Examples of Jacobi forms in many variables (part 2)

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Do curso por National Research University Higher School of Economics

Jacobi modular forms: 30 ans après

12 classificações

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 forms in many variables and the Eichler-Zagier Jacobi forms

This module is devoted to Jacobi forms in many variables. In this module we also define classical Eichler-Zagier Jacobi forms in terms of Jacobi forms in many variables. Also there is a peer review in the end of this module.

Jacobi theta-series and Jacobi forms in many variables15:03

Jacobi theta-series and Jacobi forms in many variables (part 2)15:57

Hyperbolic reformulation14:18

Examples of Jacobi forms in many variables14:55

Examples of Jacobi forms in many variables (part 2)16:20

Examples of Jacobi forms in many variables (part 3)16:29

Conheça os instrutores

Valery Gritsenko

Laboratory of Algebraic Geometry and its Applications

[MUSIC]

The next root system of the next lattice is the lattice An.

An, what is it?

This is the standard hyperplane.

In Z n + 1.

More exactly,

These all integral vectors.

In Z n+1, the sum

of coordinates of

the vector is equal to 0.

And this is not only a sublattice of the Euclidian lattice Z n+1.

But this is a sublattice of

D n+1 because the sum of

the coordinate is even.

The root system, Of this latest, this is the root,

System, An.

So, I repeat that I use the letter A, D, E for

the latest, not for the corresponding root system.

Please calculate the number of two vectors.

And this lattice.

Now how to construct a Jacobi form.

First of all, I would like to construct

the principal weak Jacoby form for these lattice.

And I can use the fact that the root

lattice An is a sublattice of D n+1.

We can construct the following function.

So for the latest D n+1,

we take this quotient,

eta / theta.

But for the last factor, we can tell more.

We have the expression for

the last coordinate in terms of the previous coordinate.

So here we have to put

theta in (-(z1+

zn)) / theta cube.

So now let us calculate what function do we get.

So we have a function,

t of weight- (N + 1) for

the root lattice An and

this is a weak Jacobi form.

So this is a weak Jacobi form,

Of weight -(n + 1), An.

And this is really the main generator of

the corresponding gradient ring of weak Jacobi form.

I don't like to go into detail, but here certainly we have to consider

symmetric Jacobi form in their end with respect to the permutation for A.

So as you see in any case for using only Jacobi theta series,

we can construct lots of interesting function.

Using the same principle, like for the latest Dm,

you can construct very many Jacobi forms.

For example, to have a Jacobi form without character,

we'll have eight factors.

And now I can construct a jacobi form for the latest A7.

How to construct amorphic Jacobi form.

Start with the product,

Z7, seven factors, times,

The theta in the sub-frame

of the Jacobi form of weight 4 for A7.

So you see then this is really

function related ways

theta series for D8.

A simple exercise which I would like to formulate in the text.

This function,

Phi4 A7.

This function is a cast form or not.

Please analyze the Fourier expansion of this Jacobi form.

So using the same principle, I would like to repeat using the same principle.

You can construct many Jacobi form for

the root system Am.

The next example, very important example for our consideration.

I would like to consider,

that even unimodular

lattice E8.

Unimodular means that this

lattice coincides with its dual.

And for this lattice, For

the following Jacobi form, we discuss this

function briefly in the introduction in

the first lecture of this course by definition.

The Jacobi thetaseries for

this lattice is equal to the sum

e to the power Pi i (the scale

is vector v tau the sum / ov and D8 + 2v Z.

It's possible to prove that this is a Jacobi

form of weight 4 for the latest E 8.

I don't like to prove this fact now, but

there's been an equation.

This is simple exercise.

And to prove the modular function equation,

you have to use the Poisson summation formula,

E8 is unimodular, so the Poisson summation formula

gives us exactly the correct functional equation so

some technical question.

And I hope that you can really do it by yourself or

I try to prepare it in the PDF file related to this lecture.

So like you see that the idea of

Jacobi theta series is really

a generalization of this idea.

We have a generalization of usual theta series,

remind you of the definition usual

theta series in one modular we're able.

This modular form of weight form with respect to S into that group.

So really, the similar argument, related to the Poisson

summation formula, gives us the functional equation for

the Jacobi theta series in 8, additional variables.

Do have any kind of relation between these function and

the original classical theta series.

Because for theta series certainly you know than the usual theta series,

Is the Eisenstein series.

It's identical to the Eisenstein series, and for

Eisenstein series, we can calculate its Fourier expansion that

gives us a lot of very nice classical arithmetic formula.

What do I have in the case of Jacobi form?

First of all, using this function,

Gives, this function gives us many examples,

Of the classical Eichler-Zagier Jacobi form.

Let me explain you this construction.

With x, A vector

u in E8 of square to m is not 0.

Now I would like to take

the following vector z = U

times z where z is a complex number.

We consider the following function.

This is by definition the pullback

Of the Jacobi theta series for

E8, for that equals to uz.

And using the functional equation, was a Jacobi theta series in 8 variable.

We get that this function is

a Jacobi form of weight 4 and

index m where m, this is

the square of vector u / 2.

This is very nice function and I can calculate its fully expansion.

The sum of a pole b and

e 8 e to the power pi i,

e to the squared,

tau +2 vuz.

This is the definition of this pullback.

And now there's the following Fourier expansion.

We start by one,

plus This is sum of n

in z positive,

l in z, a(n,l).

e to the power 2 pi i (n tau

+ l z), where a(n, l),

a(n,l) is a number

of 4 vector to v in E8

besides v to the square = m.

And the, this scalar

product of vu = n.

So as we discussed in the first lecture,

the original function gives us the distribution

of our vector controls the distribution.

And its pullback gives us very nice generated

function related with the vector of ea.

[MUSIC]

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