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 Weil representation (part 2)
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從本節課中
The Weil representation and vector valued modular forms. Jacobi forms of singular weight
This module is devoted to very useful notion of the Weil representation and vector-valued modular forms. In this module we also define Jacobi forms of singular weight. Also there is a peer review in the end of this module.
教學方
Valery Gritsenko
[MUSIC]
Now I can rewrite this
formula in terms of
the vector valued
theta series,
theta L (tau + 1,
z) = matrix U (T)
theta L, h (tau z),
where U(T) =diag
(unitary matrix).
T here, certainly, this is a translation in the modular group.
This formula is simple, but now I would like
to show you without proof, the next formula.
We can calculate, The vector valued theta series,
the vector valued Jacobi theta series after the modular transformation.
We get tau to the power
n0 over 2 (-
i) to the power n0
over 2 (det L) to
the power -one-half
(matrix e to the power
2pi i (gh)) gh
in L stop over L.
This is the matrix of dimension determinant of n times,
now maybe I have no place, so later we'll add it.
Then we have the standard
Jacobi vector e to the power
pi i z to the square over tau,
theta L (tau z).
So you see that our vector valued Jacobi
theta series transforms like modular
form like Jacobi form of weight n0 over 2 and
there's this Jacobi factor.
But we get a vector valued Jacobi form.
This matrix we denote by U(S).
S is usual, it's a modular [INAUDIBLE]
One can check and this is a simple exercise,
that U(S), Is,
A unitary matrix.
This is a unit matrix of
rank determined of L.
So, we'll have two function equation, the first one is very simple.
The second I give you without proof, but
it's verifiable if you explicit reference to the proof of this fact.
But you can do it using the the parcel summation formula.
But in the both function equation, we have,
2, Unitary matrix.
We know then the translation and
the evolution generate the full
modular group SL2(Z).
Therefore, we have the following result for
the vector valued Jacobi theta series.
For any
M=a,b,c,d SL2
(z) theta L
(M tau z over
c tau + d) =
(c tau + d) to
the power n0
over 2 U(M)
theta L (tau
z), where,
U(M) is a unitary.
Matrix.
So, for,
Even n0 for all lattices of even rank.
We get a representation U,
Of the modular group, in the unitary group,
Of order, Determinant.
This is true if the rank of the lattice n is even.
If, The rank of L = n0 is ought.
The result is true, but U will be a representation
of the double cover of this group.
In both cases, this representation is called,
The Weil, Representation
Of the lattice, L,
but for ought n0, we have to consider
the double cover of SL2 because certainly,
This square root depends on the choice of the square root.
The choice of this square root, Will determine and
project, the representation of the group, SL2.
And this is the second factor in the splitting
formula of an arbitrary Jacobi form.
Let's analyze the first formula of our lecture today.
Jacobi theta series.
Jacobi function in many variables is a product of two factors,
a vector valued function in tau and
a vector valued Jacobi theta series.
And the last result we can formulate as follows, so,
This functional equation implies
that theta L (tau, z) is a vector valued,
Jacobi modular form,
Of weight n0 over 2 for
the lattice L.
But what can we tell about the first vector?
In fact, we prove the following theorem.
Let phi be holomorphic Jacobi form of weight K.
Then phi (tau z) =
vector valued function
times the vector valued
Jacobi form, and,
Phi L (1) is
a vector-valued
modular form
with respect,
To the the full modular group S L(z),
more exactly this function satisfies the following functional equation.
Phi L (a tau + b
over c tau + d)
= (C tau + d) to
the power of k- n0
over 2 x U(M).
The conjugation, it's a complex conjugation
of the value representation of the matrix M x iL(tau).
For any, M,
And SL2(Z).
This is direct corollary from the functional equation of
the Jacobi vector-valued theta series.
Please write down the functional equation for the Jacobi form and
then you get this functional equation for the vector valued modular form.
The only fact which we use here,
this is a fact that our matrix
U is a unitary matrix.
Moreover, the corresponding vector-valued modular form is holomorphic at infinity.
L tau is holomorphic,
At infinity because for any component I have seen this formula today.
We have this,
Type of Fourier expansion where n is strictly is non-negative.
hL* over L.
From this formula falls then our
vector valued modular form is parabolic,
It means that here the condition will be a little bit stronger.
We get strictly positive n in
Fourier expansion if Jacobi
form phi is Jacobi cusp form.
This is the result of the splitting principle considered
in the last lecture, together with the fact than
the vector valued theta series of all Jacobi theta
characteristic is a vector valued modular form.
[MUSIC]
