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Theta-quarks (part 2)

Course video 35 of 71

This module is devoted to very important notion of theta-blocks and theta-quarks. In this module we also construct the first Jacobi form of weight 2. 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)

Taught By

Valery Gritsenko

Transcript

[MUSIC]

So, we construct it and it's a [FOREIGN] between two linear spaces.

Jacobi form of weight at the index 2.

And the space of weigh 2 k plus 2

with respect to full modular group.

So in this way,

R is the [INAUDIBLE] looks as follows.

This is a product,

Of a weak Jacobi form of weight minus 1 and index 2 by e.

Arbitrary [INAUDIBLE] form of a 2k plus 2.

In terms of theta blocks it looks like follows,

theta tau 2 Zed over theta.

6 tau times g, 2k plus 2, tau.

Since we can construct, is the universe map,

and this is nothing else but

the derivative, of a Jacobi form is 0.

The theorem is proved.

So you see that this idea to use corresponding function,

constructed only by two blocks.

And theta blocks is very, very useful and

constructively we can really find,

a lot of Jacobi forms explicitly.

But it's also very interesting from the theoretical point of view.

But now I would like to fix the following very serious and

very interesting question.

We can construct Jacobi forms and

Jacobi casp form of weight four.

Using the product of eight theta 0s.

Sum index and we can get the form, the usual form.

So using the product of theta function, theta, eta, product,

we can construct Jacobi form of weight four, five, six, and so on.

But what is about weight three?

Can we construct?

Jacobi forms

of weight 3?

The same question I can put about weight two using

Eta and theta functions?

And the answer on this question is positive.

[SOUND] Now I would like to formulate my theorem about theta blocks,

Or it's better to say theta cracks.

This theorem, I approved.

Or it's better to say I presented the theory first time in 2002, in my.

And the formulation of this theorem, about theta quarks, is the following.

Let a and b to positive number.

Then, is the following function,

Is a Jacobi form of weight 1 index a to

the square plus a, b plus b to the square.

Or is a character of order three.

V eta to the power 8, the practice for you.

And this function is a cusp for,

If and only if a is not congruent to b modular 3.

So this theorem shows that we can construct

Jacobi form of small weight using

theta eta quotient, not product.

Very clear corollary and very important corollary, then we can construct

Jacobi form and Jacobi cusp form of.

Let me denote it at the quark by capital theta with index a,b.

Then theta a1,

b1 times theta a2 b2

times theta, a3, b3

Is a Jacobi form of rate three index, m.

Where index m, this is the sum from one till three,

a1 to the square plus a1,

b1 plus b1 to the square.

And this is a cusp form.

If there is e such that a e is not congruent, be modular 3.

In particular, the first Jacobi

form of weight 3 we can construct,

taken as a cube of this tetraquark.

So, why we have tertaquarks?

Because, as a product of three squarks

gives us a modular form with trivial character.

So, this is a Jacobi form of phi 3.

And index 3 times 3, 9.

This function is non-cusp.

It's a very interesting Jacobi-Eisenstein series.

But the first cusp form we get, if,

we take the square of theta 1 1, times,

Theta 1 2, so let me write,

this function as a product of theta function.

So we have theta 3 times theta 2.

This is from the second factor.

And let me calculate it carefully.

Here we have theta 1 to the square theta 2 to the square.

And here we have theta 1, theta 2, theta 3.

So we have theta 2 to the power 3 times

theta to the power 4 plus 1, 5.

We have 9 theta factors, and we have to divide it by the cube of eta function.

This is Jacobi cusp form of phase 3 and

index 13, and we can prove, maybe we can prove this later,

that this is the first Jacobi form, of weight three.

And this is the first clasp form.

Jacobi clasp form of weight three.

Now I would like to prove the theorem about theta-quacks.

So I would like to prove that this theta eta quotient is holomorphic.

This is only fact, we have to prove because the fact is the weight is 1, and

the index is a to the square a b b to the square is trivial.

Maybe, you have to be sure then the character,

the Heisenberg character is always trivial.

But this is clear because we have a to the square plus b to the square plus

a plus b to the square.

This is two times our indexes, an even number.

So we have only to check that this theta,

quark is holomorphic at infinity.

And that this data quark

is Jacobi clasp form

under this condition.

[MUSIC]

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