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)
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Jacobi modular forms: 30 ans après
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Introduction to the Course
Welcome to the course! I hope you have an opportunity to reserve some time to explore the course content, course logic and our grading policy. The course consists of 12 lectures. This course will help you to start your progress in the field of the theory of Jacobi modular forms.
Best regards, Valery Gritsenko
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Valery Gritsenko
Laboratory of Algebraic Geometry and its Applications
[MUSIC]
Dear students, dear colleagues,
I invite you to follow my master course on Jacobi modular forms.
30 years ago, Martin Eichler and
Don Zagier published their book, The Theory of Jacobi Forms.
The book was published in the Progress in Math in 1985.
This is a research book, and
one of the most successful book in this Birkhäuser series.
You can find more than 1,000 reference from research papers to this book.
The reason for this and the theory of Jacobi forms is very
successful not only in number theory but also in algebraic geometry,
in particular in the theory of modular spaces surfaces.
In the theory of Lie algebras, the denominator functions,
the denominator functions function of affine
Lie algebras are written in terms of Jacobi forms.
Moreover, the data for product is a very strong modern construction in the theory
of automorphic form is also based on Jacobi forms in some sense.
Then you can find application of Jacobi modular forms in topology, elliptic
genus and generalized elliptic genus of Calabi-Yau
varieties are weak Jacobi forms of weight 0.
Then you can find application of Jacobi forms in physics, in string theory, and
more particular in quantum gravitation.
So in integrable models.
In combinatorics, this is very, very successful subject.
Moreover, from my point of view, the book of Eichler Zagier
may be the best second course in modular forms.
So you need to know only a little bit about modular forms.
The definition and two examples or three examples like the Eisenstein
series of weight 4, 6, and Ramanujan theta function.
If you know them, you are ready for my course.
But I prepare this course in 2015,
three years after Eichler and Zagier book.
What are the difference between their book and my course?
First of all, my course is the introduction, or
it's better to say the invitation in Eichler Zagier theory.
But in 2015,
the protagonist of my course is
the Jacobi theta series, theta tau z.
So this is really our main hero.
Unfortunately, you cannot find this Jacobi theta series,
due to these theories, certainly Eichler and
Zagier put this name, Jacobi forms, but
you cannot find Jacobi theta series in the book of Eichler Zagier.
The reason for this is the following.
Jacobi theta series is Jacobi form of weight one-half.
Our approach to this function is as follows.
So now I would like to tell you, what is my course about.
And now my remarks and this very, very short introduction before the course.
And mainly for the specialist or for the students who know a little bit,
something about Jacobi forms.
So my approach to Jacobi theta series is the following.
We can analyze Jacobi theta series as
a function on the Siegel upper-half plane.
So it means we add the set variable omega in the definition of Jacobi form.
And we get a function on the Siegel upper half-plane,
which is modular form with respect to a parabolic subgroup,
gamma infinity is parabolic subgroup of the Siegel modular group.
And this is our model for Jacobi modular group.
So Jacobi modular group for
us is a parabolic subgroup of Siegel modular group.
Then, in this term, Jacobi theta series in three
variables is a modular form of weight one-half and
with respect to the multiplier system v eta cubed,
certainly this is multiplier system of the Dedekind eta function
times the binary quadratic character of the so-called Heisenberg group.
Using Jacobi theta series, we can construct a lot of very interesting and
sometimes nontrivial examples of Jacobi form.
First of all, we can construct the main generator,
this is the function phi minus 2 1 from the Eichler Zagier book,
this is the main generator of the of the weak Jacobi form.
The formula is very simple.
We take the square of Jacobi theta series over the six powers of the Dedekind
theta function.
But we can construct less trivial example.
Our next example is so-called theta-quarks.
This is a product of three theta series over one eta.
We can prove that this is a Jacobi form of weight one
of the index a squared plus ab plus b to the square
with respect to a character of order three.
If we take the product of three theta-quarks,
we get a Jacobi form of weight three.
In this way, we get the first Jacobi
form of weight three and index 30.
In a similar way, but a more complicated procedure,
we can construct the first Jacobi form of weight two.
This function has index 37.
In the Eichler Zagier book, you can find a table for
some Fourier coefficient of this modular form, and
we can get the explicit formula for this function.
This idea of theta series brings us to
the idea of automorphic correction.
The automorphic correction,
this is of the Taylor expansion of Jacobi form.
So from this point of view, our approach is
very different from the approach in the Eichler Zagier book.
In the automorphic correction,
this is multiplication by the exponential function
with the quasimodular Eisenstein series of genus two.
On this page, I give you two different explanations
of the rather strange automorphic factor of Jacobi modular form.
The first explanation is related with the fact that we
can treat the Jacobi modular form as modular form with
respect to a parabolic subgroup of Siegel modular group.
This fact explains and gives us explicit Jacobi factor.
The second explanation related to this automorphic correction.
It means the Jacobi automorphic factor is related to
the quasimodular Eisenstein series.
The first part of our course is about Jacobi form in one variable.
But the second part about the Jacobi form in many variables.
And again, our main hero, the Jacobi theta-series,
gives us the main idea and some main examples.
For example, the product of eight theta series is the first Jacobi form,
that means the Jacobi form of the minimal possible weight for the lattice D8.
In the same way, we can construct the Jacobi form
of so-called critical weight for the lattice A7.
Jacobi form in many variables are Jacobi form with
respect to some positive definite lattice of rank n0.
It means that the second variable,
the Abelian variable in the Jacobi form is
a vector in the of the positive definite
lattice, so a vector of dimension n0.
And now, using Jacobi form in many variables, we explain this standard
series, the standard Eichler Zagier series Jacobi form in one variable.
For example, we can represent any Jacobi form for
any lattice as the product of vector valued modular
form in one variable and the vector of theta series.
This subject related to the so-called representation.
Then the subject to automorphic correction is very much related to
the theory of modular differential operator.
And we construct modular differential operator for
usual modular form and for Jacobi forms in many variables.
In this way, we give a new proof of Zagier theorem
about existence of some differential operator which
transform the space of modular form of weight k Into
the space of Jacobi form of weight k and index 1.
This is a very useful operator.
Using this operator, we can construct Rankin–Cohen brackets and
prove many nice results.
We give a new proof of this theorem, and our proof, our method
works very well in the case of Jacobi forms in many variables.
It's better to say we have no changes in the proof for usual modular form and
for Jacobi forms, because our proof is very algebraic.
So instead of usual differential operator, we use the corresponding
operator, and we construct a differential operator which
transform a Jacobi modular form into Jacobi form.
Using the construction,
we can construct without any problem Rankin-Cohen brackets, and so on.
So we can really formulate a lot of simple and non-simple research question.
And again, for automorphic corrections and for
differential operator, the theta function plays the most
important role, because our theta function is this
Jacobi theta series with simple zero for z equal to 0.
Its derivative is cube of the Dedekind theta function.
The logarithmic derivative of Dedekind theta function is
the quasimodular Eisenstein series.
And using this Eisenstein series, we can construct the differential operator for
modular forms or for Jacobi forms in many variables.
So you see that the Jacobi theta series is really the main tool of our course,
or as I told you, it is the protagonist of our course.
So I hope with the Jacobi theta series,
which is really the kernel function of the theory of Jacobi modular form,
we can construct this theory is very natural and very simple way.
So you are very welcome into my course.
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