A FIRST COURSE IN MODULAR FORMS
JÖRN STEUDING
Contents
1. Lattices and Elliptic Functions
2. Eisenstein Series and Further Examples of Modular Forms
3. The Modular Group and Fourier Expansions
4. The Theta Series and Functional Equations
5. Arithmetic: Sums of Squares
6. Picture Gallery
References
2
5
7
12
16
19
21
These notes are self-contained. The methods stem from elementary number theory,
complex analysis, and (a tiny bit elementary) algebra. We use standard notation
√
(as in [3]), e.g., i = −1 denotes the imaginary unit in the upper half-plane. We
give only sketches of the proofs in order to motivate the dear reader to fill the gaps
on her or his own! We highly recommend to do the exercises.
”There are five fundamental operations in mathematics: addition, subtraction, multiplication, division and modular forms.”
This quotation is ascribed to Martin Eichler, one of the pioneers in this field
in the twentieth century. Everything has started with Euler’s work on the
number of partitions p(n) of a positive integer n as a sum of positive integers.1
For example, p(10) = 42. In 1742, Euler proved
∞
X
n
p(n)q =
n=0
and
(1)
∞
Y
m=1
∞
Y
m=1
m
(1 − q ) =
+∞
X
(1 − q m )−1
1
(−1)n q 2 n(3n−1) ,
n=−∞
both valid for |q| < 1. Almost one hundred years later Jacobi derived further
identities of this flavour, approving the importance of such generating series
and products for number theory (see [1, 3]). However, we shall start our
journey somewhere else in the nineteenth century.
Date: Šiauliai, September 2013.
even with Jakob Bernoulli’s work on the lemniscate in 1718!
1or
1
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JÖRN STEUDING
1. Lattices and Elliptic Functions
A lattice Ω in the complex plane C is a discrete additive group; it can be
represented as
Ω := ω1 Z + ω2 Z := {ω = mω1 + nω2 : m, n ∈ Z}
with certain lattice points ω1 , ω2, being linearly independent over R (i.e.,
Im ωω21 6= 0). The generators ω1 , ω2 are not uniquely determined. For instance, the set of Gaussian integers Z[i] = Z + iZ is a lattice and Z[i] =
(1 + i)Z + (2 + 3i)Z is another representation.
Theorem 1. Let Ω be a lattice. The associated Weierstrass ℘-function
X 1
1
1
−
℘(z) := ℘(z; Ω) := 2 +
z
(z − ω)2 ω 2
06=ω∈Ω
is a meromorphic function with a double pole at each lattice point satisfying
(2)
℘(z; Ω) = ℘(z + ω; Ω)
∀ ω ∈ Ω.
A meromorphic function which is invariant under translation with lattice
points is called elliptic or double periodic (with respect to Ω). In view of (2) the
Weierstrass ℘-function is elliptic and the following proof of the theorem shows
that its first (and any) derivative is elliptic too.
Sketch of proof. Denote the number of lattice points from Ω inside the disc
|ω| ≤ R by N(R; Ω); for example, there are N(R; Z[i]) = πR2 + O(R) points
ω = x + iy with integer coordinates. Hence, for the partial sum we find the
estimate
X N(m; Z[i]) − N(m − 1; Z[i])
X
X
1
1
=
≤
c
1
k
k
|ω|
m
mk−1
r<m≤R
r<m≤R
r<|ω|≤R
with some constant c1 , and its limit as R → ∞ exists for k ≥ 3. It is not
difficult to see that the same reasoning holds for other lattices Ω than Z[i].
Since
1
c2
1 (z − ω)2 − ω 2 ≤ |ω|3
with some constant c2 , the series for ℘ as well as the series
℘′ (z) :=
X
∂
1
℘(z; Ω) = −2
∂z
(z − ω)3
ω∈Ω
converges on any compact subset of C \ Ω and defines a meromorphic function.
(For this and further results from Complex Analysis we refer to [10].) The
MODULAR FORMS
3
latter function ℘′ is odd and elliptic (since it is obviously invariant under
z 7→ z + ω for any ω ∈ Ω). In view of ℘′ (u + ω) − ℘′ (u) = 0 it follows that
Z z
℘(z + ω) − ℘(z) =
(℘′ (u + ω) − ℘′ (u)) du − ℘(ω) + ℘(0) = c3
0
with some constant c3 . Since ℘ is even, i.e., ℘(−z) = ℘(z), we find c3 =
℘( ω2 ) − ℘(− ω2 ) = 0. •
One can prove that every elliptic function is a rational function of ℘ and
℘ (see [1]). In some sense (2) generalizes the concept of periodic real-valued
functions to double periodic complex-valued functions, however, the additional
dimension leads to far-reaching and rather surprising applications.
′
Theorem 2. The Weierstrass ℘-function satisfies the following differential
equation
(℘′ )2 = 4℘3 − 60G4 ℘ − 140G6 ,
where the lattice constants Gk = Gk (Ω) are defined by
X 1
.
(3)
Gk (Ω) =
ωk
06=ω∈Ω
Sketch of proof. It suffices to study an elliptic function on a fundamental
domain of the lattice Ω = ω1 Z + ω2 Z, e.g., F = {µω1 + ηω2 : 0 ≤ µ, η < 1}
(since the values elsewhere are determined by translation with lattice points).
Differentiation of the geometric series expansion leads to
∞
z m
1 X
1
1
1
= 2
= 2·
(m + 1)
.
(z − ω)2
ω (1 − ωz )2
ω m=0
ω
Thus, the Laurent expansion of ℘(z) at z = 0 is given by
∞
X
1
℘(z) = 2 +
(2m + 1)G2m+2 z 2m = z −2 + 3G4 z 2 + 5G6 z 4 + O(|z|6 )
z
m=1
(note that Gk = 0 for odd k since ℘ is even, resp. Gk (−Ω) = Gk (Ω) =
(−1)k Gk (Omega)). Computing the Laurent expansions for ℘3 and (℘′ )2 up
to terms of order O(|z|), shows that f (z) := (℘′ (z))2 − (4℘(z)3 − 60G4 ℘(z) −
140G6 ) = O(|z|) is an elliptic function. It follows that f is an entire bounded
function and, hence, it is constant equal to f (0) = 0 (see Exercise 1), which
proves the differential equation. •
The differential equation yields a parametrization
(X, Y ) = (℘, − 21 ℘′ )
of all elliptic curves defined over C since any such curve is given by a Weierstrass equation of the form Y 2 = X 3 + aX + b with some complex numbers a, b
satisfying 4a3 + 27b2 6= 0 (in order to have no singularities), and for any such
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JÖRN STEUDING
pair of a, b ∈ C there exists a lattice Ω such that a = −15G4 and b = −35G6
(see [1]). This is very similar to the parametrization of discs and ellipses by
(periodic) trigonometric functions. Topologically speaking, an elliptic curve
Figure 1. Metamorphosis: from a fundamental domain for the
lattice to a donut.
over C is a donut (resp., the factor group C/Ω of C by some lattice Ω is
a torus; see Figure 1 below). In particular, the group structure implied by
addition on the torus C/Ω — a.k.a. point addition on elliptic curves — is
of interest; the finite field version of elliptic curves with this structure is an
important feature in several modern cryptosystems. Predecessors of this addition law can be found in the works of Euler, Legendre, and Gauß on elliptic
integrals (as,
R forpexample, the one giving the arc length of an ellipse). Solving
integrals dx/ p(x) with a cubic polynomial p was a major line of research
in the late 18th century and early 19th century. A systematic approach to
elliptic intergals and functions was given by Abel and Jacobi around 1827/28.
In particular, Abel showed that the inverse of an elliptic integral is an elliptic function (which clarifies their names). For more information about elliptic
functions, in particular the Liouville theorems from 1844 on general elliptic
functions, we refer to [1]. The well-known Liouville theorem that any bounded
entire function is constant has its origin in the theory of elliptic functions (see
our proof of Theorem 2).
MODULAR FORMS
5
Practise makes perfect!
Exercises are useful for learning a new subject. It might be
that the proposed tasks are (sometimes) too difficult -- sorry!
If so, we suggest the reader to consult the literature listed
in the references! Anyway, for some exercises we shall provide
explicit hints and links to the relevant literature.
Exercise 1. Prove that an elliptic function without pole (without zero) is constant. Moreover, show that in case of a non-constant elliptic function any
complex value is assumed in the fundamental domain of the lattice equally often, namely as often as zero (counting multiplicities).
Hint: for advise we refer to [1].
Exercise 2. Assume that Ω = Z + iyZ with some positive real number y.
Prove that, for 0 < r < s < 12 ,
Z ℘(r;Ω)
dx
√
= r − s.
3
4x − 60G4 x − 140G6
℘(s;Ω)
Exercise 3. For n ≥ 4 show for the lattice constants that
X
(n − 3)(2n + 1)(2n − 1)G2n =
(2k − 1)(2ℓ − 1)G2k G2ℓ ;
k,ℓ≥2
k+ℓ=n
in particular, Gk ∈ Q[G4 , G6 ] for k ≥ 8 (e.g., 7G8 = 3G24 , 11G10 = 5G4 G6 ).
2. Eisenstein Series and Further Examples of Modular Forms
Given a lattice Ω = ω1 Z + ω2 Z, we may assume that τ := ωω12 lies in the
upper half-plane H := {z = x + iy ∈ C : y > 0}. Then Ω = ω1 (Z + τ Z) and
we define the Eisenstein series of weight k by
Gk (τ ) := Gk (Z + τ Z) = ω1k Gk (ω1 Z + ω2 Z)
X
(m + nτ )−k .
=
m,n∈Z
(m,n)6=(0,0)
Thus, Gk (τ ) is a function defined on the upper half-plane and varying τ means
varying the underlying lattice. A word about the names attached to this function and ℘ from the last section as well: Weierstrass started his research already
in 1840, however, he did not publish his results before 1862, whereas Eisenstein
discovered and published the main properties of ℘ and Gk independently in
1847.
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JÖRN STEUDING
Theorem 3. For integral k ≥ 3 the Eisenstein series Gk (τ ) is analytic in H
and satisfies
a b
aτ + b
k
∈ SL2 (Z).
= (cτ + d) Gk (τ )
∀ τ ∈ H,
Gk
c d
cτ + d
Recall that SL2 (Z) is the special linear group consisting of all 2 × 2-matrices
( ac dc ) with integral entries and determinant ad − bc = 1. These matrices act
on H as Möbius transforms
a c aτ + b
τ :=
H ∋ τ 7→
∈ H.
c d
cτ + d
Sketch of proof. The series for Gk (τ ) converges uniformly for k ≥ 3 (see the
proof of Theorem 1). For the proof of the transformation formula we observe
that
X
aτ + b −k
aτ + b
m+n
=
Gk
cτ + d
|
{zcτ + d}
m,n∈Z
(m,n)6=(0,0)
=
X
+dm+bn
= (cm+an)τ
(cτ +d)
(M + Nτ )−k (cτ + d)k
M,N∈Z
(M,N)6=(0,0)
N
n
with ( M
) = ( ab dc )( m
). •
Using Eisenstein series, we can immediately define functions having similar
transformation properties.
Corollary 4. The discriminant, defined by
∆(τ ) = (60G4 (τ ))3 − 27(140G6(τ ))2 ,
is analytic in H and satisfies
aτ + b
= (cτ + d)12 ∆(τ )
∆
cτ + d
∀ τ ∈ H,
a b
c d
∈ SL2 (Z).
Sketch of proof. For M = ( ac db ) ∈ SL2 (Z) we compute
∆(Mτ ) = (60G4 (Mτ ))3 − 27(140G6(Mτ ))2
= (60(cτ + d)4 G4 (τ ))3 − 27(140(cτ + d)6 G6 (τ ))2
= (cτ + d)12 (60G4 (τ ))3 − 27(140G6 (τ ))2 = (cτ + d)12 ∆(τ ).
The regularity follows from the regularity of the Eisenstein series. •
A meromorphic function f : H → C is said to be a weak modular form of
weight k if it satisfies
a b
aτ + b
k
∈ SL2 (Z).
= (cτ + d) f (τ )
∀ τ ∈ H,
f
c d
cτ + d
MODULAR FORMS
7
Moreover, if f is additionally analytic in H and at infinity i∞, then it is
called a modular form. Thus, the Eisenstein series Gk (τ ) are modular forms of
weight k (although Gk (τ ) vanishes identically if k is odd; see Exercise 4) and
the discriminant ∆ is modular of weight 12. The notion modular form was
introduced by Klein in 1890 (as ’Modulform’ in German).
Corollary 5. Klein’s j-invariant, defined by
j(τ ) =
(60G4 (τ ))3
∆(τ )
for
τ ∈ H,
is a modular function, i.e., a weak modular form of weight k = 0.
The proof is by straightforward computation similar to the proof of the previous corollary.
A rolling stone gathers no moss...
Exercise 4. Prove that the set of modular forms of fixed weight k forms a
vector space Mk over C. Moreover, show that Mk = C (the set of constant
functions) if and ond only if k is odd; otherwise, if k is even, then Mk = C·Gk
for k = 4, 6, 8, 10 and Mk = C · Gk + C · ∆ · Gk−12 for k ≥ 12.
Hint: for advise we refer to [6, 7].
Exercise 5. Prove that Gℓk and Gk · Gℓ are modular forms of weight kℓ and
k + ℓ, respectively.
3. The Modular Group and Fourier Expansions
The matrix group behind the Möbius transforms and their action on the
upper half-plane has a rather simple structure.
Theorem 6. The modular group PSL2 (Z) := SL2 (Z)/ ± ( 10 01 ) is generated by
0 1
1 1
;
and
S=
T =
−1 0
0 1
in particular, every M ∈ PSL2 (Z) has a (not uniquely determined) representation as M = T n1 ST n2 S · · · ST nk with some integers nj .
Notice that τ 7→ Sτ = −1
is a combination of an inversion at the unit circle
τ
and conjugation, and τ 7→ T τ = τ + 1 is the translation by one. Taking the
action of the modular group into account, one can show that for any τ ∈ H
there exists a matrix M ∈ PSL2 (Z) such that
Mτ ∈ F := {τ = x + iy ∈ H : − 21 < x ≤ 12 , |τ | ≥ 1}.
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JÖRN STEUDING
Figure 2. A tiling of the upper half-plane with images of the
fundamental domain F under the modular group.
The hyperbolic triangle F is called a fundamental domain. Only for points
τ on the boundary M might be not uniquely determined by Mτ ∈ F ; e.g.,
= Si (see Figure 2 below).
i = −1
i
Sketch of proof for Theorem 6. Denote by Γ the group generated by S
and T . Given M = ( ac db ) ∈ SL2 (Z), we shall show by induction on |c| that
M ∈ Γ. If c = 0, then T m = ( 10 m1 ) = ±M with m = ±b, and in view of
S 2 = −( 10 01 ) ∈ Γ we deduce M ∈ Γ. If c 6= 0, then
∗ ∗
M ′ := ST m M = ′
c ∗
′
with suitable 0 ≤ c = a + mc < |c| and m ∈ Z (by divison with remainder).
Using the induction hypothesis, M ′ ∈ Γ, we deduce M = T −m S −1 M ′ ∈ Γ. •
Given a modular form f , in view of the regularity of f in H ∪ {i∞} and
f (τ + 1) = f (τ ) it follows that f has a Fourier expansion.
Theorem 7. For even k ≥ 3 and τ ∈ H,
Gk (τ ) = 2ζ(k) + 2
∞
(2πi)k X
σk−1 (n)q n .
(k − 1)! n=1
MODULAR FORMS
9
Here we use the abbreviation q := exp(2πiτ ) and σℓ denotes the generalized
P
divisor function, defined by σℓ (n) = d|n dℓ . Finally,
−1
∞
X
Y
1
1
ζ(s) :=
=
1− s
for Re s > 1
s
n
p
n=1
p
is the Riemann zeta-function; the product is taken over all prime numbers and
is called an Euler product. The identity between the series and the product
follows from the unique prime factorization of the integers. Notice that
ζ(2k) = (−1)k−1
(2π)2k
B2k ,
2(2k)!
where the Bm ’s are the Bernoulli numbers defined by the identity
∞
X
m=0
Bm
zm
z
1
1
=
= 1 − z + z2 + . . . .
m!
exp(z) − 1
2
12
It is not too difficult to see that the Bm ’s are all rational.
+0
Sketch of proof. Since Gk ( −τ
) = (−1)k Gk (τ ) it follows that Gk (τ ) vanishes
0τ −1
identically for odd k (see Exercise 4). We may assume that k is even. By
pairing the vertical terms n = 0 in a single series and the non-vertical terms
n 6= 0 in a double series we find
(4)
Gk (τ ) = 2ζ(k) + 2
∞
+∞
X
X
1
.
(m + nτ )k
n=1 m=−∞
Differentiation of the partial fraction decomposition of the cotangent function
+∞ X
1
1
1
π cot πτ = +
−
τ
τ +m m
m=−∞
m6=0
implies
+∞
X
π 2
1
=
,
sin πτ
(τ + m)2
m=−∞
both being valid for τ 6∈ Z; the latter series is uniformly convergent for any τ
from any compact subset of C \ Z. Moreover, for τ ∈ H,
π 2 (2πi)2 q 2
q−1
sin πτ =
, resp.
.
=
2i exp(iπτ )
sin πτ
(1 − q)2
On the right-hand side we may interpret the rational function
ative of the geometric series; this leads to
∞
X
1
2
dq d ,
=
(2πi)
2
(τ
+
m)
m=−∞
d=1
+∞
X
1
(1−q)2
as deriv-
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JÖRN STEUDING
and further differentiation gives
+∞
X
∞
(2πi)k X k−1 d
1
d q .
=
(τ + m)k
(k − 1)! d=1
m=−∞
Substituting this in (4), yields the Fourier expansion. •
As a more or less immediate consequence we may deduce
Corollary 8. For τ ∈ H,
∆(τ ) = (2π)12
∞
X
τ (n)q n ;
n=1
here n 7→ τ (n) is the Ramanujan τ -function (and shall not be confused with the
variable τ ) and always τ (n) ∈ Z.
Sketch of proof. Define
a=
∞
X
n
σ3 (n)q ,
b=
n=1
∞
X
σ5 (n)q n .
n=1
In view of the definition of ∆ and Theorem 7 we find
(2π)12 (1 + 240a)3 − (1 − 504b)2 ;
∆(τ ) =
1728
indeed, the constant term of the Fourier series expansion equals
6 2
4 3
π
π
2
3
2
3
− 27 · 280
= 0.
(60 · 2ζ(4) − 27(140 · 2ζ(6)) = 120
90
945
Therefore the discriminant is an example of a so-called cusp form, that is
a modular form that vanishes at infinity. Since d3 ≡ d5 mod 12 we have
a ≡ b mod 12, hence
(1 + 240a)3 − (1 − 504b)2 = (1 + 20 · 12a)3 − (1 − 42 · 12b)2
≡ 122 (5a + 7b) ≡ 0 mod 123 (= 1728).
This finishes the proof. •
In 1916, Ramanujan conjectured that
11
• |τ (p)| < 2p 2 for all prime numbers p, and
• τ (n) is multiplicative, i.e., τ (mn) = τ (m)τ (n) for coprime m, n.
The second conjecture was proved by Mordell in 1917 who showed
X
(5)
τ (m)τ (n) =
d11 τ ( mn
).
d2
d|gcd(m,n)
MODULAR FORMS
11
The first conjecture was proved by Deligne in 1974 (for which he was awarded
the Fields medal in 1978). The yet unsolved Lehmer conjecture claims that
τ (p) 6= 0 for any prime p. On the contrary it is known that
∆(τ ) = (2π)12 (q − 24q 2 + 252q 3 − 1472q 4 + 4830q 5 ∓ . . .)
∞
Y
12
(1 − q m )24 .
= (2π) q
m=1
Q∞
1/24
m
The product η(τ ) = q
m=1 (1 − q ) is the Dedekind η-function which is
related to the partition function by (1). Ramanujan found congruences like
τ (n) ≡ σ11 (n) mod 691
(e.g., σ11 (2) = 1 + 211 = 2049 = −24 + 3 · 691 = τ (2) + 3 · 691) as well as
p(11k + 6) ≡ 0 mod 11.
Extending work of Atkin, Ken Ono proved that for any modulus coprime with
6 there exists a Ramanujan type congruence, e.g.,
p(4 063 467 631 k + 30 064 597) ≡ 0 mod 31.
For the j-invariant one finds the Fourier expansion
1
1728 j(τ ) = + 744 + 196 884q + 21 493 760q 2 + . . . .
q
Also here the coefficients are integers and, surprisingly, they are related to the
irreducible representations of the monster group (that is the largest sporadic
simple group); this unexpected link was discovered by Mc Kay in 1978 and
had been communicated as monstrous moonshine2 before this conjecture was
solved by Borcherds in 1992 who was awarded a Fields medal in 1998. Using
the mapping properties of j(τ ) one can prove
• the existence of a lattice Ω to a given elliptic curve E(C) such that the
associated Weierstrass ℘-function ℘(s; Ω) gives the parametrization of
E(C) (as already mentioned in §1);
• Picard’s theorem: an entire function omitting two complex values is
constant;
proofs of both results can be found in [1].
Another interesting feature of the j-invariant is that certain values are algebraic or even integers. For example, j(i) = 1728 and, more impressive,
√
j( 21 (1 + −163)) ∈ Z. This result was proved by Schneider in 1937; the reason behind is of algebraic nature (namely class field theory) and anything but
trivial. In particular, a special property of an algebraic field extension√of Q is
crucial: the ring of integers of the imaginary quadratic number field Q( −163)
2Private
destillation of hard alcohol is called ’moonshine’. This attribute had been given
to this conjecture in view of the far distant topics sporadic groups and modular forms.
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JÖRN STEUDING
√
is given by the factorial ring Z[ 21 (1+ −163)]. As a consequence, the first term
√
in the Fourier series expansion of j( 21 (1 + −163)) is almost an integer which
leads to
√
exp(π 163) = 262 537 412 640 768 743 . 99999 99999 9925 . . . .
For more details, the interested reader may consult the world wide web or,
better, a library and search for the keyword Kronecker’s Jugendtraum.
Brain is better than brawn!
Exercise 6. If f is a modular function (i.e., a weak modular form of weight 0),
then the number of zeros of f is equal to the number of poles, both counted inRthe
′
fundamental domain F (counting multiplicities). Use the counting integral ff
in order to prove this theorem (and further help can be found in [1]).
√
Exercise 7. Prove G4 (ρ2 ) = 0 = G6 (i) where ρ = 12 (−1 + i 3) is a primitive
third root of unity. Moreover, deduce j(ρ) = 0 and j(i) = 123 = 1728.
Exercise 8. Prove Hurwitz’s identity
σ7 (n) = σ3 (n) + 120
X
σ3 (a)σ3 (b).
a+b=n
Exercise 9. Show that the Fourier coeffcients of j(τ ) are integers.
Exercise 10. Prove that the set of modular functions is a field, namely C(j).
4. The Theta Series and Functional Equations
The classical theta-function is given by
ϑ(τ ) =
+∞
X
2
exp(πin τ ) = 1 + 2
n=−∞
∞
X
qn
2
with q = exp(πτ )
n=1
(different to the previous value for q). In 1713, Jakob Bernoulli was the first to
study this function, Euler succeeded to go much further, as already mentioned
in the beginning; however, Jacobi proved a transformation formula for which
ϑ is often called the Jacobi theta-function.
Theorem 9. For τ ∈ H,
(6)
−1
ϑ
τ
=
r
τ
ϑ(τ ).
i
Here the branch of the square root has to be taken which is positive on the
imaginary axis; the other values are defined by analytic continuation.
MODULAR FORMS
13
Sketch of proof. We shall use a tool from Fourier analysis, the Poisson’s
summation formula: Let f : R → R be a twice continuously differentiable
function with f (z) ≪ |z|−2 as |z| → ∞. Further, assume that the integral
Z +∞
|f ′′ (z)| dz
−∞
exists. Then, for any α ∈ R,
X
X
fˆ(m) exp(2πiαm).
f (n + α) =
n∈Z
m∈Z
2
A proof can be found in [11]. Applying this with f (z) := exp(− πzx ) where
x > 0, we compute by quadratic substitution:
2
Z +∞
z
fˆ(y) =
exp −π
+ 2iyz
dz
x
−∞
Z +∞
2
= x exp(−πxy )
exp(−πx(w + iy)2 ) dw.
−∞
Consider the integral
Z
exp(−xω 2 ) dω,
R
where R is the rectangular contour with vertices ±r, ±r + iIm λ, where r is a
positive real number. By Cauchy’s theorem, the integral is equal to zero. On
the line Re ω = r, the integrand tends uniformly to zero as r → ∞. Hence,
Z +∞
I(λ) :=
exp(−πx(w + λ)2 ) dw = I(0),
independ on λ. This gives
−∞
fˆ(y) = x exp(−πxy 2 )
where
Z
+∞
√
exp(−πxw 2 ) dw = C x exp(−πxy 2 ),
−∞
C :=
Z
+∞
exp(−πz 2 ) dz.
−∞
Applying Poisson’s summation formula leads to
X
√ X
(n + α)2
=C x
exp(−πxm2 + 2πimα);
exp −π
x
m∈Z
n∈Z
here we have introduced the parameter α. Choosing α = 0 and x = 1 both
sums are equal; thus, C = 1 and the transformation formula is proved. •
Riemann built one of his two proofs of the functional equation for the Riemann zeta-function ζ(s) on Jacobi’s transformation formula. Following his
approach we start with Euler’s integral for the Gamma-function,
Z ∞
Γ(z) =
uz−1 exp(−u) du.
0
14
JÖRN STEUDING
Substituting u = πn2 x and summing up over all n ∈ N, yields
∞
∞ Z ∞
s X
X
s
1
− 2s
=
x 2 −1 exp(−πn2 x) dx.
π Γ
s
2 n=1 n
n=1 0
On the left-hand side we find the Dirichlet series defining ζ(s); hence the latter
formula is valid only for Re s > 1. On the right-hand side we may interchange
summation and integration, which is allowed by absolute convergence. We
obtain
Z ∞
∞
s
X
s
−1
− 2s
ζ(s) =
x2
exp(−πn2 x) dx.
π Γ
2
0
n=1
We split the integral at x = 1 to get
Z 1 Z ∞ s
s
− 2s
ζ(s) =
π Γ
+
x 2 −1 ω(x) dx,
2
0
1
where
ω(x) :=
∞
X
n=1
exp(−πn2 x) = 21 (ϑ(ix) − 1)
can be expressed in terms of Jacobi’s theta-function. Using the transformation
formuale, we find
√
√
i
1
1
=2 ϑ
− 1 = xω(x) + 12 ( x − 1)
ω
x
x
and the substitution x 7→ x1 leads to
Z ∞
s
s
1
− 2s
− s+1
−1
2
2
π Γ
ζ(s) =
+
x
+x
ω(x) dx.
2
s(s − 1)
1
Since ω(x) ≪ exp(−πx), the last integral converges for all values of s, and,
by analytic continuation, throughout the complex plane giving a meromorphic
continuation for ζ(s) throughout C. The right-hand side remains unchanged
by s 7→ 1 − s which proves the desired functional equation for zeta. Moreover,
1
is an entire function. Thus we arrive at
it easily follows from that ζ(s) − s−1
Corollary 10. The zeta-function ζ(s) has an analytic continuation to C \ {1}
and a simple pole with residue 1 at s = 1. For any s ∈ C,
s
π − 2 Γ( 2s )ζ(s) = π −
1−s
2
)ζ(1 − s).
Γ( 1−s
2
In the first half of the nineteenth century Hecke was the first to study modular
forms systematically. Moreover, generalizing from the Riemann zeta-function,
he investigated the associated Dirichlet series
(7)
L(s; ∆) :=
∞
X
n=1
τ (n)n−s =
Y
p
(1 − τ (p)p−s + p11−2s )−1 .
MODULAR FORMS
15
Here the product is over the primes and the identity between this Euler product
and the series follows from the multiplicativity property of the coefficients.
Both, the product and the series converge for Re s > 12. Hecke showed that
L(s; ∆) has an analytic continuation to the whole complex plane. And in the
same way as Riemann used ϑ the modularity of the discriminant ∆ implies a
functional equation for L(s; ∆) as Hecke observed:
(2π)−s Γ(s)L(s; ∆) = (2π)s−12 Γ(12 − s)L(12 − s; ∆).
Actually, Hecke proved much more, namely a bijection between modular forms
and Dirichlet series satisfying a Riemann-type functional equation (see [8]).
In 1958, Shimura & Taniyama conjectured that elliptic curves are modular,
meaning that the L-function associated to an elliptic curve coincides with an
L-function attached to a certain modular form (to subgroups of the modular
group). This so-called Shimura–Taniyama–conjecture was proved by Wiles et
al. and led (in a rather indirect way) to the proof of Fermat’s last theorem on
the very sparse set of integer solutions to the diophantine equation X n + Y n =
Z n .3 For further details we refer to [2, 4]; for the amazing (hi)story behind
read [9].
No pains - no gains!
Exercise 11. Extend the proof of Theorem 9 in order to show
r +∞
+∞
X
τ X
−1
2
2
+ 2πimλ),
exp(πim
exp(πi(n + λ) τ ) =
i n=−∞
τ
m=−∞
where λ is an arbitrary complex number.
√
Exercise 12. Use the transformation formula (6) in order to approximate y
with some positive real number y by
P
1 + 2 n≤N exp(−πn2 /y)
P
.
1 + 2 n≤N exp(−πn2 y)
√
If 2 = 1.41421 35 . . . shall be approximated with at least five correct decimals,
how big you have to choose N?
Exercise 13. Fill the gaps we left in the proof of Corollary 10 (in particular
the information about the Laurent expansion of ζ(s) at s = 1).
Exercise 14. Deduce the Euler product (7) for L(s; ∆) from (5).
3although
the illustrous mathematician Homer Simpson claimed to have found with
178212 + 184112 = 192212 a large non-trivial solution which shouldn’t exist according to
Fermat and Wiles, respectively!
16
JÖRN STEUDING
Exercise 15. For even k ≥ 4 show that
∞
X
L(s; Gk ) = c
σk−1 (n)n−s = cζ(s)ζ(s + 1 − k)
with c = 2
n=1
(2πi)k
.
(k − 1)!
Moreover, find a functional equation for L(s; Gk ).
5. Arithmetic:
Sums of Squares
By definition ϑ(τ + 2) = ϑ(τ ). Hence, taking the transformation formula
into account, the theta-function can be seen as a modular form of weight 21 to
the subgroup of PSL2 (Z), generated by
0 1
1 2
2
;
and
S=
T =
−1 0
0 1
this subgroup is called the theta group. Systematic studies of modular forms
with respect to subgroups of the modular group started with the pioneering
work of Klein and Poincaré around 1880 (and the study of subgroups of the
modular group is nowadays an important area of research). Observing that
theta-functions are generating functions for the number of representations as
sums of squares, e.g.,
!8
+∞
∞ X
X
X
X
2
m
m21 +...+m28
q
=
q
= 1 + 16
(−1)n−d d3 ,
m=−∞
n=1 d|n
m1 ,...,m8 ∈Z
Jacobi deduced arithmetical information.
Theorem 11. For τ ∈ H,
ϑ(τ )8 =
3
(16G4 (τ ) − G4 ( τ +1
));
2
π4
in particular,
8
ϑ(τ ) = 1 +
∞
X
r8 (n)q n
n=1
with
r8 (n) = ♯{(m1 , . . . , m8 ) ∈ Z8 : m21 + . . . + m28 = n}
X
= 16
(−1)n−d d3 .
d|n
Here ♯M denotes the cardinality of the set M. Notice that r8 (n) is positive
for any n ∈ N; e.g. r8 (p) = 16(1 + p3 ) for any odd prime p and r8 (2) =
112. This big numbers fro r8 (n) for already small n come from counting
the representations including all possible permutations and with respect to
symmetries; e.g. (−1)2 + 12 + 02 + . . . + 02 and (+1)2 + 12 + 02 + . . . + 02 as
MODULAR FORMS
17
well as 12 + 02 + 12 + 02 + . . . + 02 count different representations of 2 as sum of
eight squares. This is just a first example what arithmetical information can
be deduced from the analytic behaviour of modular forms.
Sketch of proof. We shall use the structure of a vector space of modular
forms; however, with respect to the theta-function we shall replace the modular
group by the theta group. Assume that f : H → C is analytic satisfying
r m
τ
−1
=
f (τ + 2) = f (τ )
and
f
f (τ )
τ
i
for all τ ∈ H with some integer m such that the following limits exist
r −m τ
1
πimτ
f 1−
exp −
=β
lim f (τ ) = α,
lim
τ →i∞
τ →i∞
i
τ
4
with some constants α, β. An example of such a function is the theta-function:
in fact, by its definition and Exercise 11,
r −1 1
πiτ
τ
ϑ 1−
exp −
= 2.
(8)
lim ϑ(τ ) = 1,
lim
τ →i∞
τ →i∞
i
τ
4
We shall show that all such f are of the form f (τ ) = αϑ(τ )m with α equal to
limτ →i∞ f (τ ).
First of all we observe that Fϑ := F ∪ T F ∪ T SF is a fundamental domain
for the theta group (that is for any τ ∈ H there exists a matrix M from the
theta group such that Mτ ∈ Fϑ ). It follows that
|ϑ(τ ) − 1| ≤ 2
∞
X
n=1
2
exp(−πn y) ≤ 2
∞
X
√
exp(− π 2 3 n) = 0.14 . . . ;
n=1
hence ϑ(τ ) 6= 0, and in a similar manner one can show the non-vanishing of
ϑ(τ + 1) and ϑ(1 − τ1 ) thorughout H. Thus,
h(τ ) :=
f (τ )
ϑ(τ )m
is analytic in H, satisfies h(τ + 2) = h(τ ) and h( −1
) = h(τ ) as well as
τ
1
lim h(τ ) = α,
lim h 1 −
=ω
τ →i∞
τ →i∞
τ
with some constant ω. Next consider
H(τ ) := (h(τ ) − α)(h(τ ) − ω).
Then, limτ →i∞ H(τ ) = 0 and limτ →1 H(τ ) = 0. Hence, H is bounded on Fϑ
and in view of
−1
H(τ + 2) = H(τ )
and
H( ) = H(τ )
τ
18
JÖRN STEUDING
it is also bounded on H. By the maximum principle it follows that H is vanishing identically. Consequently, α and ω are the only values that h assumes.
Since h is continous, we deduce that h is indeed constant, giving f = αϑm .
For arbitrary complex numbers a, b let
g(τ ) = aG4 (τ ) + bG4 ( 21 (τ + 1)).
Then, by Theorem 3,
g(τ + 2) = g(τ )
and
and, by Theorem 7,
g
−1
τ
= τ 4 g(τ )
π4
(a + b).
τ →i∞
45
We shall choose a and b such that a + b = π454 . In view of (8) we take a = π484
and b = − π34 . Since
ϑ(τ )8
f (τ ) :=
g(τ )
is a modular function (weight zero) for the theta-group hT 2 , Si, and all modular functions of the theta-group with this limiting behaviour are constant, it
follows that f (τ ) ≡ 1 (as sketched above; see also Exercise 15 below). Thus,
ϑ(τ )8 = g(τ ) and in view of Theorem 7 its Fourier expansion is given by
lim g(τ ) = 2(a + b)ζ(4) =
1 + 16
2
∞
X
n=1
n
σ3 (n)q − 16
n
∞
X
σ3 (n)(−q)n .
n=1
Comparing the coefficients at q yields the formula for r8 (n). •
As Lagrange showed, any positive integer can be written as sum of four
squares (see [3]); the theory of modular forms provides here an explicit formula
for the number of such representations too. However, it is much harder to prove
the analogous result in this case because the Eisenstein series G2 cannot be
constructed as we managed to handle the Eisenstein series Gk with k ≥ 3. One
can show that
!
∞
X
π2
1 − 24
σ1 (n)q n
G2 (τ ) =
3
n=1
is analytic in the upper half-plane and satisfies the identity
−1
= τ 2 G(τ ) − 2πiτ
G2
τ
for all τ ∈ H. Hence, G2 is not a modular form, nevertheless there is a relation
with Jacobi’s theta-function, namely
(9)
ϑ4 (τ ) =
π2
(4G2 (2τ ) − G2 (τ ))
9
MODULAR FORMS
19
Exploiting this formula one can deduce Lagrange’s theorem with a precise
formula for the number of representations as a sum of four squares.
Let a thousand flowers bloom...
Exercise 16. Give a rigorous and complete proof of (8).
Exercise 17. The sketch of the proof of the latter theorem on representations
of eight squares included the following general result: given an analytic function
f : H → C satisfying
r m
τ
−1
=
f (τ )
f (τ + 2) = f (τ )
and
f
τ
i
for all τ ∈ H with some integer m such that the following limits exist
r −m πimτ
τ
1
exp −
= β,
f 1−
lim f (τ ) = α,
lim
τ →i∞
τ →i∞
i
τ
4
then f (τ ) = αϑ(τ )m . Give a detailed proof !
Exercise 18. Fill the gaps in the just given sketch of proof for (9) and deduce
further that
X
d
r4 (n) := ♯{(m1 , . . . , m4 ) ∈ Z4 : m21 + . . . + m24 = n} = 8
1≤d≤n
4∤d|n
Use this to show that r4 (n) is positive for any n.
Hint: for solving this exercise have a look into [1].
Exercise 19. Prove that a8 (n) ≥ a4 (n) ≥ 1 for all n. Show this i) by using
the definitions, and ii) applying the representation formulae.
6. Picture Gallery
Using Mathematica one can create colorful pictures to illustrate complexvalued functions and their analytic behaviour; the recent book [12] provides
many examples. In particular, the elliptic functions and modular forms we got
to know are implemented in Mathematica and it takes only a short moment
for an ordinary personal computer to produce impressive graphics showing
their beauty.
For instance, the following figure shows the absolute value of Klein’s invariant j(x + iy) for the range x ∈ [−1.5, +1.5], y ∈ [0.1, 1.0]. This modular
function is unbounded in the upper half-plane as one can imagine by the steep
mountains showing up in direction of the real axis.
20
JÖRN STEUDING
Next is flower power: The imaginary-part of j(x + iy) for the range
x ∈ [−1.5, +1.5], y ∈ [0.1, 1.6] reveals some flowers and the pattern of the
fundamental domain under the action of the modular group.
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
A one-dimensional cut through the analytic landscape of Dedekind’s etafunction yields a periodic picture: below the values of η(x + i0.2) for x ∈
[−15, +15]. In view of the modularity of the discriminant, one has η(τ + 1) =
πi
)η(τ ) and, indeed, the eta-function is periodic with minimal period 24.
exp( 12
Plots may imply mathematical results!
MODULAR FORMS
21
1.0
0.5
-15
-10
5
-5
10
15
-0.5
-1.0
We did not mention many classical results about elliptic functions and modular forms as, for example, the works of Hardy & Ramanujan and Rademacher
on the coefficients of p(n) (resp., j; see [1]). Names of famous mathematicians
as Kronecker, Hurwitz, Fricke, Petersson, and Rankin were not mentioned at
all but made significant contributions to the theory of elliptic functions and
modular forms. In particular, the relation between modular forms and elliptic
curves has turned out to be of greatest importance for a variety of arithmetical
problems. The reader will find examples in [5, 2], e.g., the congruent number
problem, which asks whether for a given positive integer n there exists a right
triangle with rational sides and area equal to n. There are also relevant applications of elliptic functions to the real world, in particular to physics. The
monographs [4, 13] offer historically motivated introductions to elliptic functions and modular forms. For the original articles or books by the founders of
the theory of elliptic functions and modular forms — Bernoulli, Euler, Abel,
Jacobi and so on — we refer to the references of the secondary literature below (which does not mean that it is not worth to read the ancestors!). In the
author’s opinion, the best book on elliptic functions and modular forms is the
monograph [6] by Koecher & Krieg, however, this book is written in German;
my favourite online course notes on this topic are due to Milne [7]. The books
by Stein & Shakarchi [10, 11] offer very good introductions to Complex and
Fourier analysis, respectively.
References
[1] T.M. Apostol, Modular Forms and Dirichlet Series in Number Theory, Springer 1990
[2] F. Diamond, J. Shurman, A First Course in Modular Forms, Springer 2005
22
JÖRN STEUDING
[3] G.H. Hardy, E.M. Wright, An introduction to the theory of numbers, Clarendon
Press, Oxford, 1979, 5th ed.
[4] Y. Hellegouarche, Invitation tot he Mathematics of Fermat-Wiles, Elseveier 2002
[5] N. Koblitz, Introduction to elliptic curves and modular forms, Springer 1993, 2nd ed.
[6] M. Koecher, A. Krieg, Elliptische Funktionen und Modulformen, Springer 1998
[7] J.S. Milne, Modular functions and modular forms, online available at
http://www.jmilne.org/math/CourseNotes/mf.html
[8] A.P. Ogg, Modular Forms and Dirichlet Series, Benjamin, New York-Amsterdam 1969
[9] S. Singh, Fermat’s last theorem, Fourth Estate, London 1997
[10] E.M. Stein, R. Shakarchi, Complex Analysis, Princeton University Press 2003
[11] E.M. Stein, R. Shakarchi, Fourier Series and Integrals, Princeton University Press
2003
[12] E. Wegert, Visual Complex Functions, Birkhäuser 2012
[13] A. Weil, Elliptic Functions according to Eisenstein and Kronecker, Springer 1978
Jörn Steuding
Department of Mathematics, Würzburg University
Emil-Fischer-Str. 40, 97 074 Würzburg, Germany
steuding@mathematik.uni-wuerzburg.de