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Jacobi modular forms: 30 ans après

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National Research University Higher School of Economics

Jacobi modular forms: 30 ans après

12 classificações

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)

Na lição

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

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Valery Gritsenko

Laboratory of Algebraic Geometry and its Applications

So I gave you the definition of modular forms of weight k,

let's analyze this definition once more.

So in modular form, it's holomorphic function which

satisfies the modular equation M.

And would behave at infinity.

Certainly, this sum of two modular form is a modular form of the same weight.

Therefore, the set of all modular

form of a fixed weight is a space over

complex numbers, a linear space.

So we can denote by the notation.

We denote by Mk or Mk s into z if we would

like to emphasize the modular group.

The space of all modular

forms of fixed weight.

So there's a space.

Of all modular forms

Of weight k.

It's the linear space of a complex numbers and one of the most

important property of the space then this space is finite dimension.

Fact.

This space is finite dimension for any k.

You see then, We can analyze this space and

the most important modular form,

maybe the most important modular form with

respect to full modular group is the following

function, Constructed by Ramanujan.

This is a famous delta function of Ramanujan,

from which we can define as the following infinite product.

This is a modular form of weight 12 with respect to the full modular group.

Moreover, this is so-called cusp form,

cusp form means that the constant term in a Fourier expansion, is equal to 0.

So all this stuff, I mean the definition of

modular form, the delta function and so on,

you can find in the classical nice book of [FOREIGN],

A Course, Of Arithmetic.

The last chapter of this course is about modular form in one variable,

and you can find all necessary information in this book.

But later we discuss some theorem and some result of this book also in our course.

Now, I would like to show you

the formation of this delta function,

or some elliptization of this delta function, which looks very nice.

And which is also a motivation for construction of the theory of Jacobi form.

So let's consider the following infinite products in two variables.

This is q, to the power 4.

r, to the power -4.

The infinite product for M, 3 to 1.

(1- q to the power n- 1 to the power r)

to the power 8 (1- q to the power n r

to the power -1) to the power 8 (1- q

to the power n) to the power eight.

So, you see that here is q.

E to the power 2 piT.

Then r, this is e to the power 2piZ.

So we have the modular variable tau and

the elliptic, later in the second lecture,

I explain to you why this variable is elliptic.

The elliptic variable z or r.

So let's compare the Ramanujan product,

And its elliptization when we add one more variable, r.

The Ramanujan product contains

24 factor for each n.

These product also has 24 terms.

Through its kind of generalization of Ramanujan delta function.

Let's analyze this

term for n = 1.

And we get 1.

R to the power 8.

Therefore, this function g,

Vanishes for z equal to 0.

Moreover the order of 0,

z equal to 0 of this function is equal to 8.

And you can show ,this is a simple exercise,

than the 8 derivative with respect to z,

of g for 4 into z.

Z = 0 is up to a constant, the Ramanujan data function.

This function g is a Jacobi form,

Of weight, 4 and index.

This is the second characterization of Jacobi modular form.

So, Jacobi modular form is a function in two variables.

So we have the weight, and we have the index, two characteristics.

Though this is only an example, at the moment

we don't see why this function is a Jacobi form, and

moreover we don't have the definition of Jacobi form at the moment.

But my first section mainly devotes to motivation of our subject.

So now let's analyze the subject which we discussed.

First of all, we construct the generating

function of the arithmetic problem of calculation,

Of number of representation of natural

numbers by vectors, in the latest.

So, it was our generating function.

Theta, 12, 8.

I'm sorry.

Theta of the quadratic

l in total.

And this function is a modular

form, Of some weight.

Then we consider it the partition function.

The partition function with m additional complex variable.

So z here belongs to c to the power m.

Where m is the rank of the latest a.

And this function is

a Jacobi modular form.

In n variables, z.

Moreover we consider it some pull backs of this theta function.

And the pull backs, Pull

backs gives us theta function,

In one additional complex variable.

So that is in c.

And here we get Jacobi forms,

From the book of Eichler,

And Zagier.

In the next lecture, the next lecture,

we'll start by the definition of modular form of this type.

[MUSIC]

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