Automorphic correction of Jacobi forms in many variables (part 2)

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From the course by National Research University Higher School of Economics

Jacobi modular forms: 30 ans après

11 ratings

National Research University Higher School of Economics

Jacobi modular forms: 30 ans après

11 ratings

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)

From the lesson

Quasi-modular Eisenstein series. The automorphic correction of Jacobi forms and Taylor expansions

This module is devoted to Quasi-modular Eisenstein series. In this module we also define the automorpic correction of Jacobi forms and its Taylor expansion that gives us the way to construct the series of Jacobi forms. Also there is a peer review in the end of this module.

Automorpic correction15:20

Differential operator D15:03

Quasi-modular Eisenstein series15:45

Automorphic correction and Taylor expansions13:34

Automorphic correction of Jacobi forms in many variables12:07

Automorphic correction of Jacobi forms in many variables (part 2)12:22

Meet the Instructors

Valery Gritsenko

Laboratory of Algebraic Geometry and its Applications

[MUSIC]

The formula for that amorphic correction of our Jacobi form.

Capital phi.

This is zeta-morphic correction or holomorphic or

weak Jacobi form of weight k, for the lattice n.

The zeta-morphic correction depends the choice of vector v.

Then it depends on tau.

The first variable category is z1,

and the second small variable, small zeta.

By definition,

this is equal e to the power

minus 4 pi to the square.

The square of the vector v.

The quasi-modular

Eisenstein series G2 then Z,

small z, to the square.

Pi (tau, Z1 + zD).

This is zeta-morphic correction and

now the main property of this zeta-morphic correction is the following.

Let us analyze the Taylor expansion of this function in that.

I get the sum n greater equal to 0.

The Taylor coefficient f index

n depends now in tau and z1.

Z to the power n.

Then, This

is function, fn tau,

z1, are holomorphic

Jacobi form of weight.

K plus n with respect to the lattice

L1 is a regional function 5 most holomorphic.

Or, Weak,

If phi is weak.

The proof of this fact is very,

very similar to the case of fun variable.

Now you have to analyze the functional equation so

I leave this calculation for you as exercise.

So first of all you have to check.

A tau plus b you have to find the functional equation of this function.

Z1 over c tau plus d, z over c tau plus d.

You have to find this using the functional equation for

the original Jacobi form phi and to the quasi-modular Eisenstein G2.

And then the same you have to do for any,

Translation with respect to variable Z.

In the first case, a, b, c, d is SL equal to Z in for

any, and now, for any vectors, lambda and

mu, in the lattice L1, which is orthogonal

compliment of the vector B, and the lattice L.

So and you get, that this automorphic correction, has the same transformation,

like the Jacobi form of correspondent weight, and for the lattice n1.

So you see that in the case of modular form in many variables,

automorphic correction gives us Jacobi forms,

Jacobi forms,

For sub lattice of current 1 in L.

So using quan Jacobi modular form,

you can construct infinite many Jacobi form for smaller lattices.

So this is the main idea for automorphic correction,

which is very, very useful in many, many equation especially

about the greater dream of weak Jacobi form and so on.

And now I don't like to write the proof because this is the right calculation.

In the first and the second equation you have to use only the function equation.

And, please analyze this idea of automorphic correction because

you cannot find it in the book of.

Moreover, these fact about automorphic

correction of Jacobi form in many variables.

Maybe I don't know, maybe my paper you cannot fight it and

then says well so this is very simple but

nevertheless really new method, and nearly new formulation.

And in PDF file related to this lecture,

I add some number of exercises related to this method.

But now I would like to finish this lecture with classical formulas

about theta function.

So automorphic correction maybe the last small subsection,

Automorphic, Correction, Of Jacobi theta series.

Because the Jacobi theta series is really the main hero of our cause.

First of all, the first fact, you can find this in the literature.

Because the falling formula for the Taylor expansion of the Jacobi theta series.

This is equal to 2pi iz, we know,

Jacobi theta series, has 0 of order 1 for

z equal to 0 times the Dedekind

eta-function function to the power 3.

This is the derivative of theta series, z equal to 0, and then,

E exponent minus the sum 2 over

2k factorial, the summation

of all k greater or equal to 1.

Then we have Eisenstein series g2k for k equal 1 over half quasi-modular.

Eisenstein series g2 2

pi i z to the power 2k.

So this is more or less explicit exponential formula for

the Taylor expansion of the Jacobi theta series.

This is the first fact that I would like to match.

I'll give you explicit reference where you can find the proof but

the second fact is very, very important if you would like

to understand not only the Jacobi theta series but

some other important special function related to this function.

And now, one can prove the following

formula for the double algorithmic

derivative of Jacobi theta series,

one can proof that this is up to minus

Weierstrass a function plus 8 p squared

our quasi-modular Eisenstein series.

So you can get the quasi-modularized Eisenstein series as logarithmic

derivative of the Dedekind eta-function, or as the second,

a part of the second logarithmic derivative of the Jacobi theta series.

Rho, this is the famous Weierstrass p function, p tau z.

This is Jacobi form.

Metamorphic Jacobi form of weight 2 and index 0.

And this Weierstrass p function has the following Taylor expansion.

One over z to the square, so this function has a pull of 42,

times the sum k greater than equal to 2, 2 pi i to the power 2k.

This is like in this formula but without any exponent.

G2k tau times z to

the power 2k minus 2 over 2k minus 2 factorial.

And maybe to fix all formulas.

I remind you, the Fourier expansion of

the Eisenstein series G2 to with G normalization.

This is minus the Bernoulli number

B2k over 4k plus the sum sigma

2k minus 1 at nq to the power n,

n great equal 2.

So, if you would like to understand, really to understand

the theory of special function of the series Jacobi theta function,

please try to prove or to find in the literature,

first of all the formula for the Taylor expansion of this function.

And also the formula for

the double logarithmic derivative of the Jacobi theta series.

If you can understand these two formula then

the Jacobi theta series becomes your best friend.

[MUSIC]

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