Examples of Jacobi modular forms

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来自 National Research University Higher School of Economics 的课程

Jacobi modular forms: 30 ans après

11 个评分

National Research University Higher School of Economics

Jacobi modular forms: 30 ans après

11 个评分

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)

从本节课中

Zeros of Jacobi forms. The Jacobi theta-series, the Dedekind eta-function and the first examples of Jacobi modular forms

This module is devoted to zeros of Jacobi modular forms, their Taylor extensions and the first examples of Jacobi forms. Using classical Jacobi theta-series and Dedekind eta-function we construct a series of Jacobi forms. Also there is a peer review in the end of this module.

The zeros of elliptic functions13:30

The zeros of Jacobi forms (part 2)11:08

Taylor expansion of Jacobi forms12:47

Taylor expansion of Jacobi forms (part 2)13:27

Dimensions of some spaces of Jacobi forms10:31

Examples of Jacobi modular forms14:24

与讲师见面

Valery Gritsenko

Laboratory of Algebraic Geometry and its Applications

[SOUND]

[MUSIC]

>> To construct the generators of the correspondent

pace of Jacobi cusp form, we'll study Jacobi theta series and

this is our next section, Jacobi,

Theta series and examples

of Jacobi modular forms.

But first I recall you the property of the classical Dedekind eta function.

So this section, for

this section we can add another title.

Jacobi theta and Dedekind,

Eta.

The Dedekind

eta function.

The Dedekind eta.

Function eta is defined by

the following infinite product.

Q to the power 1 over 24,

the product of the following factors,

where n is greater or equal to one.

Eta one is the well defined function on the upper half plate.

And this function has the following invariant properties.

First of all, this function

is quasi-periodic.

More exactly, adding tao plus 1 is equal to e to

the power pi over 12.

Theta.

This is clear from the definition of Jacobi form.

The next property is more complicated.

Eta in minus one

of it is equal

square root tau

b eta and tau.

So this, the first equation, this a modular equation for the translation.

The second equation is a modular equation with respect to the evolution.

The only problem that we have the square root from tau.

We fix the branch using this condition.

Square root is positive if tau is positive.

But these two model equation

have the following corollary.

Eta function is a modular form

of half integral weight.

Means that this functional equation is true for

any m equal to a, b, c, d and s, z.

Where V eta M is the root of

what a 24 from 1, and

this is a projective character.

This means that the product.

Of two values of the v eta is

equal to +1 or -1, v eta,

from the product of two matrices.

We call this function multiplier system.

This is multiplier system.

Of the modular group s of 2 z.

Certainly the square, of eta is the usual character.

Usual multiplicative,

a character of order 12 and

the corresponding even power

of Dedekind eta function.

Will be modular form of integral weight.

Okay.

With respect to the full modular group, s l to zed, with a character

For any k greater or equal to 1.

In particular, eta function to the power of 24.

This is Delta function.

This is the cusp form of weight 12 with respect to the whole modular group.

For this function,

we have two nice identity of Euler.

One can calculate the Fourier expansion of

the Dedekind eta function, have the following formulas.

Minus 12 over n, this so called Kronecker symbol.

Q to the power N square over 24, where n is greater or

equal to 1 and 12 over

n is equal to 1.

If n is congruent to plus or

minus 1 L, minus one,

if n is congruent to plus or

minus pi, [INAUDIBLE] 12.

And 0, if n [INAUDIBLE] are not [INAUDIBLE]

This is the famous Euler formula who found also

The following expression for the cube of the Dedekind eta function.

Minus 4 n times n q to the power

n squared over 8 n greater to 1.

Where -4 over n is another chronicle

symbol, this is + or- 1,

if n is congruent to + or -1 model 4 or

0, if n is congruent to zero

modular 2 if n is even.

This is 2, formula of Euler.

You can prove these formulas by yourself but certainly this is not so easy.

But, some standard can be notarial argument

could help you to get both formulas.

Maybe in the better file,

I'll give you some more detailed argument about those two formulas.

But now the main hero of this section,

[SOUND] Jacobi,

Theta function.

Jacobi theta series is a function in two complex variables.

Tau is in H, z in C.

By definition, this is the following function.

-1 to the power n-1 over 2,

E to the power pi times n

squared over 4 tau plus nz,

where summation is taken over all odd numbers.

But we can give you another expression for the same function.

This is the sum -4 over n,

q to the power n square over 8,

r to the power n over 2 as usual q,

e to the power two P-I tau.

R-E to the power two P-I-Z.

When summation is taken over all that.

Please compare two formulas then you

can check then this expression gives

us another expression for the Kronecker symbol.

And what is very important and was proved by Jacobi, then

We have another expression for

the same function, this is

famous Jacobi product formula.

Then for the same function,

we have an expression in terms of

infinite products q minus, sorry.

Minus q to the power 1 over 8, r to the power minus 1 over 2.

The product of n 2 greater than 1.

One minus q to the power n minus one r.

One minus q to the power n r to the power -1.

One minus q to the power n.

So we have two different formulas for Jacobi theta series.

And the properties,

the modular properties of Jacobi theta

series will start in the next lecture.

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