Theta-quarks as pull-backs of Jacobi forms

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Из курса от партнера National Research University Higher School of Economics

Jacobi modular forms: 30 ans après

12 оценки

National Research University Higher School of Economics

Jacobi modular forms: 30 ans après

12 оценки

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)

Из урока

Jacobi forms in many variables and the splitting principle. Theta-quarks as a pull-back. Weak Jacobi forms in many variables

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.

Theta-quarks as pull-backs of Jacobi forms14:37

Theta-quarks as pull-backs of Jacobi forms (part 2)16:18

Theta-quarks as pull-backs of Jacobi forms (part 3)13:07

The splitting principle15:42

The splitting principle (part 2)16:40

Weak Jacobi forms in many variables13:18

Познакомьтесь с преподавателями

Valery Gritsenko

Laboratory of Algebraic Geometry and its Applications

[MUSIC]

Good afternoon, dear colleagues.

Lecture eight.

The subject of our discussion today.

The first subject is theta-quarks.

As pullbacks.

Of Jacobi modular forms.

In many variables.

We constructed theta-quarks two lectures ago,

and they are very interesting,

Jacobi form in one variable.

Jacobi forms of Eichler–Zagier type of weight 1 and

index a2 + ab + b2 to the square

with a character of order 3.

Using these theta-quarks, we constructed for

example, the first cusp form of weight 3 and index 13.

In the last lecture, we introduced

Jacobi modular form in many variables.

A Jacobi form.

Of Jacobi form of weight k and

index a has the following free expansion.

Z now is an element of the complex vector

space of dimension n0 where the rank.

Of a is equal to n0.

So we have the following

Fourier expansion a, n,

a, n is not negative.

A in the dual lattice a star

equals the power 2pi (n tau +

the scalar product of (n,d)).

This is a Jacobi form.

Weight k with respect to the lattice n.

This modular form, Jacobi modular form, is holomorphic, it's infinity.

If we have non-zero Fourier

coefficient only for indices n, a,

satisfying the following condition.

In the last lecture, we gave the following interpretation of this condition.

If we consider the index,

Of these Fourier coefficient as a vector

in the hyperbolic lattice U + l(0)

negative, l negative different.

Then this condition, Means that our vector.

Vector f for example,

is inside the positive cone of

this hyperbolic lattice.

Or on the boundary of this cone.

If the hyperbolic noam of this

vector is equal to 0, moreover we,

Consider the following example.

I remind you the definition of Jacobi theta series of the lattice.

So your modular lattice E8, today we consider more general function.

The following function, the summation is taken

of a whole vector in the even modular lattice.

E8 is a Jacobi form of weight for

the lattice E8.

Please analyze the definition because today, we consider more general functions.

In particular, we have the following correlation between,

Jacobi form with many variable, variables and

the Jacobi forms of Eicher-Zagier type.

We'll have a map from the space of

Jacobi form of weigh k to the lattice

a to the space of Eichler-Zagier Jacobi

form in one variable with the index

u to the square / 2 for any vector u(l).

L is even integral,

therefore u to the square = 2m.

And in this map we get Jacobi modular forms

of Eichler-Zagier type of index m.

How we construct this map, for arbitrary,

Jacobi modular form.

In many variables, we can construct its restriction.

On the line, zu = phi (tau,

u times z) where z is

a complex number.

In particular, it works very well if our vector is equal to 0.

In this case we simply take If u = 0,

then we get phi (tau,0),

which is the modular form of weight k

with respect to the full modular group.

Now, I would like to consider

the restriction, or the pull back,

of Jacobi form phi on some sublattice more

exactly on a sublattice of corank 1.

We take now nonzero vector u in n and

we take its orthogonal

complement in the lattice l.

We get a sublattice in l.

Moreover their diagonal sum of the sublattice m and

the one-dimensional lattice

generated by u is a sub lattice a finite index in L.

L is sublattice in the dual lattice in the dual lattice of N.

Sublattice of the direct sum

of the dual lattices of M and U.

The dual lattice of U Is

generated by one vector,

U over the scalar product of U.

If I take now that in

the complex vector space,

By L then we can write down

this vector is the sum orthogonal

sum of two components ZM and

small z times u,

Where small z is a complex.

The pullback which I would like

to analyze Is zeros friction.

Apology Jacobi form.

For z = 0.

We denote this function by

phi in this u tau times ZM,

and this is clean.

This is in Jacobi form of the same

weight K for the lattice M.

I remind you that use it M is a.

M is orthogonal content of U.

What type of re-expansion

has this function?

To analyze the Fourier expansion,

I would like to write down in

habit through vector L.

In the dual lattice

as a sum of two vectors,

lm+ lprime.

Now I can write the expected formula for this vector.

So let's analyze this question more carefully.

lis a vector in the dual lattice

which is sublattice of M* + Z

times u / (scalar square of u).

So then l = LN + orthogonal projection of

L on this one-dimensional sublattice.

So we get (l, u) times u / (u, m).

And I would like to denote this vector as l and there's u.

Now we are ready to write down the Fourier expansion of,

the Jacobi form phi,

consider it at the point

small z = 0.

[MUSIC]

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