A new proof of Cohen-Kuznetsov-Zagier theorem

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Из курса от партнера National Research University Higher School of Economics

Jacobi modular forms: 30 ans après

12 оценки

National Research University Higher School of Economics

Jacobi modular forms: 30 ans après

12 оценки

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)

Из урока

Jacobi type forms and the generalisation of the Cohen-Kuznetsov-Zagier operator

The last module is devoted to Jacobi type forms. In this module we also consider the generalisation of the Cohen-Kuznetsov-Zagier operator. Also there is a peer review in the end of this module.

A new proof of Cohen-Kuznetsov-Zagier theorem15:12

A new proof of Cohen-Kuznetsov-Zagier theorem (part 2)9:09

Cohen-Kuznetsov-Zagier theorem in case of many variables14:04

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Valery Gritsenko

Laboratory of Algebraic Geometry and its Applications

[SOUND].

Dear colleagues, dear participants of the course.

Today, we are studying the last lecture.

The last section Is about generalization

Of Cohen-Kuznetsov-Zagier operator.

This operator was defined by Kunetzov, by Cohen and

by Zagier in different context.

Let me formulate the definition of this

operator in the context of Jacobi form.

Let f be a modular form, of weight K,

with respect to the full modular group.

Then, the coin

Kuznetzov-Zagier operator

Transforms, maps f into the space

of the Jacobi type forms.

We defined the space in the last lecture.

By definition this is,

A function in two variables Which is equal to k- 1,

factorial, where k is the weight.

Then the sum of all non-negative,

n 2 pi i to the power n.

Over k-

1 + n.

So we have to write

it.

(k- 1 + n) factorial,

then n factorial, then the nth

derivative of modular form

times z to the power 2h.

This is a Jacobi type form of weight k and index 1.

More exactly this means that the function,

the image of the Cohen-Kuznetzor-Zagir differential

operator satisfies the following functional equation.

This is the functional equation

of a Jacobi type form,

is the only modular equation,

of weight K, was a Jacobi factor,

to the power by IC,

z to the square, over C tau + d.

The function, in tau and z.

We have this equation for any elements from

the modular group S of 2 [INAUDIBLE].

So you see that we can construct a Jacobi

tie form studying a modular form of weight k.

What application do I have?

This operator is very useful in different context.

But now I would like to discuss only one application.

More exactly, we can construct now, a special operation with modular form,

the so called a Cohen Modular Bracket.

Let's consider two modular form.

One modular form of weight k and

the second modular form of weight A.

Then we can take two functions.

The first coin Zagier

operator related to the function F.

And then we multiply this function by the coin

Kuznetsov-Zagier operator of the function, G.

But for that, I put the argument, E Z.

Then as a functional equation for

the first function, we have C Z to the square.

But for the second function,

-C Z to the square.

Therefore, this product

is Jacobi type form of

weight k + l of index 0.

In other words, if we write

the Taylor expansion of this problem

All Taylor

will be modular forms,

of weight k + l.

See function, I'm sorry, here it's clear,

then we'll have Taylor expansion only,

with even power with it because our function is

symmetric with respect to formation from Z to minus that.

And the, then the N's Taylor expansion

corresponding to the power, that was a power to 2n.

Has weight k + l plus 2n.

And we can write down the formula for this coefficient.

This is the ranking bracket is the sum.

Over E from 0 till n- 1 to the power e.

Then we have two binomial coefficient.

K = 1 + n e c l- 1

+ n, n- i,

f i times g n- i.

You can check without any problem that this product is a casp form, or

have no zeros for any coefficient.

It's a Casp form of k + l + 2n.

This operation is called

a Rankin–Cohen bracket.

So by two function, two modular form of k and

l, we can construct the karst form of k + l +

2n using standard differential operator.

Now I would like to generalize the definition

of the Cohen-Kuznetzor-Zagier

operator to the case of Jacobi

modular forms in many variables.

And first of all, I would like to give a new proof of this theorem.

This theorem of Cohen-Kuznetzov-Zagier, then this series

is a Jacobi type form of weight k and index 1.

The next step of our lecture is to give,

A new proof, a new purely algebraic proof,

Of Cohen-Kuznetsov-Zagier theorem.

Which works without any changes.

In the case of Jacobi modular forms in many variables.

Our idea is the following.

First, we use the automorphic correction of Jacobi form.

The automorphic correction,

this is our approach to the Taylor expansions of Jacobi forms.

So, if we have Jacobi

form of weight k and index 1.

Jacobi or Jacobi type form, this does not matter,

we get the automorphic correction by multiplication by the function, e to

the power -8

p to the square G2 (tau) Z to the square.

And we get Jacobi type form, weight k, and index 0.

Certainly, we can construct

multiplying by e to the power 8 p to the square,

G2 times Z to the square.

This is our first idea.

The second idea is to construct in some way

exponential function of the differential operator, D,

where D is our differential operator.

1 over 2 by I, D or D tao.

This works very well in the case of the modular form.

This is the case, gives the latest l is trio.

But if the latest l is non-trivial,

we consider exponential function of the theta variable.

H is the the heat operator.

In our normalization,

h is equal to -1 over 8 p to

the square 4 pi i d over d tao

minus the square over d over dz.

These two operators helps us very much.

[SOUND]

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