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)
Theta-function
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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.
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Valery Gritsenko
Laboratory of Algebraic Geometry and its Applications
[SOUND] What properties
have this generating function,
the theta function,
of the quadratic lattice L?
[SOUND] By the definition, The data function, is.
This is evident because we define this function as a function in Q.
But it is very, very important that this generating function has a hidden symmetry.
More exactly, this function satisfies a function equation of the following type.
We can calculate the theta
function at 1 over tau.
And we can calculate this series.
One can prove that this is equal to
tau over e to the power n over 2.
I remind you that n is the rank,
Sorry, n is the rank.
Of the quadratic lattice.
So and the function at -1 over tau
= (tau over e) to the power n over
2 x (det lative a) to the power -one-half
x this same function function tau,
but for the dual lattice.
The dual lattice, what is it?
L*= is by definition the lattice of all,
Element u in the ration quadratic space generated by the lattice.
l such that for any v and
L, scale of product of u and
v is different.
Our lattice is integral, so is the larger list.
L is integral.
So you see in our equation we can compare theta
function of the lattice a at the point -1 over tau.
And theta function of the it's a point tau.
What is this transformation?
Let's consider once more, The complex upper half plane.
This is zero, this is on the real axis.
If you go to infinity, We go to the point,
i infinity and- 1 over tau interchange
this to, i infinity.
N, 0, this is
transformation from
22- one-half.
So you see?
When we're working with our generating function at infinity
we get another generating function at 0.
This type of effect we have in physics more exactly in
some mural symmetry consideration when we compare two different forces,
strong and weak forces, per account.
So this functional equation, Which
is really some hidden symmetry of how a theta function
has very nice form if the lattice a is unimode.
Let's assume.
That L u is equal to that.
This is equivalent to the fact that determinate of L = 1.
We have such positive definite even integral lattice,
but only if the rank of our lattice n,
if the rank of it = 0 modularly.
So let's assume that we have a unimodular lattice.
Then the functional equation has the following very nice form.
Data function at (- 1 over
2) = 4 to the power n over 2 x
the theta function of L( tau).
I add the evident property, then our function is periodic.
And we get a function,
which is usually called
modular form of weight,
k with respect to special
linear group SL2(z).
So the question, which you can put immediately,
why we have the group, F, into that here,
which we usually call the modular group.
The explanation is very simple.
So the group, SL2(z),
Or more generally SL2 (R), and there's group,
Of real, sorry, we add some notation here.
This is a group of matrices,
a b c d with real elements
of determinate value.
This group act on the upper half plane,
By fractional linear transformation.
And the integral special linear group,
Is generated by two elements, T and s,
Where T is the so called translation lattice.
And T transform T tau and tau + 1.
And S, Is the involution.
Since tau is -1 over tau.
So now if we analyze more carefully the functional equations,
Of the theta function, we see then the first
equation is related to the matrix S.
And the second, To the matrix T.
If we combine, these two equations, with respect to S, and
with respect to T, we can get, All
translation with respect to
the group SL2(z), so S into z.
This is very important for our consideration,
discrete group acting on the upper half plate.
A little bit later, I give you the formal definition or
the correct definition of modular form of weight k on the upper half plane.
But now I would like to analyze our arithmetic problem once more.
What's the problem?
We put we have, Even integral,
positive definite quadratic lattice L,
and we try to analyze this quantity.
We would like to analyze the number of representation
of an integer to n, by vectors in this lattice.
We define the generating function of this letters but
now I would like to consider a more general function,
a function which is more general then this generating function,
the so-called partition function.
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