Jacobi forms of critical weight

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Do curso por National Research University Higher School of Economics

Jacobi modular forms: 30 ans après

11 classificações

National Research University Higher School of Economics

Jacobi modular forms: 30 ans après

11 classificações

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)

Na lição

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

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Valery Gritsenko

Laboratory of Algebraic Geometry and its Applications

[SOUND] Our

next sub

title is

Jacobi.

Modular, or

cusp forms of critical aid.

According to the theorem of there are no Jacobi form of critical weight.

In the case of but,

we constructed a Jacobi form of critical weight in our course.

Let's take

a vector, v,

In the latest D8.

And we analyzed

the pull back of

Jacobi form for

D8 on the orthogonal

complement of

the vector v.

This is a product of 8 Jacobi

tethered series data zed 1 to

data zed 7 times the 8 factor

minus zed 1 plus zed 2.

And so on to zed 7, this is Jacobi form,

of weight 4 for the latest,

A7, because the latest

A7 is by definition.

The orthogonal complement, it was a vector V and

D8 4 is the critical weight.

For A7, I put you a question,

this is a cusp form or not?

Please try to answer but now I would like to give you

an example of Jacobi cusp form of critical weight and

we know, we're sure that this is a cusp form.

So now the first example, Of cusp form.

We're taking as a vector u =

2(1, and 2).

This is a primitive vector in the lattice D8.

This vector is primitive because the sum

of the coordinate without the first factor 2 is odd.

Then, according to the serum proved in the last lectures,

the pullback of Jacobi theta function,

theta D8 on the orthogonal complement.

Of this vector u is a modular form Jacobi form of this with respect to this lattice.

Moreover, we can calculate the order of this function

at infinity that remind you since the order of Jacobi form, it infinity.

This is the minimal

values of hyperbolic.

Nomes two M minus eight to two squared.

Taking overall non-zero for

your coefficient and we'll prove than

the order on this pool back is great or

equal to one over u to the square.

And this is equal to 1 over 4 times 7 plus 4,

1 over 44, this is positive.

Therefore, this

is Jacobi casp form of weight 4,

with respect to this lattice.

The rank of this lattice is equal to 7,

so generally this function is not 0.

Identically, because we know all 0s of the Jacobi so

you see, if we have more than one variable.

Then it might happen that we can construct Jacobi cusp form of critical weight.

So this interests in question to define this dimension and

in principle we can put this question now by the dimension formula.

So is the next sub-title,

dimension formula for J K, L.

From our splitting principle,

immediately we'll have that the dimension

of this space is equal to the dimension

of the vector valued modular four

ofrade K minus N zero over two cotol.

With respect to the complex conjugation

of the right representation or

maybe I write simply here as

representation complex conjugation.

And like for the case of modular form in one variable,

you can calculate this function.

You can find this formula in the recent Freitag's paper.

[SOUND] Formula [SOUND] so

I'll give you the explicit reference.

It was this Frei tag's paper, and the bit there fine, and

rate it to the scores but I would like to emphasize that.

Usually, Frei Tag's use the method of use the formula

to calculate the Dimension Formula for the vector of the modular form.

This formula is very explicit and very useful, but

for some small Weight, like critical or singular.

Weight, we certainly have some difficulties with application

of Riemann Roch Formula.

Moreover, what

weights are small for us from many points of view.

Let me fix the least

of small weights for

the lattice l

of rank n0.

The minimal possible weight are certainly n0 over 2,

this is singular of the minimal possible.

Then, we'll have the critical weight.

Then we can go to the weight in zero and

this is so called Canonical.

Sorry, as is this not N 0 I am sorry but

this is M 0 plus 2.

This is canonical.

Now I explain it, why this weight is canonical.

For example, I assume that n 0,

understand this, n0 is equal to 1.

Then using Jacobi form of weight k and

index m, we can construct, using my lift in construction.

In modular form on

the plane, H2.

Moreover, let me

repeat this, so

Phi is Jacobi cusp

form of weight 3 and

index m.

So, 3 is exactly here,

1 + 2 is equal n0 + 2, then.

Some constructions, the lift construction of

this function gives us, a Siegel Modular form,

a cusp form of weight 3, with respect to so

called paramodular group, Gm.

Where Gm is a subgroup of the group of the group

operational point of simplistic group of genus 2.

[SOUND] And in particular, if denote this

function is phi, in the F index phi.

Then the following function is that,

is that a standard differential on the signal for

uplane is canonical differential form on Sigel trifled.

This is why this weight is called canonical.

Moreover, now I am very sketchy, without any proof,

it follows from this construction than if the space of Jacobi cusp form,

Isn't real, then H3,0 of this,

modular three fault.

Is non trivial or in the [INAUDIBLE] of this para modular

group is non [INAUDIBLE].

So you see then the dimension formula for

Jacobi cast form plus some construction of modular form for larger group.

Which we can get using Jacobi form, are very, very useful in mathematics.

So coming back to the dimensional formula.

You can analyze this paper.

The exact reference I'll give and to analyze what

type of Jacobi form you can really construct Using this dimensional formula.

But to finish this lecture, I would like to analyze

the component of the modular form, so our problem now to analyze.

The combatant fh

tau as modular

forms with respect

to congruence.

So we need one more formula.

[SOUND]

[MUSIC]

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