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Automorpic correction

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Automorpic correction

This module is devoted to Quasi-modular Eisenstein series. In this module we also define the automorpic correction of Jacobi forms and its Taylor expansion that gives us the way to construct the series of Jacobi forms. Also there is a peer review in the end of this module.

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)

Visualizar o programa do curso

Ministrado por

Valery Gritsenko

Transcrição

[SOUND]

[MUSIC]

Hi there colleagues, we are starting the lecture 10

Our current title, auto morphic direction of Jacobi form,

This construction Works very well in

the case of Jacobi forms in one variable,

the case of [INAUDIBLE] Jacobi forms.

But also we can use it for Jacobi forms in many variables but

let me start was the following theorem.

You can find it in the paragraph of the book.

On Jacobi forms, the formulation of this theorem is the following.

Let phi be a weak Jacobi form

of weight, k, and index m so

we consider this innovation

described in [INAUDIBLE].

Then, we can consider Its Taylor

expansion in Zed, for Zed = 0.

And all theta coefficient,

coefficient FF tau are quasi modular form

N plus K,

I give you today

The different proof of this fact, our method,

I call this method automorphic correction.

Works very well in this case,

it gives different proof of this result of all the vectors I give.

Moreover, we can generalize this method without any problems

to the case of Jacobi forms in many variables.

And in the case of many variables this result is

our form of automorphic correction gives very nice and very constructive result.

Which is better than this formulation in the case of Jacobi forms in operator.

But first of all, I would like to discuss Quasi-modular forms, what are they?

And our first sub section today will

be modular differential operator.

We start.

With a differential operator.

In tao with this normalization 1 over 2 pi i.

We use this normalization because we can derive this

differential operator in terms of our formal variable q,

so q, this as usual, 2 to the power of 2 pi i, tau.

And this operator, act on any holomorphic or

miramorphic function.

In the following we'll have to multiply

four coefficients by its index error.

This is why I used this normalization.

So z differential operator transform holomorphic form into holomorphic

but this operator is not modular in general so it means then

if I denote this by d then d doesn't keep.

The property to be modular but

in one special case, this is still true.

Let's consider the action of this operator

on Jacobi modular function of weight 0.

So we can consider weak modular form.

Weak means that we accept forms at infinity.

In the Fourier expansion of this function has the following form.

The index n in greater equal to minus N, where N is a natural number.

So, this function has a pull of order capital M infinity and

now I would like to analyze the modular behavior of

the action of the differential operator on f,

maybe I can use other slide.

F is a modular function.

It means that this function is

invariant with respect to all

element in the who ordered group.

Now we can apply at the rate,

D to the left and to the right to the side and of the support it.

On the left hand side we get C tau

plus D to the power minus 2,

D F zero A tau plus B over C tau plus

D is equal to d of 0 and tao.

So the differential

operator d transforms weak

modular functions modular function was infinity.

Into the space of weak modular

form of weight 2, with respect for modular.

So for the weight 0, the is a differential operator, D is a module operator.

What can we do, For

the classical modular form of weight k,

I would like to consider the following operator.

So we fix.

A modular form of weight K.

Then.

We can define the following.

We divide the modular form f

by The [INAUDIBLE] function to

the power 2k then we apply the differential operator d.

And then, we'll multiply the result by the same power of the data function.

I would like to analyze this operator and to calculate what function you will have.

The element for calculation

shows that it's equal to df

times eta to the power of 2k

Minus f 2k eta to the power

of 2k minus 1 D eta and

we have to divide by the power,

not 2, by the power of 4k,

the square of the denominator.

The result Is the following (Df) tau- 2k,

then we have the logarithmic

derivative of the function

times Is a modular form f.

So this is our DK.

What type of modular transformation do we have for this function?

The theta function is modular,

with respect to the full modular group.

Therefore, this function.

Is a modular form.

Of weight 2.

Weak modular form maybe with some

infinity with respect to the full modular group, S h2 zed.

But with a character, the character

will be the multiplier system of the Dedekind function

to the power -2k.

This is a second factor but

we have to multiply this function to the first factor.

So we see that this product in

fact belong to the space of maybe

weak modular form of weight 2,

of weight 2 + k with

To have a better formula for this function, I have to analyze

the logarithmic derivative of the data function.

And this is [SOUND] our next task, so

I would like to calculate, [INAUDIBLE] derivative,

of the decay data function, where,

The function is given by the following infinite product.

We have used this product several times in our course.

Our problem now is to calculate the logarithmic derivative

of the Is that

maybe I write down that it's modular property.

This is a modular form with weight one-half,

with respect to full modular group Zed,

to Zed with multiply system,

where v eta is the root for the 24 from 1.

[SOUND]

[MUSIC]

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