generating function for this lattice.

Or in another word, the data function of this lattice.

Let's consider the following arithmetic question.

I denote by rL(2n)

the number of vectors

in L of the fixed

scalar square 2m.

I use here 2m because the latest is even.

Moreover, the lastest is positive definite.

Therefore, this number is finite.

Now we can construct the generating function for

this quantity, or we can define the theta function of the latest a.

This is a falling formal sum,

rL(2n) q to the power n.

And we have, The summation,

over all non-negative, integral n.

Certainly the first term, of this generated function, is 1.

Because the latest is positive definite.

This is a formal function, but now, I would like to construct

a holomorphic function related to this generating function.

We put q equal to the e to

the power 2 by i tau,

where tau is a point in

the so-called upper half-plane.

This is a set of complex numbers with positive definite imaginary part.

And we have the following expression for the same generating function.

E to the power pi v to the square tau,

the summation is taken over all the v in L.

Another notation for the same function, theta a tau.

This function is holomorphic,

On the upper half-plane.

Moreover, How to,

Understand this transformation from tau to q.

So tau is an element from the upper half-plane.

And, q,

Is an element of the unit disc,

Because the absolute value of q,

Is equal to e to the power- pi y,

circle this, then what?

What do I have here at 0?

This is the image of the infinite point e,

infinity of, The upper half-plane.

So we can trade the same function,

the same generated function as a function on the unit open disc.

Or is a function in the variable

tau on the upper half-plane.

[MUSIC]

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