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)
The splitting principle (part 2)
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从本节课中
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.
教学方
Valery Gritsenko
[MUSIC]
Let me analyze this rather complicated formula once more.
So we use the property, it's a general property
of Fourier coefficient of all Jacobi forms.
Let me fix it once more.
Fourier coefficient depends only on the hyperbolic nome of its index and
the class of the vector in the lattice in the dual modular latus.
So in the left-hand side and the right-hand side,
we have the same Fourier coefficient with the different indices.
So we can modify h,
fix element h by any vector v in L.
Using this fact, we rewrite
the Fourier expansion, first of all,
using the summation over all classes,
In the discriminate group.
There are exactly determinate to fail classes.
Then we modify the hyperbolic nome
of Fourier coefficient because the hyperbolic
nome of the index is exactly 2 times n.
Then we fixed a summation with respect to vector v.
And we get our splitting principle,
we get this function depends only on, wait, sorry.
We have to add here the variable tau.
This function depends only on tau and
this is exactly our modular form.
Fh in tau.
Maybe I write the formula for
the Fourier expansion of this function,
Once more.
So this is summation of all hyperbolic nomes.
N in general is rational, but
N + h square / 2 is integral
e to the power 2 pi r N tau.
This is the new function.
We have this function for any class in the lattice modular L.
So we proved.
Let me formulate now this result.
Theorem, For
any weak Jacobi form of weight k and
index, the lattice L phi
in tau z = the sum of this
finite sum contains
determinant of L terms,
fh tau times tau Lh (tau,zeta ).
And the Fourier expansion of these two functions are the following.
This is Jacobi theta series with characteristic h.
We simply that this translation
to any vector V is the latest N and
this is Fouier expansion
of the function F(h).
Now I would like to analyze the Fourier
expansion or function of f(h).
Fourier expansion have this form.
So I would like to analyze
a (N + h square / 2h).
I would like to understand for
each indices this coefficient is not equal to 0.
First of all, let's assume that phi is holomorphic Jacobi form.
Then a(n,l)
Is not 0 only if the hyperbolic
nome of the index and this is equal
to 2N because the hyperbolic
nomes is 2N is non-negative.
It means then in our function O and
R are non-negative,
so let's fix it here.
N is greater to, equal to 0,
if phi is holomorphic.
It means that this function is holomorphic at infinity.
Fh tau is holomorphic,
At phi infinity
for any h.
Certainly even if Imodify our
condition a little if we take for class form.
Then we'll have better estimation,
and our function fh tau vanishes at infinity.
Now I would like to analyze the property of
Fourier coefficient for weak Jacobi form.
To do this, I would like to
introduce the notion of the nome
of the class modular a.
Let's consider an element in this finite group.
Then by definition,
h square is the minimum,
Scalar square of all element in h.
And I would like to take the following
invariant of the latest a.
By definition, this is the maximum,
Nome of the classes,
Of this final group.
This is max over
h of the minimal
square of vector in h
Now using this notion, I would like to analyze Fourier expansion.
So analyze the same coefficient
a (N + h square / 2, h).
If our Jacobi form is weak,
We know then this Jacobi form is
a non-vanishing coefficient
only if n is greater or equal to 0.
But according to our definition,
n is equal to N + L square / 2.
In other words,
a (N + h2 / 2,
h) is not equal to 0.
It follows then, N is greater or
equal to -h square / 2 or
in terms of our new notation, mu(L).
That's formulated as this fact, this proposition.
Phi is a weak form, then a(n,
l) is not equal to 0.
Follow then, 2
n- l to the square.
I am sorry.
In this inequality,
I forgot certainly the coefficient one-half,
because here we have h square / 2.
And now here,
we have -M(h).
So this is a property of the weak Jacobi forms.
And now let's consider running them both.
Let's take a weak Jacobi
form of factors z and r.
Rate k and Index M,
which means weak form of weight k for
the latest 2M where 2M
is the latest of when1
times mu square = 2m.
Mu 2m, easy to see, we have to analyze
all classes of the dual lattice.
So we have to analyze
x square / 2 m.
And this is clear when mu Is greater or
equal to -M / two because
the maximum of nome we have
exactly if x here is equal to M.
Maybe I can write it more carefully but
I hope you can prove it by yourself.
And now what we can create Fourier coefficient If we write down
the Fourier coefficient in the terms of the lattice.
So we have here x times,
U square, but
U square is 2m is
not equal to 0,
then 2m- 2n,
I'm sorry,
-x2 / 2m is greater or
equal -m / 2.
Therefore if we use the usual notation of
the Eichler-Zagier modular form,
this greater equal to -m to the square.
And this is true If the Fourier coefficient
of weak Jacobi form is not 0.
This fact was proved in
the Eichler Zagier book,
Jacobi Form on the page 105.
You see that our proposition gives us
the proof of this classic effect.
[SOUND]
