Lattices, Eisenstein Series and Modular Forms
D.J. Basson
Talk on 5 March 2012 at Stellenbosch University for the
Postgraduate Seminar
Abstract
The nineteenth century is, among other things, known for the huge development in complex function theory that took place. The properties of
many fundamental functions like the exponential, gamma and zeta functions were explored. In this talk I would like to concentrate on functions
that have some relation with lattices in the complex plane, in particular,
on modular forms. To relate this to my research, I would like to draw
some parallels between the classical theory over the complex numbers,
and a characteristic p theory.
1
Complex Numbers — Classical and in Characteristic p
The rings Z and A := Fq [T ] are very similar. In precise terms, both are one
dimensional Dedekind domains with finite residue fields. This already implies
many things like analogues of the theorems of Fermat, Euler and Wilson. There
are also many deeper connections like class field theory and Riemann’s hypothesis. In this talk, however, we shall construct an analogue for the complex
numbers over A and consider certain “complex functions”. This “complex”
field will be required to be both algebraically closed and complete with respect
to some absolute value.
In order to construct the analogue of the complex numbers over A we recall
how C is constructed over Z. First, one passes to the fraction field Q of Z.
Then we define an absolute value (the usual one) and complete the field Q with
respect to this absolute value to obtain the complete field R. Then we pass to
1
C which is an algebraically closed field. Also, since the degree of C over R is
finite, C remains complete.
Analogously we pass from A to its field of fractions F := Fq (T ). Then we
need to define an absolute value1 , so that we may complete the field with respect
to this absolute value. Any non-zero element can be written as
f (T )
g(T )
where f (T )
and g(T ) are polynomials in A. We define an absolute value by 0 = 0, and
otherwise
f (T )
= q deg f
g(T )
deg g
,
where deg f refers to the degree of f as a polynomial in T .
The completion of F with respect to this absolute value is Fq (( T1 )) which
is shorthand for Fq [[ T1 ]](T ), the fraction field of the ring of formal power series
in
1
T
. We denote this complete field by F1 . If we follow the construction of
C directly, we should pass to the algebraic closure of F1 at this time, which is
what we do. However, this field fails to be complete, because its degree over
F1 is infinite. Therefore we need to take the completion once more. Luckily
this field remains algebraically closed. Denote it by C1 . We shall not describe
this field more explicitly, but only remark that this field is the complete and
algebraically closed analogue of C we were looking for.
One may observe that this construction does not really depend on starting
with the ring Fq [T ]. We may, in fact, start with any ring that is endowed with
a non-archimedean absolute value. So we may, for example, perform this construction again to Z, but with a p-adic absolute value. This gives an analogous
complete algebraically closed field Cp . Of more interest to us is the possibility to
start with any ring of regular functions of an algebraic curve (or more concretely
a finite extension of Fq [T ] with certain properties). However, for simplicity we
shall continue working only with A = Fq [T ] in the remainder of this talk.
Let us end this section by summarizing this analogy in a table:
1 Recall
that an absolute value is a function | · | : F ! R which is positive for all non-zero
elements of F , which is also multiplicative and satisfies the triangle inequality.
2
Z
A = Fq [T ]
Q
F = Fq (T )
R
F1 = Fq (( T1 ))
Fq (( T1 ))alg
C
2
C1 = (Fq (( T1 ))alg )^
Lattices
2.1
Lattices in C
Originally a lattice point was defined to be a point in R2 which has integer coordinates. Identifying R2 with C the set of lattice points is simply the Gaussian
integers Z[i]. However, it is more flexible to include any copy of Z2 in C in this
definition.
Definition 1. A lattice is a discrete additive subgroup of C. As abstract groups,
a lattice is isomorphic to Zr for some non-negative integer r. This r is called
the rank of a lattice.
Here, discrete refers to the subspace topology on the lattice, induced from
C.
Example 1. The lattice Z[i] has rank 2, while the simple inclusion Z ⇢ C is a
lattice of rank 1.
For the rest of the talk we shall restrict ourselves to the rank 2 case for
lattices in C. We define an equivalence relation on the set of (rank 2) lattices
by declaring two lattices ⇤1 and ⇤2 to be similar if ⇤1 = c⇤2 for some non-zero
p
p
complex number c. For example, (1 + i)(Z +
2Z) = (1 + i)Z + ( 2 +
2)Z,
p
p
so the lattices Z +
2Z and (1 + i)Z + ( 2 +
2)Z are similar.
Any lattice can be written in the form ⌧1 Z+⌧2 Z, where ⌧1 and ⌧2 are complex
numbers. However, for this group to be discrete, we need ⌧1 and ⌧2 to be linearly
⌧1
independent over R (or equivalently
2
/ R). Under the stated equivalence, the
⌧2
⌧1
⌧2
Z + Z and Z + Z. Exactly one of
lattice ⌧1 Z + ⌧2 Z is equivalent to both
⌧
⌧1
2
the numbers ⌧⌧12 and ⌧⌧21 has positive imaginary part. We call this ⌧ and think
of the lattice Z + ⌧ Z as the representative of the equivalence class containing
the lattice ⌧1 Z + ⌧2 Z. So the equivalence class of a lattice can be described by
3
giving one element ⌧ in the upper half plane
H = {z 2 C | Im(z) > 0}.
Lattices associated to two di↵erent ⌧ can turn out to be similar, but we shall
not worry about that.
+
We also define an
! action of GL2 (R) on the set of lattice classes by letting a
a b
matrix =
act by (⌧1 Z + ⌧2 Z) = (a⌧1 + b⌧2 )Z + (c⌧1 + d⌧2 )Z. If we
c d
choose to work only with lattices of the form Z + ⌧ Z, ⌧ 2 H, this induces an
a⌧ + b
action of GL+
Z. It
2 (R) on the set of these lattices by: (Z + ⌧ Z) = Z +
c⌧ + d
turns out that if we choose 2 SL2 (Z), then the lattice class is always fixed.
+
This
! discussion suggests that we may define an action of GL2 (R) on H by
a b
a⌧ + b
(⌧ ) =
. It can be checked that this does indeed define an action.
c⌧ + d
c d
!
1 1
Example 2. The matrix
takes Z + iZ to Z + (1 + i)Z, but these are
1 0
!
0
1
exactly the same lattice. The matrix
takes Z + ⌧ Z to Z + ( 1/⌧ )Z,
1 0
which is similar to ⌧ Z + ( 1)Z = ⌧ Z + Z under multiplication by ⌧ .
2.2
Lattices in C1
We can define lattices in C1 as well, but they will not be copies of Zr . Remem-
ber that the base ring in this case is A and not Z. So we want a lattice to be a
copy of Ar inside C1 .
Definition 2. A lattice in C1 is a projective A-module M which is strongly
discrete in the topology of C1 .
In our case (A = Fq [T ]) projective means the same as free, but the definition
as it stands generalises better. We also need the stronger notion of strong
discreteness, since F1 /F is an infinite extension. The main di↵erence with the
classical case is that there exist lattices of rank higher than 2. So a lattice
can be represented as ⇤ = !1 A + !2 A + · · · + !r A. Once again we define an
equivalence relation where two lattices are similar if ⇤1 = c⇤2 for some c 2 C1 .
There are also analogues of the upper half plane (which now is r 1-dimensional
over C1 ) and an action of GLr (F ) on it. We omit the details, because of time
constraints.
4
3
Eisenstein Series
3.1
Classical case
The functions we are considering in this talk are functions with domain the set
of lattices 2 . We define the weight k Eisenstein series for the lattice ⇤ by
Gk (⇤) =
6=0
X
k
.
2⇤
This series converges whenever k > 2.
In particular, we would like to view Gk as a function on H by Gk (⌧ ) :=
Gk (Z + ⌧ Z). Recall!the action of SL2 (Z) on H, which we defined by (⌧ ) =
a b
a⌧ + b
if =
. If 2 SL2 (Z), then the lattices Z + ⌧ Z and Z + ⌧ Z are
c⌧ + d
c d
similar, so one might expect the values of the function Gk to be related when
evaluated at ⌧ and ⌧ .
We use the fact that if
a
b
c
d
!
2 SL2 (Z) then Z+⌧ Z = (a⌧ +b)Z+(c⌧ +d)Z
as lattices, so
Z + ⌧Z =
1
1
(a⌧ + b)Z + (c⌧ + d)Z =
(Z + ⌧ Z).
c⌧ + d
c⌧ + d
Then
Gk ( ⌧ ) =
6=0
X
2Z+ ⌧ Z
3.2
k
=
6=0 ⇣
X
2Z+⌧ Z
c⌧ + d
⌘
k
= (c⌧ + d)k Gk (⌧ ).
Characteristic p case
We make exactly the same definition in this case. Define the weight k Eisenstein
series as
Gk (⇤) =
6=0
X
k
.
2⇤
In this case we have convergence when k > 0, because the absolute value is
non-archimedean.
If we define the action of GLr (F ) on a basis ! = (!1 , . . . , !r ) for a lattice
by viewing this vector as a column matrix and multiplying by the matrix
2 Not
the set of equivalence classes of lattices
5
2
GLr (F ) on the left. If we call the last entry in this product j( , !) we obtain a
similar relation between Gk (!) and Gk ( !). We have
Gk ( !) =
⇣ j( , !) ⌘k
Gk (!),
det
where j( , !) is an expression similar to c⌧ + d in the classical case. Once again,
we leave this discussion without further details due to time constraints.
4
Modular Forms
4.1
Classical case
We have seen that the Eisenstein series have remarkable symmetry properties. It
is natural to ask whether there are any other holomorphic functions f : H ! C
for which f ( ⌧ ) = (c⌧ + d)k f (⌧ ) holds for every ⌧ 2 H and every
2 SL2 (Z).
Such a function is called a weakly modular function for SL2 (Z). It turns out
that there are too many such functions, so we want to restrict ourselves to those
that are also holomorphic at infinity. This condition means that the value of
f (⌧ ) stays bounded as Im(⌧ ) ! 1. There is a more precise definition for this,
but for now this is good enough.
The reason why the “holomorphic at infinity” condition makes the space of
modular forms finite dimensional may be explained by looking at the similar
situation of holomorphic functions on C. There are infinitely many functions
that are holomorphic on C (take any function with an everywhere convergent
Taylor series). However if we also require the function to be holomorphic at
infinity (bounded), the only functions left are the constants.
Let us finish by describing the set of modular forms. Denote by Mk the set
of modular forms of weight k. It is clear that this forms a vector space over C
(if f and g are modular forms, then so is f + g, and if c 2 C, then cf is also
a modular form). Furthermore, if f is a modular form of weight k1 and g is a
modular form of weight k2 , then f g is a modular form of weight k1 + k2 . So it
makes sense to denote by M the graded ring
k 0 Mk .
It turns out that M is algebraically generated by G4 and G6 , or more ex-
plicitly M = C[G4 , G6 ] where there is no relation between G4 and G6 . So for
example, the modular forms of weight 12 are G34 , G26 and linear combinations
of these two functions.
6
4.2
Drinfeld modular forms
Denote the analogue of the upper half plane by ⌦r .
Definition 3. A holomorphic function f : ⌦r ! C1 is called a Drinfeld weakly
modular function of weight k if, for every ! 2 ⌦r and every
2 GLr (A),
f ( !) = j( , !)k f (!).
If f is also holomorphic at infinity, it is called a Drinfeld modular form.
Here, holomorphic is the suitable analogue of holomorphic complex functions. The easiest way to describe them is as those functions with a everywhere
converging power series expansion. Once again holomorphic at infinity is somewhat subtle to define, but can be thought of as “stays bounded as the variable
tends to infinity”.
Similarly to the classical situation, we may also describe the graded ring of
modular forms as C1 [g,
], where g and
are two modular forms similar to
G4 and G6 in the classical case. In this case the weight of g is q
weight of
is q
2
1.
7
1 and the