Jacobi forms of critical weight (part 2)

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來自 National Research University Higher School of Economics 的課程

Jacobi modular forms: 30 ans après

12 個評分

National Research University Higher School of Economics

Jacobi modular forms: 30 ans après

12 個評分

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)

從本節課中

The Weil representation and vector valued modular forms. Jacobi forms of singular weight

This module is devoted to very useful notion of the Weil representation and vector-valued modular forms. In this module we also define Jacobi forms of singular weight. Also there is a peer review in the end of this module.

The Weil representation11:58

The Weil representation (part 2)15:49

Jacobi forms of singular weight14:03

Jacobi forms of singular weight (part 2)15:34

Jacobi forms of critical weight15:34

Jacobi forms of critical weight (part 2)12:14

與講師見面

Valery Gritsenko

Laboratory of Algebraic Geometry and its Applications

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To describe this behavior, we need the following definition.

The level.

Of the latest L, by definition, as this is,

The minimum positive q such that,

q times the joule lattice is a sub lattice of L.

So, you can really analyze at this level, or

in [FOREIGN], for different lattices,

but using the structure of the dual lattice.

The fact that this is determinant of L,

you can prove that q is always

a divisor of 2 determinant, L.

Now, we need the Hecke

congruence group,

gamma 0 (q) of s into Z.

A group of elements such that c is congruent to 0 modular q.

And now we have the following fact.

To prove it, we have to analyze very careful

the modular behavior of the Jacobi theta series or

general Siegel theta series, with group heuristic, so.

Follows, From

the modular, Properties,

Of theta L, h.

The fact is the following.

For any h in the discriminant

group, the component.

f h of the vector-valued modular form

satisfies the following functional equation.

So maybe I need more place for this equation.

Fh (a tau + b / c tau + d),

that is a modular form

of weight k- n0/2.

With some naturally character but multiply system.

L of order, 8 of the metrics.

M this our a, b, c,

d in gamma 0 q times

the following constant,

minus pi i a b h to the square.

And then we have,

The component with index of our vector.

So we'll have the following functional equation for

the component of the vector-valued modular form SL2z vector-valued modular form.

But this is functional equation with respect to the Hecke congruence subgroup.

If n0, the realm of a is even,

then we have the following explicit

formula for this function.

In this case, this is not of order 8, but

simply a quadratic character of the following form.

The sign of d means plus minus 1 to the power n0 / 2.

And then the chronic symbol,

Of the discriminate of a over the absolute value of d.

a, b, c, d are elements of the metrics

from the Hecke congruence subgroup.

So you see that in fact you finalize as a component of our

vector modular form with respect to the Hecke congruence subgroup.

We have this orbit a relation between the component of our lattice.

So, in some simple cases,

let us use then,

the discriminate as a group

is a sightly group of prime order.

Then there was a chance, I formulate it like this,

To write all components

of our vector related

modular form in terms

of fo, and tau.

So in some cases, this is really true.

For example, you can can analyze the Eiglier Zigea book.

And to understand, for example, the structure of

the Jacobi form of index 1.

It means p is equal to 2 in this case and for some other prime indices.

So if this is true then we can prove

sometimes that the space of Jacobi

modular form, it's just isomorphic,

through the space of vector-valued.

I have here the symbol of vector,

vector-valued modular form of weight k- n0 / 2, (tau).

But in some cases, the space,

if the last argument involves the component is true.

And if you can write down all component in terms of the 0s

component then will be isomorphic to some space.

M which is subspace.

In the space of modular form of weight k- n0

/ 2 with respect to the congruence (gamma o (q)).

This is exactly the case of Eiglier Ziga modular form.

You can find this theorem in the book of factors again.

Where we can prove that the space of Jacobi

modular form is isomorphic to some subspace, so

called Kohnen plus subspace of the modular form of half integral index.

So, in this direction, you can start a realize

structure of the space of Jacobi modular form for

the lattice L and the main conclusion of this lecture.

Then the main object, For

this space is the discriminate group.

Group D A defines discriminate group, and I can add.

Now, I can not prove it, cannot give illustration,

but in the PDF file to this lecture maybe I add some remark on this.

Then you can add also

the property of the finite octagonal group of this finite quadratic model.

For example, the action of this group could give

us a very nice decomposition of this space.

With respect to some characters of this finite octagonal group.

So you see then the Jacobi modular form and

vector-valued modular forms are equivalent object.

But sometimes Jacobi modular forms has a lot of very nice features.

We can take the multiplication of Jacobi form.

We can write down polynomials in term for Jacobi form.

So this type of effect, Jacobi forms as Jacobi forms, but

not as vector-valued modular form,

I would like to study in the last part of our course.

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