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)
Modular and abelian transformations
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來自 National Research University Higher School of Economics 的課程
Jacobi modular forms: 30 ans après
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Jacobi modular forms: motivations
This module is devoted to motivations to study Jacobi forms. We provide some first examples including theta-functions. Also there is a peer review in the end of this module.
與講師見面
Valery Gritsenko
Laboratory of Algebraic Geometry and its Applications
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Now we consider the partition function,
Of the same problem, theta a tau zeta.
As before, l is an even integral
positive definite, like this.
Tau this is an element in the upper half plane.
But this calligraphic zeta is a vector in
the complexification of the lattice L.
This is a complex quadratic space of rank n.
By definition,
the partition function
theta L tau zeta,
is defined as the falling
sum of all vectors in the lattice L
e to the power pi(v,v)
tau + 2(v,z).
This is a new function now.
Certainly like in the first generating function,
we have this term, v to the square tau.
But now we have an additional term.
In principle, we cannot simplify these Fourier expansion.
So partition function is defined by its Fourier expansion and
all Fourier coefficients are 1 here.
So we cannot construct, we cannot simplify the Fourier expansions.
So let's analyze once more the definition of partition function.
So in the partition function, we, in that generating function of
the lattice L, the generating function is also defined the similar way.
But we can combine different
terms in this sum with the same
scalar square of the vector v.
Now, in the partition function we cannot do this.
[SOUND] So it means that we
can use this function to
analyze the distribution.
Distribution of
v with fixed,
Norm.
So in this function, what is exactly an example
of so-called Jacobi Modular Form.
So what property has this function?
First of all dot theta dot z of the lattice L is holomorphic,
On the direct product of H1 and
the complex quadratic space,
L C of dimension m.
And as holomorphic function, this function also has some hidden symmetries.
Or it's better to say it satisfies two functional equations.
The first equation is similar to the modular equation
of the generating function, but it has the following form.
We analyze the value of this function 1/2,
but we have modified the vector z also by 2.
And then this = tau to the power n over 2,
like in the case of the function,
times e to the power pi the scalar square of
the complex vector z over 2, theta tau z.
Certainly we have this property for an even unimodular lattice.
So where's U?
Since it's unilattice, it's equal to the original lattice.
So this is a module equation of this theta function but this theta function,
s theta function, if you worked a little bit with them,
also has another function equation it satisfies.
The following equation for the variable z.
For lambda and mu in the lattice L,
we can calculate this way.
So it means that we can consider theta function
of the lattice L as needed double periodic function with respect to that.
So it means that the period here.
I'm sorry, we have another lattice
in this case certainly, this is tau L + L.
So the functional equation
is the following,
e to the power- pi lambda,
to the square 2 + 2 lambda z.
Those are just two frontal equations of the partition function of this lattice.
I call this function the partition function because we can
control the distribution of our vector with fixed norm.
I can formulate the first exercise,
To prove this identity.
So this is elementary question.
But to prove this identity we have to use so-called poisson summation formula.
Now I would like to understand the relation between
the partition function of the lattice L and the usual theta function,
or the arithmetic distribution function.
[SOUND] So for this, we have to consider the pull backs,
Of the petition function theta L tau z.
The first pullback is quite natural.
Let's put, z = 0.
This is a 0 vector in the complex quadratic space.
Then theta L tau
0 is equal,
To the generating function or
to the usual theta function, theta L tau.
So from this point of view, the partition function theta
is a generalization of the generating function for
the number of the representation of natural
numbers by the quadratic lattice L.
But we can consider another pullback.
Let u be a vector of L with square 2m,
which is positive.
Then we can define the following pullback of the partition function.
So it means that, we put z equal to U
times z where is complex number.
Then, this function is equal
to the following sum of all vectors in v.
And here we get the scalar
product of v and u times z.
I can denote this function
by theta L u(tau, z).
And this is the so-called
Jacobi Theta series,
Of Eichler, And Zagier.
Now, we considered this function in some details.
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