Modular differential operators for Jacobi forms

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From the course by National Research University Higher School of Economics

Jacobi modular forms: 30 ans après

12 ratings

National Research University Higher School of Economics

Jacobi modular forms: 30 ans après

12 ratings

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)

From the lesson

Modular differential operators. The graded ring of the weak Jacobi modular forms

This module is devoted to Modular differential operators. In this module we also consider the Jacobi forms as the space with the structure of the bigraded ring. Also there is a peer review in the end of this module.

Modular differential operators for Jacobi forms15:32

Heat operator14:12

Heat operator (part 2)15:35

Applications of differential operators for Jacobi forms11:59

The graded ring of the weak Jacobi modular forms17:45

Jacobi type forms in many variables14:32

Meet the Instructors

Valery Gritsenko

Laboratory of Algebraic Geometry and its Applications

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Dear colleague, hi, we're starting our 11th lecture.

Its title is Modular Differential Operators for

Jacobi Modular Forms.

So we discussed modular differential operators for

usual modular forms, but now we would like to construct

similar operators for Jacobi modular forms.

Let me fix the standard setup.

First of all let M,

Be a quadratic lattice.

Of rank n.

Then a complex variable, a complex vector,

from the complexication of this

quadratic space we can represent as

a sum z e times the vectors of the basis

ei where z e is a complex number.

Then we define The vector d/dz,

this is by definition the co-vector and

we write co-vector in the dual basis.

Means that scalar product of ei and

ej* is equal to Kronecker delta symbol.

So you can check then if we take the quadratic

differential operator dz, d z.

Let denoted by Delta this differential operator of order two.

Then I would like to apply this operator

to the following standard function

e to the power 2 pi i (l, z).

Where l is a vector,

l is a vector from real,

let's see, real lattice,

real space M and

that is a complex vector.

Then you can calculate,

you'll get 2 pi i to the square,

l to the square, e to the power 2 pi i (l,z).

Please check this.

This is a standard formula, or related to Laplacian,

but in our case I don't fix any signature of the matrix M.

So we have a differential operator of order two for

any non-degenerate quadratic lattice M and

you see that we can apply this differential

operator to the standard exponential function.

But now I would like to write down this operator in more concrete context.

Now we take

a positive-definite

lattice L of rank n0.

Then we take

The following hyperbolic lattice

U plus L renormalized by 1.

Then the signature of this lattice

L1 = (1, n0+1).

I remind you that by U, we denote

the hyperbolic plane with the following

Gram matrix, so this is a unimodular hyperbolic plane of signature 1,1.

Now, we can take an arbitrary vector,

that from the complexification of L1,

And I write down this vector as a vector,

(tau, omega) in the natural basis of the unimodal

hyperbolic plane + we have orthogonal sum here also,

by plus I denote the orthogonal sum.

U and L- 1 are orthogonal

to each other,

then the vector z (tau,

omega) is in U and

z is in L under C.

Certainly, for U also we're taking its complexification.

Let's write down the differential

operator Delta for the lattice L1.

It would equal,

in this particular basis,

to 2 delta/delta tau times

delta/delta omega-

Delta over delta calligraphic z, its square.

We have minus here because we renormalized

the positive-definite lattice by minus 1.

So this is the differential operator of order two and

I would like to understand its action on the space of Jacobi form.

Now, in the first part of our lectures,

we explained the functional

equation of Jacobi form of weight k and

index m using the function f tilde

on the Siegel upper half plane.

Where we added the third formal variable in the following way.

We add the factor, e to the power 2 pi i, m omega,

where m is the index of the corresponding Jacobi form.

The similar procedure works in this

case of modular form in many variables.

If we consider a Jacobi form of weight k with respect to a lattice L,

then we can construct the corresponding function

depending on the additional variable omega

e to the power of two pi i omega.

And in this case, we use a Siegel half plane of genus two and

the fact that the Jacobi group is a parabolic

subgroup of Siegel modular group of genus two.

Totally in the same way, we can consider

the second function as a function on

homogeneous domain of type IV and

Jacobi group will be subgroup of special

orthogonal group of signature (2, n0+2).

This really is a full analogy of the first construction.

You can find this construction in my papers, and

I'll give you explicit reference in the PDF file related to this lecture.

Now I would like to apply the differential operator Delta L1 and

I would like to consider its action

On this function in three variables,

tau, z, and omega.

What we get?

We get 4 pi i,

I guess now I would like to analyze

first of all the action with

respect to this variable.

Then we get 2 pi i from this factor,

an additional factor 2 from this coefficient.

Then, we have,

The differential operator with respect to

tau- the square (d/dz, d/dz).

Its action phi tau z e to the power 2 to the power 2 pi i omega.

Where this vector now, we can to put as coefficient here,

so, we get the following definition.

I can define the operator,

H, which acts on Jacobi

form of weight k and index.

And with respect to the lattice L and

its action up to the factor -1 over 8

pi to the square is exactly the operator,

which we have in this line,

(4 pi i d/dtau- (d/dz, d/dz)

Action on function phi.

What do we have?

And why I denote this operator by H because

this is the classical heat operator,

The heat operator related to positive dividend, lattice L.

So we can rewrite, certainly this is the rater, after r normalization, as follows.

1/2 pi i, d over d tau, minus the Laplacian

related to the positive-definite

lattice with coefficient, I'm sorry,

here we'll have plus, let me change it.

With our additional factor, we've got + 1/8 pi to the square and

we act by this operator phi(tau, z).

This is our main object for this lecture.

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