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The splitting principle

Course video 47 of 71

In this module we continue studying Jacobi forms in many variables. Among other things we discuss splitting principle and realize theta-quarks as a pull-back. Also there is a peer review in the end of this module.

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)

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教学方

Valery Gritsenko

脚本

[SOUND] The second

part of this

lecture.

Is very, Very important for

Jacobi forms in many, or in one, variable.

This is the splitting principle.

I would like to represent.

An arbitrary

weak or holomorphic or

cusp Jacobi form of [INAUDIBLE]

for the lattice L As a product.

Of the vector valued,

modular form x the vector our Jacobi

theta series with characteristic

with summation is taken over all classes of

the geo latest module is the latest.

It means that, the jacobi form I,

we can represent as a scalar product of vector,

valid modular tone, times the vector.

Of theta serif.

A H.

Some remark about this formula.

First of all, the order of this group,

is equal to the determinate of L.

So we'll have a finite sum in this summation.

Jacobi Theta series with characteristic h,

Is by definition the following function: E to the power pi i

V plus H to the square,

plus V plus HZ.

[INAUDIBLE] efficient

V the summation is

over the For

example, if L

then this group has only one element,

and in this case,

we have the usual Jacobi theta series of the lattice L.

We use this formula fore the even new modular latest E8.

And also with an additional character for the Jacobi series of dm.

So to prove the splitting principle

we have to recognize the Fourier

expansion of the Jacobian data series.

We can work with weak Jacobian.

Reorganization.

Of Fourier expansion.

First of all, I would like

to write down the second

function of equation for

Jacobi form, for

any Lam and nu and

L pi into z plus lambda 2

plus mu into the power

2 pi i 1 half lambda to

the square 2 plus lambda

zed is equal phi tau zed.

I simply put the standard factor

to the lefthand side, and I modify it a little bit.

This factor putting 2 here,

because I would like to write down the Fourier expansion of our module of 4.

So now the the 3 expansion of the function in the left hand side has a falling form.

This sum, ANA,

each is a problem,

2 PI N Thor plus A,

that is more complicated.

And we have to add the additional effector

This is equal to, The summation is over n.

Now I would like you to write This Fouriam expansion,

Putting all

we get n +lλ

+1/2λ to

the square.

Comes, from this term.

This is coefficient N tau

plus zed l plus lambda.

And this is equal to the function pi (tau, zed).

n, l. Now we can compare the Fourier coefficient

in the lefthand side and in the righthand side of this formula.

So we get the following.

We see the a(n,l) = a(n+l

lambda) + lambda

squared over 2, l + lambda.

The hyperbolic of the index of the fourier

coefficient on the left hand side is equal to heterobolic, no.

Of this index and we prove the fallen proposition.

The Fourier coefficient.

A (n,l) depends

only on the hyperbolic norm

of its index, and on the class.

N modular (l,l) is N,

This is true for all the Jacobi form, form.

Anamorphic or weak Jacobi form doesn't matter.

So, you see that in principal we can organize

a resumation in the fully expansion of Jacobi form.

And now, I would like to

organize it, first of all I,

would like to denote by N one

half of the hyperbolic non of

the index of jacobi form is equal

to N minus l squared/2.

Then a (n,l)

= a Capital N +

l square over 2, l).

With l, we can consider modular L.

Let us fix A vector h,

Modular L.

Then, We

get the following property.

According to the last Fact

about fourier coefficients or

Jacobic form.

We have that, we have the same fourier coefficient.

If we modify h by any vector v,

in l, and now I

would like to write the Fourier coefficient

of Jacobi form using this new index.

First of all, we'll take a finite sum over

all classes Of L star over L.

Then, what takes the summation over all N.

But N is rational in general, because

the vector L is so L squared Is rational in general.

So we take N congruent

to minus H squared over two, modular Z.

We need this condition to have an interger here.

Then we take a, N plus h square over 2, h.

And then, we

add the summation over all vector V.

In L.

E 2 pi i (N+ h+v

to the square over 2.

So we use this fact.

+ (h + v, zeta)).

But now, We can put these coefficient here,

because it doesn't depend on v.

I get here e to the power 2 pi i N.

And we can cancel with N here.

Then this function There's nothing else but

Jacobi that the series with characteristic h.

[SOUND]

[MUSIC]

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