Theta-blocks

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En provenance du cours de National Research University Higher School of Economics

Jacobi modular forms: 30 ans après

12 notes

National Research University Higher School of Economics

Jacobi modular forms: 30 ans après

12 notes

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)

À partir de la leçon

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

Rencontrer les enseignants

Valery Gritsenko

Laboratory of Algebraic Geometry and its Applications

[MUSIC]

Hi there, participants.

So today, we'll start our lecture number six.

We're nearly in the middle of our course,

and today we can start to do mathematics.

First of all, let me recall you what we have considered,

what has been done in our course.

First of all, after a short introduction about generating and

partition function, we introduced Jacobi modular

forms as usual modular forms with parameter.

So Jacobi form, Jacobi modular forms.

The first definition, we

gave using two relations.

Modular relation, which shows that Jacobi form is a modular form,

with respect to the first variable.

And the so-called elliptic equation, which shows,

then our function is a billion function with respect to the variable theta.

Then, we gave another interpretation of this object.

We consider it Jacobi form as a function on the plane,

where the third variable is rather formal,

when added with index M, this coefficient

M is exactly the index of Jacobi form.

And we considered this function

as gamma g modular form

on z upper half plane H2.

Gamma J in this context, Jacobi group.

Is amorphic to the semi-direct product of the usual modular group,

SL2(Z) times the circle Heisenberg group.

And we consider it, this Jacobi group,

as a parabolic sub-group of group, of genus two.

Using this interpretation, for example, we can reformulate

the definition of Jacobian modular form to be holomorphic at infinity.

Because we had two functional equation,

the modular equation and the elliptic equation,

plus some condition to be holomorphic at infinity.

But this third condition is equivalent

to the fact that our Jacobi form

consider it as a modular form on H2,

is simply holomorphic on H2

In particular, from this we immediately get the fact

that all Jacobi forms form a gradient ring because

the product of two holomorphic functions is holomorphic.

So the second interpretation of our modular group,

a subgroup Sp2Z, is very useful.

Especially when we consider the Jacobi theta series.

So we defined Jacobi theta series.

And we proved then the Jacobi theta series with

the additional factor, E to the power pi i, orbing here.

So, you see that corresponding

index m here is equal to one-half.

Then, this modular form, Jacobi theta series

considered on the upper-half

plane is a modular form of weight one-half and

index one-half with the following multiplier system.

This is the multiplier system of the function to the power of 3 times,

the binary character of the Heisenberg group.

VH with this the binary

character of the Heisenberg group.

The explicit formula for

it is given by -1 to

the power mu + lambda

+ mu times lambda + R.

Exactly this factor, mu times lambda,

this term, we don't see if we consider Jacobi

theta series as a function of two variables, tau and zed.

Because these additional terms Cams

from the variable m omega.

So this is a binary character, and this character, this is very important,

is invariant with respect to the conjugation

by any element from the module group S.

So, and now I would like to put the following question.

In what sense we will do mathematics today.

I would like to pose the following question or problem.

To construct

Jacobi forms

using only

two blocks.

The first block is Dedekind eta function.

The second block is the Jacobi theta series.

And to construct as many Jacobi modular form as possible.

This is a problem which I would like to consider today.

I would like to add then you cannot find the Jacobi theta series

in the book of idea Jacobi modular forms and really,

the subject of this book we discuss in our course.

But this interpretation of Jacobi

modular form as a function on the Ziegel

upper half-plane was used very much in my

joint paper with Viacheslav Nikulin.

Please see the exact reference in the PDF file.

This is a paper about automorphic forms and [INAUDIBLE] algebra.

So this is a paper, from '96, '98,

and now, I would like to show you this

technique to put some new questions.

Moreover, now we can construct Jacobi forms which

you cannot find in any books or papers.

We're doing mathematics.

Now I would like to fix the type of blocks which we, which we'll consider today.

First of all, we proved in the last lecture the following fact.

If we multiply Jacobi theta series by,

if we multiply the second variable by a,

we get a Jacobi form of weight one-half and

index a to the square over 2 with respect to the following multiplier system,

the cube was [INAUDIBLE] function times the power a of the Heisenberg character.

So, we proved this fact in the last lecture and

I would like to use the following notation.

I denote this function simply as theta of the index a.

So certainly we can use this function

as a standard block, especially because if A is even,

the Heisenberg character will be [INAUDIBLE].

So the most general function which we consider at the moment

has the following form.

Add the function the power

d times Theta a1,

Theta a2, so on,

theta ak where ae is

strictly positive.

This is not a restriction because Jacobi theta series are old functions.

But certainly, if we put here a equal to 0, the product will be 0,

because we know that Jacobi theta series as the only divisor.

This is an integral point point of the latest Z Tau + Z.

So and we would like to get a Jacobi

modular form with trivial character.

Let us control the character of this product.

First of all, the multiplier

system has the form d + 3 times k.

Assume that this is equal to 1, identically.

Moreover, I can calculate the Heisenberg part of the character.

So VH to the power a1 + so

on + ak which is also equal to.

So the first condition gives

us d + 3k = 0 module 24.

In particular, d is 3e

Is divisible by 3.

And the second condition gives us the following restriction.

The sum of the powers is even.

Now, I would like to consider all product with this condition.

[MUSIC]

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