MODULAR FORMS
WEI ZHANG
NOTES TAKEN BY PAK-HIN LEE
Abstract. Here are the notes I took for Wei Zhang’s course on modular forms offered at
Columbia University in Spring 2013 (MATH G4657: Algebraic Number Theory). Hopefully
these notes will appear in a more complete form during Fall 2014. I recommend that you
visit my website from time to time for the most updated version.
Due to my own lack of understanding of the materials, I have inevitably introduced
both mathematical and typographical errors in these notes. Please send corrections and
comments to [email protected].
Contents
Classical Modular Forms
1. Lecture 1 (January 23, 2013)
1.1. Introduction
1.2. Basic Definitions
1.3. Eisenstein series
2. Lecture 2 (January 28, 2013)
2.1. Eisenstein series
3. Lecture 3 (January 30, 2013)
3.1. Modular Curves
4. Lecture 4 (February 4, 2013)
4.1. Modular Curves
4.2. Elliptic curves with level structures
5. Lecture 5 (February 6, 2013)
5.1. Heegner points
5.2. Dimension formulas
6. Lecture 6 (February 11, 2013)
6.1. Hecke operators
7. Lecture 7 (February 13, 2013)
7.1. Hecke operators for level 1
7.2. Hecke operators for level N
7.3. Petersson inner product
8. Lecture 8 (February 18, 2013)
8.1. Petersson inner product
8.2. L-functions
9. Lecture 9 (February 20, 2013)
Last updated: September 6, 2014.
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9.1. Hecke operators
9.2. Atkin-Lehner theory
10. Lecture 10 (February 25, 2013)
10.1. Atkin-Lehner theory
10.2. Rationality and Integrality
10.3. Plan
Warning: The following lectures have not been edited.
11. Lecture 11 (February 27, 2013)
12. Lecture 12 (March 4, 2013)
13. Lecture 13 (March 6, 2013)
14. Lecture 14 (March 11, 2013)
15. Lecture 15 (March 13, 2013)
p-adic Modular Forms
16. Lecture 16 (March 25, 2013)
17. Lecture 17 (March 27, 2013)
18. Lecture 18 (April 1, 2013)
19. Lecture 19 (April 3, 2013)
20. Lecture 20 (April 8, 2013)
21. Lecture 21 (April 10, 2013)
22. Lecture 22 (April 15, 2013)
23. Lecture 23 (April 17, 2013)
24. Lecture 24 (April 22, 2013)
25. Lecture 25 (April 24, 2013)
26. Lecture 26 (April 29, 2013)
27. Lecture 27 (May 1, 2013)
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1. Lecture 1 (January 23, 2013)
1.1. Introduction. Last year this course was about class field theory. This year we will
focus on modular forms. The two main topics are:
(1) classical (holomorphic) modular forms for the full modular group SL2 (Z) and its
congruence subgroups. We will be interested in both the analytic and arithmetic
theory.
(2) p-adic properties and p-adic modular forms.
We will not follow any specific textbook closely, but a list of references includes:
(1) Diamond, Shurman, A First Course in Modular Forms. Homework will mainly be
assigned from this book (among other resources).
(2) Serre, A Course in Arithmetic. The last part of this book has a concise explanation
of modular forms for SL2 (Z).
(3) Shimura, Introduction to the Arithmetic Theory of Automorphic Functions.
(4) Lang, Introduction to Modular Forms.
1.2. Basic Definitions. We begin with some basic definitions.
Definition 1.1. The upper half plane is H = {τ ∈ C : Im(τ ) > 0}.
We will reserve the letter zfor any complex number not necessarily contained in H.
a b
Any γ ∈ SL2 (R) =
: a, b, c, d ∈ R, det = 1 acts on H via
c d
aτ + b
.
cτ + d
Check that this defines a group action. The key point is
Im(τ )
Im(γ · τ ) =
|cτ + d|2
γ·τ =
using the fact that det(γ) = 1. Moreover this action defines a biholomorphic automorphism
of H. The set of biholomorphic automorphisms of H is
Aut(H) ∼
= PSL2 (R) = SL2 (R)/{±I}.
We will be interested in discrete subgroups of SL2 (Z). More specifically,
Definition 1.2. Congruence subgroups are subgroups Γ ⊂ SL2 (Z) that contain Γ(N ) for
some positive integer N , where Γ(N ) is the principal congruence subgroup of level N defined
by
a b
a b
1 0
Γ(N ) =
∈ SL2 (Z) :
≡
(mod N ) .
c d
c d
0 1
We can define various other subgroups that have finite index in SL2 (Z), for example, by
changing the condition into
a b
a b
∗ ∗
Γ0 (N ) =
∈ SL2 (Z) :
≡
(mod N ) .
c d
c d
0 ∗
Definition 1.3. A weakly modular function of weight k is a function f : H → C such that:
(1) (meromorphicity) f is meromorphic on H;
3
(2) (transformation rule) f (γ · τ ) = f
aτ + b
cτ + d
a b
= (cτ + d) f (τ ) for all γ =
∈
c d
k
SL2 (Z).
The group SL2 (Z) is generated by two elements, which are the matrices corresponding to
translation τ 7→ τ + 1 and inversion τ 7→ − τ1 .
1 1
0 −1
Lemma 1.4. SL2 (Z) =
,
.
0 1
1 0
With this we can show
Lemma 1.5. A fundamental domain of H modulo SL2 (Z) is
1
1
1
D = τ ∈ H : − ≤ Re(τ ) < , |τ | > 1 ∪ τ ∈ H : − ≤ Re(τ ) ≤ 0, |τ | = 1
2
2
2
i.e. under SL2 (Z)-transformation, every point in H is equivalent to exactly one point in D.
(Diagram here)
A proof can be found in Serre’s book. The key idea is to maximize Im(γ · τ ). Maximum
is achieved precisely when it lies in the fundamental domain.
Since −I acts trivially on H, we consider PSL2 (Z) = SL2 (Z)/{±I}.
Suppose
f is a weakly modular function that is holomorphic on H. Transformation by
1 1
γ=
shows that f satisfies f (τ + 1) = f (τ ), i.e. f has period 1. We can then consider
0 1
its Fourier expansion: for any τ = x + iy, viewing f as a function of x gives
X
f (x + iy) =
an (y)e2πinx
n∈Z
where the Fourier coefficients are given by
Z 1
Z
−2πinx
−2πny
f (x + iy)e
dx = e
an (y) =
0
1+iy
f (τ )e−2πinτ dτ
0+iy
So far we have not used the holomorphicity Rof f . By contour integration over the rectangle
1+iy
with vertices iy, 1+iy, iy 0 , 1+iy 0 , we have that 0+iy f (τ )e−2πinτ dτ is independent of y because
the integral over the two vertical edges cancel. Hence we can write
X
f (τ ) =
an e2πinτ .
n∈Z
It is convenient to introduce q = e
2πiτ
. Then the Fourier expansion of f is given by
X
f (τ ) =
an q n .
n∈Z
Geometrically, the map τ 7→ q = e2πiτ sends the upper half plane H into D − {0} where
D = {z ∈ C : |z| < 1} is the open unit disk. In fact this induces a biholomorphism
H/Z ∼
= D − {0}.
Here we see that the singularity around the origin is important. A priori we could have any
Fourier expansion.
4
Definition 1.6. f is meromorphic at τ = ∞ if there exists M ∈ Z such that an = 0 for all
n < M . f is holomorphic at τ = ∞ if an = 0 for all n < 0.
This is equivalent to saying that f (q) extends to a meromorphic (holomorphic) function
on the unit disk D.
Definition 1.7. A modular form of weight k is a function f : H → C such that:
(1) f is holomorphic on H∪{∞} (i.e. the Fourier expansion only has nonnegative powers
of q);
a
b
(2) f (γ · τ ) = (cτ + d)k f (τ ) for all γ =
∈ SL2 (Z).
c d
Let us look at some examples, otherwise we would just be doing function theory.
1.3. Eisenstein series. We associate to τ ∈ H the lattice Λτ = Z + Zτ ⊂ C, which is free
abelian of rank 2 since τ has positive imaginary part. Define
X
X0
Gk (τ ) =
w−k =
(mτ + n)−k
(m,n)∈Z2
w∈Λτ −{0}
P
where 0 means we are summing over (m, n) 6= (0, 0).
Gk satisfies the following transformation property: if we replace τ 7→ γ · τ =
aτ +b
,
cτ +d
then
Gk (γ · τ ) = (cτ + d)k Gk (τ ).
Gk is identically zero when k is odd, so it is only interesting when k is even.
We have not addressed convergence issue yet.
Lemma 1.8. Gk (τ ) is absolutely convergent if k ≥ 4, and uniformly convergent on any
compact subset of H, so it defines a holomorphic function on H.
Proof (Sketch). To prove convergence, we can assume τ ∈ D is in the fundamental domain.
Then
τ ∈D
|w|2 = |mτ + n|2 = m2 τ τ + 2mn Re(τ ) + n2 ≥ m2 − mn + n2 = |mρ + n|2
(where ρ is a cube root of unity). Therefore we can bound
X0
X0
|mρ + n|−k
|Gk (τ )| ≤
|w|−k =
m,n
m,n
which is convergent when k ≥ 3. This proves convergence and uniform convergence over
compact subsets.
By absolute convergence, we can evaluate limits term-wise.
X0
Lemma 1.9. lim Gk (τ ) =
n−k = 2ζ(k) where ζ is the Riemann zeta function.
τ →∞
n∈Z
Corollary 1.10. For k ≥ 4 even, Gk is a modular form of weight k, called the Eisenstein
series of weight k.
5
Later we will see that these are essentially the only modular forms.
Next we consider the Fourier expansion of Gk (τ ). It is well-known that
∞ 1
1
cos πz
1 X
+
+
= π cot πz = π
z d=1 z + d z − d
sin πz
which has simple
at the integers and is holomorphic elsewhere. Be careful not to write
P poles
1
the sum as d∈Z z+d because this is not convergent!
For a proof of this identity, Diamond-Shurman gives the hint
Y
z2
sin πz = πz
1− 2
n
n≥1
but this is like cheating because the two identities are equivalent – the identity we want is
just the logarithmic derivative of this one. Instead we invoke a general theorem.
Lemma 1.11 (Special case of Mittag-Leffler). If f : C → C is a meromorphic function
having simple poles a1 , a2 , · · · , ai , · · · with residues b1 , b2 , · · · , bi , · · · respectively such that
(1) the sequence 0 < |a1 | ≤ |a2 | ≤ · · · ≤ |ai | ≤ · · · goes to ∞;
(2) there exist closed contours Cm with length lm and distance Rm to 0 such that lm /Rm <
N1 , Rm → ∞, and |f |Cm | < N2 for given positive numbers N1 , N2 , then
∞
X
1
1
f (z) = f (0) +
bi
+
z − ai ai
i=1
The proof uses the Cauchy integral formula and is left as an exercise.
2. Lecture 2 (January 28, 2013)
2.1. Eisenstein series. Last time we defined the space of modular forms and gave the
example of Eisenstein series
X0
X0
(cτ + d)−2k
ω −2k =
G2k (τ ) =
(c,d)∈Z2
ω∈Λτ
where Λτ = Z + Zτ . For k ≥ 2, this is absolutely convergent and uniformly convergent on
compact sets. Thus G2k ∈ M2k , the space of modular forms of weight 2k. Also recall that
G2k (∞) = 2ζ(2k).
We want to find the Fourier expansion. We make use of the following identity
∞ ∞
1 X
1
1
1 X 2z
π cot πz = +
+
= +
z d=1 z − d z + d
z d=1 z 2 − d2
(the way to remember this is to look at the zeroes and poles of cot πz) which is convergent.
Multiplying z on both sides gives
∞
∞
X
X
cos πz
z2
πz
=1+2
=
1
−
2
ζ(2k)z 2k
2 − d2
sin πz
z
d=1
k=1
∞
X
z2
z2 1
z2 k
where we expanded at z = 0 using the geometric series 2
=
−
=
−
(
) .
2
z − d2
d2 1 − dz 2
d2
k=1
6
Incidentally we found a way to evalute the Riemann zeta function! This relates to the
Bernoulli numbers as follows. The left hand side of the above equation can be written as
t
πiz
−πiz
= 2t eet +1
where t = 2πiz, so
πiz eeπiz +e
−e−πiz
−1
∞
X
t
2
ζ(2k) 2k
1+ t
=1−2
t
2
e −1
(2πi)2k
k=1
Recall that
Definition 2.1. The Bernoulli numbers are defined by
∞
X
Bk k
t
=
t .
t
e − 1 k=0 k!
As a corollary,
ζ(2k) = −
(2πi)2k B2k
∈ π 2k Q×
2 (2k)!
for k ≥ 1.
Let us go back to the question of finding the Fourier expansion of the Eisenstein series
∞ X
X0
X
1
G2k (τ ) =
(cτ + d)−2k = 2ζ(2k) + 2
.
2k
(cτ
+
d)
c=1 d∈Z
(c,d)
The series expansion of π cot πz becomes 1
X
−1
(z + d) = π cot πz = πi 1 +
d∈Z
2
e2πiz − 1
= πi −1 − 2
∞
X
!
e2πinz
n=1
2πiz
which is valid for Im(z) > 0 (so that |e | < 1). Taking (2k − 1) times derivative with
respect to z, we have
∞
X
X
−2k
2k
(z + d)
= −(2πi)
n2k−1 e2πinz .
−(2k − 1)!
n=1
d∈Z
Taking z = cτ and suming over all positive integers c, we obtain
∞
(2πi)2k X
G2k (τ ) = 2ζ(2k) +
σ2k−1 (n)q n
(2k − 1)! n=1
P
where the divisor sum function is defined as σk (n) = 1≤m|n mk .
The normalized Eisenstein series of weight 2k is defined to be
∞
G2k (τ )
2k X
E2k (τ ) =
=1−
σ2k−1 (n)q n .
2ζ(2k)
B2k n=1
For example,
E4 = 1 + 240
∞
X
σ3 (n)q n ∈ Z[[q]];
n=1
1The
left hand side is written this way for simplicity. There will not be any convergence issues after
differentiation.
7
E6 = 1 − 504
∞
X
σ5 (n)q n ∈ Z[[q]].
n=1
In general we only have E2k (τ ) ∈ Q[[q]]. The fact that E2k has rational coefficients is
important in the second half of the course when we discuss p-adic modular forms.
X
χ(d)
We can generalize modular forms for SL2 (Z) to Γ(N ), for example
.
(cτ + d)k
c≡0 (mod N )
We will prove that, in a certain sense, the Eisenstein series generate all the modular forms.
If f is a weakly modular function of weight k for SL2 (Z) that is meromorphic everywhere
on H ∪ {∞}, we define its order of vanishing vτ (f ) at τ ∈ H ∪ {∞} as follows. At τ ∈ H, this
is just the routine definition for complex-valued functions. For τ = ∞, consider the Fourier
expansion
!
X
X
an q n
f (τ ) =
an q n = q n0 an0 +
n>n0
n−∞
where an0 6= 0. Then the order of vanishing is defined to be
v∞ (f ) = n0 .
We can check that vτ depends only on the SL2 (Z)-orbit of τ .
Lemma 2.2.
X
mτ vτ (f ) =
τ ∈D∪{∞}
where
mτ =


k
12
(1)
1
2
1
3
if τ = i
if τ = ρ

1 if τ = ∞ or τ ∈ D\{i, ρ}
Remark. The weights are present because i and ρ have non-trivial stabilizers. Later we will
give a more geometric proof using modular curves and Riemann-Roch.
Proof (Sketch). Use contour integration along the boundary of D truncated by a horizontal
segment which bounds all the zeroes and poles in D, detouring around ρ, i, ρ + 1 along
small circular arcs. We first assume there are no zeroes or poles on the boundary. By the
argument principle,
Z 0
X
1
f (τ )
dτ =
vτ (f ).
2πi
f (τ )
τ ∈D\{i,ρ}
1 π
Around τ = ρ, ρ + 1, the contribution from the two arcs is 2(− 2π
v (f )) = − 31 vρ (f ).
3 ρ
Similarly, around τ = i, the contribution is − 12 vi (f ).
k
Under τ 7→ − τ1 , f (− τ1 ) = τ k f (τ ). The arc on the unit circle has contribution 12
.
2πiτ
Under τ 7→ q = e , the horizontal path gives −v∞ (f ).
If there are zeroes or poles on the boundary, we detour along small circular arcs such that
each equivalence class of zeroes and poles is counted exactly once inside the contour.
This handy formula can help us find the dimension and basis of modular forms for SL2 (Z).
Theorem 2.3.
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(1) The graded algebra of all modular forms for SL2 (Z) is
∞
M
Mk = C[E4 , E6 ]
k=0
i.e. any modular form is a polynomial in E4 and E6 .
1
(E 3 − E62 ) is a cusp form of weight 12. ∆(τ ) 6= 0 for τ ∈ H and has a
(2) ∆ =
1728 4
simple zero at ∞.
Proof. If k ≤ 2, there is no integral solution to Equation (1), so there are no non-zero
modular forms of weight ≤ 2 for SL2 (Z). The same fact for Γ0 (2) is important to the proof
of Fermat’s Last Theorem!
If k = 4, the only solution is vρ (E4 ) = 1 and E4 doesn’t vanish at any other point. We
have M4 = CE4 .
If k = 6, the only solution is vi (E6 ) = 1. We have M6 = CE6 .
If k = 8, the only solution is vρ (f ) = 2. Since E42 and E8 have the same constant term 1,
we have E42 = E8 (which implies a relation between σ3 and σ7 ).
If k = 10, we have E4 E6 = E10 .
If k ≥ 12, we see that Mk = Sk ⊕ CEk where Sk is the space of cusp forms of weight k.
f
If f ∈ Sk , by definition v∞ (f ) ≥ 1. But v∞ (∆) = 1 and vτ (∆) = 0 for all τ ∈ H, so ∆
is holomorphic everywhere and the map Mk−12 → Sk given by f 7→ f ∆ is an isomorphism.
Thus
Mk = ∆Mk−12 ⊕ CEk ,
so it suffices to prove Ek is a polynomial of E4 and E6 , and the theorem will follow by
induction on k.
But the above splitting is not unique. We only required that Ek is not a cusp form. Since
the equation 4a + 6b = k always has a solution in non-negative integers if k ≥ 4 is even, we
rewrite
Mk = ∆Mk−12 ⊕ CE4a E6b
which finishes the proof of (1). (2) is trivial.
We set
E4 (τ )3
1
= + ··· .
∆(τ )
q
Then j is holomorphic for τ ∈ H with v∞ (j) = −1. Next time we will see that it defines an
isomorphism SL2 (Z)\H ∪ {∞} ∼
= P1 .
j(τ ) =
3. Lecture 3 (January 30, 2013)
3.1. Modular Curves. We will discuss modular curves, which are Riemann surfaces over
C. So far we have studied modular forms over SL2 (Z), but the definition works if we replace
it by congruence subgroups2, for example
Γ(N ) ⊂ Γ1 (N ) ⊂ Γ0 (N )
2Holomorphicity
at cusps requires some care. See Chapter 1.2 of Diamond-Shurman for details.
9
for any N ≥ 1, where
a
Γ(N ) =
c
a
Γ1 (N ) =
c
a
Γ0 (N ) =
c
b
a
∈ SL2 (Z) :
d
c
b
a
∈ SL2 (Z) :
d
c
b
a
∈ SL2 (Z) :
d
c
b
1
≡
d
0
b
1
≡
d
0
b
∗
≡
d
0
0
1
(mod N ) ;
∗
1
∗
∗
(mod N ) ;
(mod N ) .
Note that the principal congruence subgroup Γ(N ) is the kernel of the map
mod N
SL2 (Z) −→ SL2 (Z/N ).
More generally, if we define the Borel subgroup
∗ ∗
B0 (Z/N ) =
⊂ SL2 (Z/N )
0 ∗
and denote
1 ∗
B1 (Z/N ) =
⊂ SL2 (Z/N );
0 1
1 0
B(Z/N ) =
⊂ SL2 (Z/N );
0 1
then Γ∗ (N ) is the preimage of B∗ (Z/N ) ⊂ SL2 (Z/N ), where ∗ = 0, 1 or nothing.
These subgroups all have finite indices. To find [SL2 (Z) : Γ∗ (N )] for ∗ = 0, 1 or nothing,
note that it is equal to the indices
e ∗ (Z/N )]
[SL2 (Z/N ) : B∗ (Z/N )] = [GL2 (Z/N ) : B
e ∗ is the corresponding subgroup of GL2 (Z/N ). It is a standard exercise to
where the latter B
find the order of GL2 (Z/N ).
For N = p a prime, the order is (p2 − 1)(p2 − p), the number of bases of (Z/p)2 .
For N = pk a prime power, the order is ( Np )4 (p2 − 1)(p2 − p) = N 4 (1 − p−1 )(1 − p−2 ).
Q
In general, | GL2 (Z/N )| = N 4 p|N (1 − p−1 )(1 − p−2 ).
e 0 (Z/pk )| = (N (1 − p−1 ))2 · N = N 3 (1 −
On the other hand, for N = pk a prime power, |B
Q
−1 2
e 0 (Z/N )| = N 3
p−1 )2 , so in general |B
p|N (1 − p ) .
Therefore, we have
Y
e 0 (Z/N )] = N
[SL2 (Z) : Γ0 (N )] = [GL2 (Z/N ) : B
(1 + p−1 ).
p|N
Let Γ ⊂ SL2 (Z) be a congruence subgroup containing ±I. We equip YΓ = Γ\H with the
quotient topology given by
π : H → Γ\H,
i.e. U ⊂ YΓ is open if and only if π −1 (U ) ⊂ H is open. It follows that π is an open map.
Lemma 3.1. Under this topology, YΓ is Hausdorff.
10
Proof (Sketch). We show that the action of Γ on H is properly discontinuous, i.e. for any
τ1 , τ2 ∈ H (possibly τ1 = τ2 ), there exist neighborhoods U1 and U2 of τ1 and τ2 respectively
such that for γ ∈ Γ, γU1 ∩ U2 = ∅ if and only if γτ1 = τ2 . Thus π(U1 ) and π(U2 ) separate
any π(τ1 ) 6= π(τ2 ).
If this is true for Γ = SL2 (Z), then it is true for any subgroups. If τ1 and τ2 are in the
interior of the fundamental domain, then it is clearly true. If they are on the boundary, we
use two half-disks.
Definition 3.2. The stabilizer group of τ ∈ H is Γτ = {γ ∈ Γ : γτ = τ } ⊂ Γ.
Lemma 3.3. Γτ is finite cyclic.
Proof. Consider H = SL2 (R)/ SO(2) (SO(2) is a compact abelian group isomorphic to S 1 ).
Then Γτ = Γ ∩ γ −1 SO(2)γ is the intersection of a discrete group and a compact group, hence
is finite. But any finite subgroup of S 1 must be cyclic.
We will see later that Γτ /{±I} is isomorphic to one of {1}, Z/2, Z/3. Up to SL2 (Z)equivalence, the only points with non-trivial stabilizers are i and ρ.
Definition 3.4. τ ∈ H is an elliptic points if Γτ / ± I is nontrivial.
Lemma 3.5. There are only finitely many elliptic points modulo Γ.
Proof. This is true for SL2 (Z), hence true for any subgroup of finite index.
Recall that a Riemann surface is defined to be 1-dimensional complex manifold, and charts
are local coordinates for each point of YΓ which are compatible.
It is a completely routine process to check that YΓ is a Riemann surface. Every point of
YΓ is of the form Γτ for τ ∈ H. If τ is not elliptic, applying the proof of Lemma 3.1 to
τ1 = τ2 = τ shows that τ has a neighborhood U such that γU ∩ U 6= ∅ implies γτ = τ . Then
U gives a local coordinate at τ .
If τ is elliptic, locally its neighborhood is D/µ2 or D/µ3 with local coordinate given by
z 7→ z 2 or z 3 .
(Diagram here)
Example 3.6. For Γ = SL2 (Z), YΓ = SL2 (Z)\H. Recall that the j-function is defined as
E43
. Viewing j as a function on YΓ , the order of vanishing is given by
∆
1
vπ(ρ) (j) = vρ (j).
3
We will simply accept compatibility of these local charts...
YΓ is a non-compact Riemann surface. For Γ = SL2 (Z), j is holomorphic everywhere on
YΓ .
We want to compatify YΓ . Recall “one-point compactification” from topology: if X is a
e = X ∪ {∞} where the open sets of X
e are the open sets of X
topological space, consider X
e is compact.
together with {∞}∪ cocompact open sets of X. This implies X
We will do some sort of one-point compactification for H modulo SL2 (Z). Define
H∗ = H ∪ P1 (Q) = H ∪ Q ∪ {∞}.
11
Note SL2 (Z) acts on P1 (Q): for [x, y] ∈ P1 (Q),
a b
x
ax + by
=
c d
y
cx + dy
so P1 (Q) = SL2 (Z) · ∞.
The topology on H∗ is generated by open sets of H, sets of the form RM = {∞} ∪ {τ ∈
H : Im(τ ) > M } for all M > 0, and all possible SL2 (Z)-translates of RM . Then H∗ is
compact. In terms of the fundamental domain, D ∪ {∞} is compact (same proof as onepoint compactification). This implies
XΓ := Γ\H∗ = YΓ ∪ (Γ\P1 (Q)).
The points of Γ\P1 (Q) are called cusps.
Since π : H∗ → XΓ is surjective, XΓ is compact for Γ = SL2 (Z). For congruence subgroups
Γ, XΓ is a union of finitely many translations of D ∪ {∞}, hence compact as well.
consider the local coordinate at ∞. Since the
stabilizer
SL2 (Z)∞ is given by
Next
we
1 ∗
1 NZ
±
, in general we have Γ∞ = Γ ∩ SL2 (Z)∞ ⊃
for some positive
0 1
0 1
1 h∞
integer N . Let h∞ be the minimal positive integer such that
∈ Γ∞ , called the
0 1
width or period. Give local coordinate at ∞ by q = e2πiτ /h∞ . In general, for τ ∈ P1 (Q), hτ
is defined by moving τ to ∞ by SL2 (Z).
To summarize, we have shown that Γ\H = YΓ ⊂ XΓ = Γ\H∗ are Riemann surfaces, and
XΓ is compact.
We are interested in understanding the invariants of XΓ as a Riemann surface, e.g. its
genus.
Notation. For Γ = Γ∗ (N ) where ∗ = 0, 1 or nothing, we denote Y∗ (N ) = YΓ and X∗ (N ) = XΓ .
Theorem 3.7. The map j : X0 (1) → P1 sending SL2 (Z)τ 7→ j(τ ) and ∞ 7→ ∞ is an
isomorphism.
Proof. This is a degree 1 map between two compact Riemann surfaces.
We use this to study the genus of modular curves via coverings.
Suppose ±I ∈ Γ1 ⊂ Γ2 and write X1 = XΓ1 , X2 = XΓ1 . Then φ : X1 → X2 is a covering
map of degree deg(φ) = [Γ2 : Γ1 ], since for a non-elliptic point τ we have #(Γ2 \Γ1 τ ) = [Γ2 :
Γ1 ], and there are only finitely many elliptic points.
To calculate the genus, we use the Riemann-Hurwitz formula. It is enough to calculate
ramifications, which is where we need to use the local coordinates.
2g1 − 2 = (2g2 − 2) deg(φ) + deg Ram
P
P
where deg Ram = x∈X1 (ex − 1) = y∈X2 deg Ry , deg Ry = x∈φ−1 (y) (ex − 1), and ex is the
ramification degree at x ∈ X.
Note that the ramification points are a subset of the fibers over elliptic points of X2 (but
the preimage of an elliptic point of X2 may or may not be an elliptic point of X1 since Γ1
is smaller). Thus the ramification degrees must divide 2 or 3, the only possible orders of
elliptic points. If there were elliptic points of order 4 we would be dead!
To compute the genus of any modular curve, we take Γ2 = SL2 (Z) and let d = deg(φ).
P
12
For h = 2, 3, define eh to be the number of elliptic points in the fiber over the order h
elliptic point Ph of SL2 (Z)\H∗ , where P2 = i and P3 = ρ. Then the number of non-elliptic
h
h
points in the fiber over Ph is d−e
, so Rh = (h − 1) d−e
.
h
h
(Diagram here)
P
Define e∞ to be the number of cusps in X1 . Then R∞ = (ex − 1) = d − e∞ .
Plugging in everything,
X h−1
1
2
(d − eh ) = −2d + (d − e2 ) + (d − e3 ) + (d − e∞ )
2g1 − 2 = −2d +
h
2
3
h=2,3,∞
So we have proved
Theorem 3.8. g(XΓ ) = 1 +
d
1
1
1
− e2 − e3 − e∞ .
12 4
3
2
Next time we will compute this X0 (N ), somewhat the most useful case. In this case,
the elliptic points are interesting. We will apply the so-called “moduli interpretation” using
elliptic curves.
In general it is hard to calculate eh . It is easy if Γ is a normal subgroup, e.g. Γ(N ), because
every point in a fiber will either be simultaneously elliptic or non-elliptic. But Γ0 (N ) is not
normal.
4. Lecture 4 (February 4, 2013)
4.1. Modular Curves. Last time we constructed the modular curve XΓ , which is a compact
Riemann surface with genus given by
d
e2 (Γ) e3 (Γ) e∞ (Γ)
−
−
−
12
4
3
2
where d is the degree of the natural map XΓ → XSL2 (Z) , e2 , e3 are the number of elliptic
points of order 2, 3 respectively, and e∞ is the number of cuspidal points of XΓ .
If Γ is normal, e.g. Γ(N ), everything is easy to compute.
Today we will do the example of X0 (N ) = Γ0 (N )\H∗ .
g(XΓ ) = 1 +
Theorem 4.1. For Γ = Γ0 (N ), we have
(
0
if 4 | N,
e2 (Γ0 (N )) = Q −4
if 4 - N,
p|N 1 +
p
(
√
0
if (Q( −1), N ) does not satisfy the Heegner condition,
√
=
2#{odd p|N } if (Q( −1), N ) satisfies the Heegner condition,
(
0
e3 (Γ0 (N )) = Q
if 2 | N or 9 | N,
p|N
1+
(
0
=
2#{odd p|N }
−3
p
otherwise,
√
if (Q( −3), N ) does not satisfy the Heegner condition,
√
if (Q( −3), N ) satisfies the Heegner condition,
13
−4
2
where we define
= 0; otherwise p∗ is the Legendre symbol, and
X N
e∞ (Γ0 (N )) =
φ gcd d,
,
d
= 0 and
−3
3
d|N
where φ(n) =
P
c|n
1.
Definition 4.2. The “Heegner condition for (K, N )” means for every p | N , either p is split
in K or p is ramified with p || N .
Remark. This condition is important for the development of the Gross-Zagier formula.
To prove these formulas it is better to use the “moduli interpretation of Y0 (N )”, at least
as a set.
For N = 1, we can give meaning to SL2 (Z)\H as the “moduli” of elliptic curves over C, i.e.
the set of 1-dimensional complex tori C/Λ for some lattice Λ modulo isomorphism classes as
Riemann surfaces. Equivalently, this is the set of lattices Λ ⊂ C modulo homothety, namely
Λ1 ∼ Λ2 if and only if there exists λ ∈ C× such that λΛ1 = Λ2 .
Indeed, we can
a bijection by sending τ ∈ SL2 (Z)\H to Λτ = Z + Zτ . Replacing
establish
a b
τ by γτ where
∈ SL2 (Z), the lattice becomes
c d
Z+Z
aτ + b
1
1
=
(Z(cτ + d) + Z(aτ + d)) =
(Z + Zτ ),
cτ + d
cτ + d
cτ + d
i.e.
1
Λτ
cτ + d
and so the map is well-defined. Bijectivity can then be verified easily.
Λγτ =
Remark. A function f on H satisfying the weight k condition
f (γτ ) = (cτ + d)k f (τ )
can be interpreted as a function g on the set of lattices of homogeneous degree −k
g(λΛ) = λ−k g(Λ).
Given g, we can define f on H by f (τ ) = g(Λτ ). Then
1
Λτ ) = (cτ + d)k g(Λτ ) = (cτ + d)k f (τ )
f (γτ ) = g(Λγτ ) = g(
cτ + d
and vice versa.
We can further interpret this as a function h on the set of equivalence classes (E, ω) of
elliptic curves with a 1-form 0 6= ω ∈ H 0 (E, Ω1E/C ) such that
h(E, λω) = λk h(E, ω).
Given a function h, how do we get f ? To any given τ ∈ H we associate the elliptic curve
E = C/Λτ and the canonical 1-form dz, which descends to E because it is invariant under translation, and define f (τ ) = h(E, dz). Then C/ cτ1+d Λτ is isomorphic to C/Λτ via
multiplication by cτ + d, so
1
h(C/Λγτ , dz) = h(C/
Λτ , dz) = h(C/Λτ , (cτ + d)dz) = (cτ + d)k h(C/Λτ , dz),
cτ + d
14
i.e.
f (γτ ) = (cτ + d)k f (τ ).
Note that we have not put any holomorphic conditions on H or at the cusps. We are just
trying to interpret the weight k condition by considering the points of the modular curves
as a set.
4.2. Elliptic curves with level structures. By definition, an elliptic curves with Γ∗ (N )level structure (∗ = 0, 1 or nothing) is a pair (E, ι∗ ) where


if ∗ = 0,
cyclic subgroup of order N (of E[N ])
3
ι∗ = order N point (of E[N ])
if ∗ = 1,

a Z/N -basis of E[N ]: (α, β) ∈ E[N ]2 with Z/N α + Z/N β = E[N ] if ∗ = nothing,
where E[N ] is the subgroup of N -torsion. If E = C/Λ with group structure induced by the
complex numbers, then E[N ] = N1 Λ/Λ ∼
= (Z/N )2 as abelian groups.
The equivalence relation is given in the obvious way: two pairs (E1 , G1 ) and (E2 , G2 ) are
φ
isomorphic if there is an isomorphism E1 → E2 with φ(G1 ) = G2 .
∼
Lemma 4.3. There is a bijection between Y∗ (N ) and {(E, ι∗ )} modulo equivalence given by
Γ∗ (N )τ 7→ (C/Λτ , ι∗ )
where
(For E = C/Λτ , E[N ] =

1

 N + Λτ
ι∗ = N1 + Λτ

( 1 , τ )
N N
1
Λ /Λτ
N τ
= ( N1 Z +
if ∗ = 0,
if ∗ = 1,
if ∗ = nothing.
1
Zτ )/Λτ .)
N
Proof (Sketch). Let us check this
for the Γ0 (N )-level structure, which is the
is well-defined
a b
only case we will use. Let γ =
∈ Γ0 (N ). We have
c d
1
1
cτ + d
d
1
cτ +d
C/
Λτ ,
−→ C/Λτ ,
= C/Λτ ,
= C/Λτ ,
cτ + d
N
N
N
N
since N | c and gcd(d, N ) = 1. In fact the same calculation proves injectivity as well: if
τ and γτ go to the same pair, then γ must be in Γ0 (N ). Surjectivity can be proved by
changing basis.
Now we return to the interpretation of elliptic points of X0 (N ) or Y0 (N ). The correspondence above implies that the stabilizer group of any point in the orbit Γ0 (N )τ is isomorphic
to
Γ0 (N )τ ∼
= Aut(E, G)
where G is an order N cyclic subgroup. In the case N = 1, this corresponds to the fact that
Aut(E) can only be Z/2, Z/4, Z/6, with the last two cases occurring when E has complex
multiplication structure by the Gaussian integers Z[i] or the Eisenstein integers Z[e2πi/3 ]
respectively. In general we have
3A
Γ(N )-level structure is also called the full level structure. In fact the basis (α, β) is required to have
Weil pairing equal to a fixed root of unity. See Section 1.5 of Diamond-Shurman.
15
Theorem 4.4. There is a bijection between the set of elliptic points of X0 (N ) of order
h = 2 and the set of ideals N ⊂ Z[i] such that Z[i]/N ∼
= Z/N , and a bijection between the
set of elliptic points of X0 (N ) of order h = 3 and the set of ideals N ⊂ Z[e2πi/3 ] such that
Z[e2πi/3 ]/N ∼
= Z/N .
Thus the number of elliptic points is equal to the number of ideals with certain properties,
which are easy to count since Z[i] and Z[e2πi/3 ] are Dedekind domains (even PID!). Assuming
this, we can give a proof of Theorem 4.1.
Proof of Theorem 4.1 (Sketch). It suffices by the Chinese Remainder Theorem to consider
N = pa . For a quadratic field K, we want to find ideals N such that OK /N ∼
= Z/pa is cyclic.
There are three situations, depending on whether p is ramified, inert or split.
If p is ramified, then p = p2 and N = pb . If b ≥ 2, then p | N and OK /N cannot be cyclic,
since OK /(p) is (Z/p)2 . Thus N = p. Now p must exactly divide N because OK /p ∼
= Z/p.
2
∼
If p is inert, then pOK is a prime ideal with OK /(p) = (Z/p) , so OK /N cannot be cyclic.
If p is split, then pOK = pp. Then the only choice of N is either pa or pa , because if
N = pi pj then it is contained in (p)min(i,j) , but Ok /(p) is not cyclic.
This is how we get precisely the factor 1 + ( dp ), where d is the discriminant of K.
The cusps of P1 (Q) are counted by brute-force.
Now we explain how to get the bijectivity in Theorem 4.4. The key point is that the extra
automorphisms of the pair (E, G) force certain lattices to be fractional ideals.
Proof of Theorem 4.4 (Sketch). In the order 2 case, the lattice is Λ = Z[i] = OK for the
quadratic field K = Q(i). Our goal is to count the number of subgroups G ⊂ C/OK such
that G is cyclic of order N and the pair (C/OK , G) admits non-trivial automorphisms.
Since the automorphism group of the lattice OK is generated by multiplication by i (iOK =
OK = Z[i]), we want [i] : C/OK → C/OK to preserve G. Writing G = M/OK , where
M ⊂ K = Q ⊗ OK is contained in N1 OK , the condition is translated as iM = M. Hence
M is a Z[i]-module of K, i.e. a fractional ideal. Therefore, it suffices to count the number
of ideals N = M−1 ⊂ OK such that OK /N = M/OK = Z/N .
Corollary 4.5. If N = p is a prime, then
(
b p+1
c − 1 if p ≡ 1 (mod 12),
12
g(X0 (p)) =
p+1
b 12 c
otherwise.
Note
√ the congruence condition on p precisely depends on its decomposition in Q(i) and
Q( −3).
Corollary 4.6. g(X0 (N )) = 1 if and only if N ∈ {11, 14, 15, 17, 19, 20, 21, 24, 27, 32, 36, 49}.
These are important because they are both modular curves and elliptic curves.
5. Lecture 5 (February 6, 2013)
5.1. Heegner points. Last time we used the moduli interpretation on the modular curve.
Let us talk a bit more about the Heegner condition defined last time.
The points of Y0 (N ) = Γ0 (N )\H can be interpreted as elliptic curves with level structure,
i.e. pairs (E, G) where E/C is an elliptic curve and G ⊂ E[N ] is a cyclic subgroup of order
16
N . If E = C/Λ, then the set of N -torsion points is E[N ] = N1 Λ/Λ. Write G = Λ0 /Λ where
Λ0 ⊂ N1 Λ.
An equivalent but more symmetric way is to consider pairs of elliptic curve (E1 , E2 ) with
an (surjective) isogeny φ : E1 → E2 with ker(φ) ∼
= Z/N . Under the notations above, this is
C/Λ → C/Λ0 .
Define EndC (E) to be {φ : E → E : φ(0) = 0}. Then EndC (C/Λ) = {λ ∈ C : λΛ ⊂ Λ}
(so that λ induces the multiplication √
map [λ] : C/Λ
√ → C/Λ).
Last time we considered K = Q( −1) or Q( −3). In general if K is any imaginary
quadratic field with ring of integers OK , then EndC (C/OK ) = {λ ∈ C : λOK ⊂ OK } = OK ,
since (λ)OK ⊂ OK means λ ∈ OK by inverting ideals.
The modular curve X0 (N ) contains pairs of the form (C/OK , C/Λ0 ) where OK ⊂ Λ0 ⊂
1
O and Λ0 /OK ∼
= Z/N .
N K
Definition 5.1. A Heegner point attached to OK on X0 (N ) is such a pair (C/OK , C/Λ0 )
where both curves have endomorphism ring OK .
This condition can be translated into
Lemma 5.2. EndC (C/Λ0 ) = OK if and only if Λ0 is an ideal of K, i.e. a fractional ideal of
OK .
Proof. If Λ0 is an ideal, then λΛ0 ⊂ Λ0 ⇔ λ ∈ OK by inverting ideals. The converse is easy,
since {λ ∈ C : λΛ0 ⊂ Λ0 } ⊃ OK means Λ0 is an ideal.
We are reduced to the question of finding ideals Λ0 such that Λ0 /OK ∼
= Z/N . Last time
we showed
Lemma 5.3. A fractional ideal Λ0 such that Λ0 /OK ∼
= Z/N exists if and only if p | N implies
p ramifies with p || N or p is split. In particular, if p has an inert factor, then Λ0 does not
exist.
This is called the Heegner condition. We summarize our work last time as follows. Let Eh
be the set of elliptic points on X0 (N ) of order h. We proved
Theorem 5.4. E2 (resp. E3 ) corresponds to the set of Heegner points attached to Z[i] (resp.
Z[e2πi/3 ]).
Remark. X0 (N ) has a canonical model over Q. X0 (N ) → E/Q is always parametrized by
a modular form of weight 2 (using the modularity theorem as a black box!). Mapping the
Heegner points into E shows that they are defined over the Hilbert class field of K, and
taking trace pushes them down to Q. This is the only systematic way of producing rational
points on elliptic curves over Q and provides evidence for the BSD conjecture.
5.2. Dimension formulas. We derive the dimension formulas for modular forms and cusp
forms, which is almost trivial after the genus formula, at least for even weights. The idea is
to interpret the space of modular forms as the space of global sections of line bundles over
Riemann surfaces. For simplicity, assume k is even.
Fix a congruence subgroup Γ ⊂ SL2 (Z), and X = XΓ which is a compact Riemann surface.
We will use local charts to compute orders of vanishing. Let ΩX be the canonical line bundle
(holomorphic differential 1-forms on X), Mk = Mk (Γ) (resp. Sk = Sk (Γ)) be the space of
modular forms (resp. cusp forms) of weight k.
17
Lemma 5.5 (Key Lemma).
⊗k/2
(1) The map Mk → H 0 (X, ΩX
(k )) given by
f 7→ ωf = f (τ )(2πidτ )k/2 ,
where k := b k4 c2 + b k3 c3 + k2 ∞ ∈ Div(X), is an isomorphism. More precisely, this
is the space of meromorphic forms ω of degree k2 with poles div(ω) ≥ −k .
∼
⊗k/2
(2) The same map gives an isomorphism Sk → H 0 (X, ΩX (k − ∞ )).
Proof. For any γ ∈ Γ ⊂ SL2 (Z), define the automorphy factor j(γ, τ ) = cτ + d. Then
d(γτ ) = j(γ, τ )−2 dτ . Since f (γτ ) = j(γ, τ )k f (τ ), ωf is invariant under Γ and indeed descends
to a meromorphic form of degree k2 on X.
Let us express the condition of f being holomorphic in terms of ωf . Choose local coordinate
z at elliptic points so that τ = z 1/h . Then
1
k
f (τ )(dτ )k/2 = f (z 1/h )(dz 1/h )k/2 = z ( h −1) 2 f (z 1/h )(dz)k/2 .
k
c. (Note that invariance
Hence f (τ ) is holomorphic on H if and only if vz (ωf ) ≥ −b h−1
h 2
under stabilizers shows that f (τ ) has only h-th powers of τ .)
∼
At the
cusps, let us only consider ∞. Let h be the width at ∞, i.e. Γ∞ /{±I} =
1 h
. The local coordinate is given by q = e2πiτ /h , so dq/q = 2πi/hdτ . If f has
0 1
Fourier expansion f (τ ) = g(q) = a0 + a1 q + · · · , then
ωf = f (τ )(2πidτ )k/2 = g(q)(h
dq k/2
) = Cq −k/2 g(q)(dq)k/2 .
q
Therefore, f holomorphic (resp. zero) at ∞ if and only if vq=0 (ωf ) ≥ − k2 (resp. − k2 + 1). Corollary 5.6. S2 ∼
= H 0 (X, ΩX ), so dim S2 = gX .
With this identification, we can apply Riemann-Roch. For any line bundle L on X,
χ(L) = h0 (L) − h0 (Ω ⊗ L−1 ) = deg(L) + (1 − gX )
where h0 (L) = dim H 0 (X, L).
For k ≥ 2, we have
k
k
k
k
deg Ωk/2 (k ) = (2g − 2) + b ce2 + b ce3 + e∞ ≥ (2g − 2) + e∞ > 2g − 2.
2
4
3
2
For k ≥ 4 (to study cusp forms), we have
k
k
k
k
(2g − 2) + b ce2 + b ce3 + ( − 1)e∞ ≥ 2g − 2 + e∞ > 2g − 2.
2
4
3
2
k/2
k/2
Applying to L = Ω (k ) and Ω (k − ∞ ) respectively, we conclude
deg Ωk/2 (k − ∞ ) =
k
k
k
Corollary 5.7. For k ≥ 2, dim Mk = (k − 1)(g − 1) + b ce2 + b ce3 + e∞ .
4
3
2
k
k
k
For k ≥ 4, dim Sk = (k − 1)(g − 1) + b ce2 + b ce3 + ( − 1)e∞ .
4
3
2
For k odd, the same idea works but the calculations are more complicated. It is most
difficult to apply Riemann-Roch to weight 1 forms.
18
6. Lecture 6 (February 11, 2013)
6.1. Hecke operators. Today we will discuss Hecke operators. For congruence subgroups Γ
(so Γ(N ) ⊂ Γ ⊂ SL2 (Z) for some N ≥ 1), we have defined Mk and Sk , the spaces of modular
forms and cusp forms of weight k respectively. We want to study the endomorphisms of
these vector spaces.
Denote GL2 (Q)+ = {γ ∈ GL2 (Q) : det γ > 0}. It defines an action on H and has the
following property: if γ ∈ GL2 (Q)+ and Γ is a congruence subgroup, then γΓγ −1 ∩ SL2 (Z)
is still a congruence subgroup (possibly for a bigger N ), and in particular has finite index.
Consider the pairs (G, H) where H is a subgroup of G satisfying the following hypothesis:
for every γ ∈ G, γ −1 Hγ and H are “commensurable”, i.e. γ −1 Hγ ∩ H is of finite index in
H and γ −1 Hγ.
Example 6.1. (GL2 (Q)+ , Γ), where Γ is a congruence subgroup.
Example 6.2. (GL2 (Qp ), GL2 (Zp )), where p is a prime number.
Let C[H\G/H] = C[G//H] be the space of functions φ : G → C which are bi-H-invariant
(i.e. φ(hgh0 ) = φ(g) for all h, h0 ∈ H and g ∈ G) and supported on finitely many double
cosets. The last condition is the analogue for φ to have compact support when G is equipped
with a topology, e.g. Example 6.2.
We can define an algebra structure on C[G//H] by
X
(φ ∗ ϕ)(g) =
φ(gx−1 )ϕ(x)
x∈H\G
Z
which can be interpreted as the integral
φ(gx−1 )ϕ(x)dx where H\G is given the counting
G
measure 4. Then (C[G//H], ∗) is an algebra with identity 1H , the characteristic function of
H. In general this may not be commutative. In the case where C[G//H] is commutative,
there is essentially only one idea to prove so, which is the Gelfand trick.
Theorem 6.3. For (G, H) = (GL2 (Qp ), GL2 (Zp )), C[G//H] is commutative.
Proof. The idea is to find an anti-involution σ : G → G, i.e. (xg)σ = g σ xσ , such that every
double coset has a representative fixed by σ.
In the case of GL2 (Qp ), there is a natural choice – take σ to be transposition on GL2 (Qp ).
α
a
p
0
Lemma 6.4. GL2 (Qp ) =
GL2 (Zp )
GL2 (Zp ).
0 pβ
α≥β∈Z
This immediately implies the second requirement that every double coset has a representative fixed by σ.
Note σ defines an action on C[G//H] as well by φσ (g) = φ(g σ ), and is an anti-involution
(φ ∗ ϕ)σ = ϕσ ∗ φσ .
Since each double coset has an invariant element, σ acts trivially on C[G//H]. Combining
these, we conclude that φ ∗ ϕ = ϕ ∗ φ, so C[G//H] is commutative.
4The
hypothesis on (G, H) implies every double coset HgH is a finite union of cosets in H\G. See Lemma
5.1.2 of Diamond-Shurman.
19
Remark. (G, H) is called a Gelfand pair, i.e. for all irreducible representations π of G, the
space HomH (π, C) is at most 1-dimensional.
How do we deal with Example 6.1 where (G, H) = (GL2 (Q)+ , SL2 (Z))? We will only
consider the subalgebra generated by functions supported in M2 (Z) ∩ GL2 (Q)+ . This set is
certainly bi-H-invariant.
Theorem 6.5. This subalgebra is commutative.
Lemma 6.6. For n ≥ 1, M [n] = M2 (Z)[n] = {γ ∈ M2 (Z) : det γ = n} is equal to the
disjoint unions
a α 0 a n/β x
Γ
Γ=
Γ
0 β
0 β
β|α>0
αβ=n
β|n
0≤x<β
where Γ = SL2 (Z). In particular, we have
#Γ\M [n] =
X
β = σ1 (n).
β|n
(In order to find a double coset representative for any matrix in M [n], note that β will be
the gcd of the four entries.)
Proof (Sketch). The same proof works by taking σ to be transposition again.
Consider the characteristic function 1M [n] for n ≥ 1.
Theorem 6.7.
(1) 1M [n] is multiplicative, i.e. 1M [n] ∗ 1M [m] = 1M [nm] if gcd(n, m) = 1.
(2) For p prime, 1M [p] ∗ 1M [pn ] = 1M [pn+1 ] + p1pM [pn−1 ] as equality in C[G//H].
Proof.
(1) We can write
M [nm] = M [n] · M [m]
0 −1
0
γm has deter, then γn0−1 γn = γm
in an essentially unique way: if γ = γn γm = γn0 γm
1
1
minant 1 and is contained in n M2 (Z) ∩ m M2 (Z) = M2 (Z), hence is in Γ = SL2 (Z).
0
Thus γm
∈ Γγm differ by an element in Γ.
(2) We can easily check 1M [p] ∗ 1M [pn ] ≥ 1M [pn+1 ] + p1pM [pn−1 ] , so it is enough to prove
they have the same integral, i.e.
vol(M [pn ]) vol(M [p]) = vol(M [pn+1 ]) + p vol(M [pn−1 ])
⇔
⇔
σ1 (pn )σ1 (p) = σ1 (pn+1 ) + pσ1 (pn−1 )
pn+2 − 1
pn − 1
pn+1 − 1 p2 − 1
·
=
+p·
.
p−1
p−1
p−1
p−1
Today we will study Mk and Sk for the full modular group Γ = SL2 (Z). For γ ∈ GL2 (Q)+ ,
define the weight-k γ-operator
f [γ]k (τ ) = (f |γ, k)(τ ) = (det γ)k−1 j(γ, τ )−k f (γτ )
a b
where for γ =
, j(γ, τ ) = cτ + d is the automorphy factor. Using this notation, the
c d
weight-k condition for γ ∈ SL2 (Z) means f [γ]k = f . We can verify that f [αβ]k = (f [α]k )[β]k .
20
The underlying fact is that the automorphy factor satisfies the cocycle condition j(αβ, τ ) =
j(α, βτ )j(β, τ ).
For φ ∈ C[GL2 (Q)+ //Γ], define
X
f [φ]k =
f [γ]k φ(γ)
Γ\ GL2 (Q)+
Z
which is a finite sum. Again this can be interpreted as a formal integral
f [γ]k φ(γ)dγ,
GL2 (Q)+
by putting the counting measure on Γ\ GL2 (Q)+ .
Lemma 6.8. [φ]k defines operators Mk → Mk and Sk → Sk .
Proof (Sketch). If f [γ]k = f for all γ ∈ Γ, then f [φ]k satisfies the same property. We can
also check holomphicity at the cusp.
Definition 6.9. The Hecke operator Tn is defined to be the operator [φ]k for φ = 1M [n] .
Thus Tn is contained in End(Mk ) and End(Sk ). Note that φ = 1pM [pn−1 ] corresponds to
p Tpn−1 because
k−2
f [pγ]k = det(pγ)k−1 j(pγ, τ )−k f (γτ ) = p2(k−1) p−k f [γ]k
since γ and pγ act on H in the same way. With this, Theorem 6.7 translates into
Theorem 6.10.
(1) Tn Tm = Tnm for gcd(n, m) = 1.
(2) For p prime, Tp Tpn = Tpn+1 + pk−1 Tpn−1 .
P
n
Lemma 6.11 (Effect on q-expansion). If f = ∞
n=0 an (f )q ∈ Mk , then Tm f ∈ Mk with
n-th Fourier coefficient given by
X
an (Tm f ) =
dk−1 anm/d2 (f ).
d|(n,m)
In particular, we have a1 (Tm f ) = am (f ).
Example 6.12. Let ∆ = q
∞
Y
n 24
(1 − q )
n=1
=
∞
X
τ (n)q n = q + · · · , where τ is known as the
n=1
Ramanujan tau-function. Since S12 ∼
= C∆ is one-dimensional, ∆ must be an eigenform for
Tm with eigenvalue τ (m), i.e. Tm ∆ = τ (m)∆ . The eigenvalues must satisfy the properties
in Theorem 6.10, so
(
τ (mn) = τ (m)τ (n) if (m, n) = 1,
τ (pn+1 ) = τ (p)τ (pn ) − p11 τ (pn−1 ).
This is Ramanujan’s conjecture, proved by Mordell.
7. Lecture 7 (February 13, 2013)
7.1. Hecke operators for level 1. Last time we considered the double coset space Γ\ GL2 (Q)+ /Γ,
where Γ = SL2 (Z).
Theorem 7.1. Z[Γ\ GL2 (Q)+ /Γ] is a commutative algebra.
21
We defined the Hecke operator
X
Tn (f ) =
f [β]k
β∈Γ\M [n]
(which is a finite sum) for f ∈ Mk and n ≥ 1, where
f [β]k (τ ) = (det β)k−1 j(β, τ )−k f (β(τ ))
P∞
n
Lemma P
7.2 (Effect on q-expansion). Let f =
∈ Mk (SL2 (Z)), and denote
n=0 an q
∞
n
Tm (f ) = n=0 an (Tm f )q . Then
X
an (Tm f ) =
dk−1 anm/d2 .
d|(m,n)
In particular, a1 (Tm f ) = am and a0 (Tm f ) = σk−1 (m)a0 .
From this, we can show
an (Tm Tm0 f ) = an (Tm0 Tm f )
which verifies that the subalgebra of End(Mk (SL2 (Z)) generated by Tn for all n ≥ 1 is
commutative. Note this is an immediate consequence of Theorem 7.1.
Proof of Lemma 7.2. Recall M [m] = {γ ∈ M2 (Z) : det(γ) = m} can be decomposed as
a
a b
Γ
.
0 d
ad=m
0≤b≤d−1
Hence
Tm (f )(τ ) = mk−1
X
j(β, τ )−k f (β(τ ))
β
a b
. Since β(τ ) =
where β runs through the coset representatives
0 d
aτ +b
e2πin d and j(β, τ ) = d, we have
Tm (f )(τ ) = mk−1
∞
X X
d−k
ad=m n=0
k−1
=m
X X
d−1
X
aτ
aτ +b
,
d
e2πinβ(τ ) =
b
an e2πin( d + d )
b=0
−k
aτ
d dan e2πin( d )
ad=m d|n
∞ X X
m k−1
0
=
adn0 e2πin aτ
d
ad=m n0 =0


∞
X X

=
qn 
ak−1 adn0  .
n=0
n0 a=n
ad=m
Example 7.3 (Eisenstein series). Recall that the Eisenstein series of weight k is given by
∞
X
(2πi)k X
−k
Gk (τ ) =
w = 2ζ(k) +
σk−1 (n)q n
(k − 1)! n=1
w∈Λτ −{0}
22
where Λτ = Z + Zτ . It turns out to be an eigenform for Tm , i.e.
Lemma 7.4. Tm (Gk ) = σk−1 (m)Gk .
Proof. It is clear that if Gk is indeed an eigenform, then the eigenvalue must be σk−1 (m) by
comparing the constant terms. In fact we can directly check the formula
nm X
dk−1 σk−1
= σk−1 (m)σk−1 (n).
d2
d|(n,m)
Remark. Recall that functions on H satisfying the weight-k condition can be interpreted
as functions on lattices of homogeneous of degree −k. With this, we can give a simpler
definition of Hecke operator
X
f (Λ0 ).
(Tn f )(Λ) = nk−1
Λ0 ⊂Λ
[Λ:Λ0 ]=n
For a fixed lattice Λ = Λτ = Z + Zτ ⊂ C, there is a bijection
{Λ0 ⊂ Λ of index n} ←→ SL2 (Z)\M [n]
a b
where we associate to each β =
∈ M [n] the lattice βΛ = span(a + bτ, c + dτ ), which
c d
is indeed a sublattice of Λ of index n.
7.2. Hecke operators for level N . So far we have been working over SL2 (Z). We can
extend Hecke operators to Γ∗ (N ) where ∗ = 0, 1 or nothing. It is enough to study Γ1 (N ),
because Γ1 (N ) ⊂ Γ0 (N ) and
−1
N 0
N 0
= Γ1 (N 2 )
Γ(N )
0 1
0 1
a b
a b/N
(
is sent to
). Recall Γ1 (N ) is normal in Γ0 (N ): there is an exact
c d
cN
d
sequence
1 → Γ1 (N ) → Γ0 (N ) → (Z/N )× → 1
a b
where the last map is given by
7→ d (mod N ).
c d
Let χ : (Z/N )× → C× be a fixed Dirichlet character, called the Nebentypus. Define
Mk (Γ1 (N ), χ) = {f ∈ Mk (Γ1 (N )) : f [β]k = χ(β)f for all β ∈ Γ0 (N )}
where χ is lifted to Γ0 (N ) → C× . Then Mk (Γ0 (N )) can be identified as Mk (Γ1 (N ), χ0 ) for
the trivial character χ0 . It is almost trivial to see that
M
Mk (Γ1 (N )) =
Mk (Γ1 (N ), χ).
χ:(Z/N )× →C×
Remark. Here are three interpretations of Hecke operators for Γ0 (N ).
(1) For (n, N ) = 1, we can view f ∈ Mk (Γ0 (N )) as a function
{(Λ1 , Λ2 ) : Λ1 ,→ Λ2 with Λ2 /Λ1 ∼
= Z/N } → C.
23
Then
X
(Tn f )(Λ1 , Λ2 ) = nk−1
f ((Λ02 ∩ Λ1 ), Λ02 ).
Λ02 ⊂Λ2
[Λ2 :Λ02 ]=n
(2) Equivalently, in the language of elliptic curves, viewing f : {(E, G, ω)}/ ∼→ C gives
X
f (E/H, (G + H)/H, ωH )
Tn (f )(E, G, ω) = nk−1
H⊂E subgroup
of order n
(because (n, N ) = 1) where ωH is the pullback of ω under the dual isogeny of E →
E/H.
a b
∗ ∗
(3) In terms of double cosets, let MN =
≡
(mod N ) and MN [n] =
c d
0 ∗
{γ ∈ MN : det γ = n} for (n, N ) = 1. Then we have the coset decomposition
a
a b
Γ0 (N )
MN [n] =
.
0 d
ad=n
0≤b≤d−1
We define Hecke operators for Γ = Γ∗ (N ) where ∗ = 0, 1 as follows. For p prime, define
1 0
Tp = Γ
Γ
0 p
and
Tpi = Tp Tpi−1 − pk−1 hpiTpi−2
(the diamond operator hpi will be defined below). Note when p - N , the second equation
reduces to Tpi = (Tp )i . We then extend the definition to Tn by multiplicativity.
We can
the algebra
generated by these is commutative, by using the involution
prove
−1
N 0
N 0
Xt
σ(X) =
on Γ∗ (N ).
0 1
0 1
Lemma 7.5. Let f ∈ Mk (Γ1 (N ), χ) with Nebentypus χ : (Z/N )× → C× . Then
X
an (Tm f ) =
χ(d)dk−1 anm/d2
d|(n,m)
(as usual, we extend χ to Z → C by 0 at integers not coprime to N ).
Now we define the diamond operator, which acts on the space Mk (Γ1 (N )). Since Γ1 (N ) is
a normal subgroup of Γ0 (N ), any coset satisfies Γ1 (N )γ = γΓ1 (N ) for γ ∈ Γ0 (N ) and hence
Γ1 (N )γΓ1 (N ) = Γ1 (N )γ = γΓ1 (N ). This defines a double coset operator which depends
only on the image of γ under
Γ0 (N ) → (Z/N )× → 1
a b
sending γ =
to d (mod N ). This double coset operator is denoted by hdi and called
c d
the diamond operator.
Theorem 7.6. The operators Tn and hdi, where n ≥ 1 and d ∈ (Z/N )× , are commutative
on Mk (Γ1 (N )).
24
7.3. Petersson inner product. If f, g ∈ Mk (Γ∗ (N )), then f (τ )g(τ ) is of weight (k, k) in
the following sense.
Definition 7.7. φ : H → C has weight (m, n) if
n
φ(γτ ) = j(γ, τ )m j(γ, τ ) φ(τ ).
Example 7.8. Im(τ ) = y =
1
(τ
2i
− τ ) is of weight (−1, −1) since
Im(γτ ) =
Im(τ )
j(γ, τ )j(γ, τ )
.
Define the Petersson inner product: for f, g ∈ Sk (Γ∗ (N )), set
Z
dτ dτ
hf, giΓ =
f (τ )g(τ )y k 2
y
Γ\H
Claim. If one of f or g is a cusp form, then limy→∞ f (τ )g(τ )y k = 0.
Claim. hGk , Gk i diverges.
Note that the Eisenstein series can be given by
X
X0
(cτ + d)−k =
j(γ, τ )−k
Gk =
γ∈Γ∞ \Γ
c,d
1 ∗
. Then for any g ∈ Mk (Γ),
0 1
Z
Z
X
dτ dτ
k dτ dτ
Gk (τ )g(τ )y
j(γ, τ )−k j(γ, τ )−k g(γτ ) Im(τ )k 2
=
2
y
y
Γ\H
Γ\H γ∈Γ \Γ
∞
Z
X
dτ dτ
g(γτ ) Im(γτ )k 2
=
y
Γ\H γ∈Γ \Γ
∞
Z
dτ dτ
g(τ ) Im(τ )k 2
=
y
Γ \H
Z ∞
dτ dτ
=
g(τ )y k 2
y
Γ∞ \H
where Γ = SL2 (Z) and Γ∞ =
is divergent unless g(τ ) → 0 as y → ∞, i.e. unless g is a cusp form. This implies the second
claim.
Next time we will study the inner product space (Sk (Γ∗ (N )), h·, ·i), and show that Hecke
operators are normal, i.e. they commute with their adjoints X ∗ X = XX ∗ . This implies X is
diagonalizable. Therefore we can simultaneously diagonalize the family of Hecke operators
and decompose the space as direct sum of Hecke eigenforms.
8. Lecture 8 (February 18, 2013)
dxdy
on H which is
y2
invariant under SL2 (R). If f and g are modular forms of weight k for a congruence subgroup
8.1. Petersson inner product. Consider the hyperbolic measure
25
Γ, then the product f (τ )g(τ ) Im(τ )k is Γ-invariant. Define
Z
dxdy
f (τ )g(τ )y k 2 .
hf, giΓ =
y
Γ\H
Lemma 8.1. If one of f or g is a cusp form, then the integral converges absolutely.
Proof. Let us check this for Γ = SL2 (Z) and D the fundamental domain (in general the
fundamental domain will be finitely many copies of D). Write f (τ ) = a0 + a1 q + · · · and
g(τ ) = b0 + b1 q + · · · . Since a0 b0 = 0, there exists some constant C > 0 such that
|f (τ )g(τ )y k | < Ce−2πy y k .
Hence the integral converges on y 0. The remaining region is compact.
Remark.
If f and g are both not cusp forms, then the integral diverges whenever k ≥ 1 since
Z
dxdy
y k 2 does.
y
y0
Therefore, hf, giΓ is a well-defined inner product on Sk (Γ), the space of cusp forms, so it
makes sense to talk about adjoint operators. For Γ = Γ∗ (N ) where ∗ = 0, 1, consider the
Hecke operators Tn for (n, N ) = 1.
(Morally speaking, they already determine Tp for primes p dividing N . Also we don’t lose
too much just by considering Γ0 (N ), since Γ1 (N ) is normal in Γ0 (N ) with abelian quotient.
Last time we introduced the diamond operators hdi for (d, N ) = 1. In fact we don’t lose too
much just by considering SL2 (Z). The same proof techniques work. Anyway we’ll assume
Γ = Γ∗ (N ) where ∗ = 0, 1.)
Let Tn∗ be the adjoint operator of Tn on Sk (Γ), characterized by
hTn f, giΓ = hf, Tn∗ giΓ .
Theorem 8.2. Let Tα be the operator associated to ΓαΓ, where α ∈ M2 (Z) with det(α) 6= 0.
Then
Tα∗ = Tα∗
where α∗ = det(α)α−1 ∈ M2 (Z).
Corollary 8.3. For Γ = Γ∗ (N ) and (n, N ) = 1, we have
Tn∗ = hni−1 Tn .
In particular, Tn is self-adjoint if ∗ = 0 (because the diamond operators are trivial on Γ0 (N )).
Thus Tn is “almost” self-adjoint. In fact Tn is normal.
Assuming the theorem, let us deduce the corollary.
Proof of Corollary 8.3. For n = p prime with (p,
= 1, Tp corresponds
N) to the double coset
1
0
p
0
ΓαΓ and Tp∗ corresponds to Γα∗ Γ, where α =
and α∗ =
. We claim that
0 p
0 1
p 0
1 0
∈ Γ1 (N )
Γ0 (N ).
0 1
0 p
0 −1
(Note: In SL2 (Z) we could have simply conjugated by
.)
1 0
26
If β ∈ Γ1 (N ) and γ ∈ Γ0 (N ) are such that α∗ = β −1 αγ, then αγ(α∗ )−1 = β. Writing
a b
γ=
∈ Γ0 (N ), we want
c d
−1 1 0
a b
p
0
a/p b
=
0 p
c d
0 1
c pd
to be contained in Γ1 (N ). We can pick a = p, and solve for N | c with pd = bc + 1.
Therefore, Tp∗ corresponds to the double coset
1 0
1 0
1 0
Γ
γΓ = Γ
Γγ = Γ
ΓγΓ
0 p
0 p
0 p
where γ ∈ Γ0 (N ), i.e.
Tp∗ = Tp hγi = Tp hdi = Tp hp−1 i.
We will not prove the theorem, which amounts to changing basis. The complexity of the
theory is just about finding double cosets.
Corollary 8.4. The operators Tn for (n, N ) = 1 are a commuting family of normal operators,
so there exists an orthogonal eigenbasis of Sk (Γ).
Definition 8.5. f ∈ Sk (Γ) is an eigenform if it is an eigenvector for each Hecke operator
Tn and hdi, where (n, N ) = 1 and (d, N ) = 1. (For Γ0 (N ), we just need to consider Tn since
hdi is trivial.)
Remark. For Γ = SL2 (Z), Mk (SL2 (Z)) = CEk ⊕ Sk (SL2 (Z)). Thus Ek is an eigenform with
eigenvalue σk−1 (n) for Tn . The same conclusion in Corollary 8.4 holds for Mk (SL2 (Z)), with
“orthogonal” suitably interpreted (since hEk , Ek iΓ diverges).
8.2. L-functions. Next we will
an application to L-functions. Let f ∈ Mk (Γ)
Pdiscussion
∞
n
have Fourier expansion f (q) = n=1 an q . Define
L(s, f ) =
∞
X
an n−s .
n=1
To address convergence issues, note that we have the following trivial bounds on Fourier
coefficients.
Lemma 8.6. If f is a cusp form, then |an | ≤ Cnk/2 for some constant C. In general, we
have |an | ≤ Cnk .
Proof. For simplicity, assume Γ = Γ∗ (N ) where ∗ = 0, 1, so the local parameter is given by
q = e2πiτ (for general congruence subgroups we need to take into account the width of Γ at
∞). Then
Z
Z 1
1
−n dq
an =
f (q)q
=
f (x + iy)e−2πin(x+iy) dx
2πi
q
0
is independent of y > 0 by holomorphicity. Choosing y = n1 , we have
Z 1 i
2π
an = e
f x+
e−2πinx dx
n
0
27
Recall that |f (τ )|2 | Im(τ )|k is invariant under Γ. If f is cuspidal, |f (τ )|2 | Im(τ )|k ≤ C, so
|f (x + ni )| ≤ C( n1 )−k/2 = Cnk/2 and so
|an | ≤ Cnk/2 .
For the Eisenstein series Ek for SL2 (Z), we have
X
an = σk−1 (n) =
dk−1 ≤ Cnk .
d|n
Since any modular form for SL2 (Z) can be written as a sum of an Eisenstein series and a
cusp form, we obtain the trivial bound.
We omit the proof for general congruence subgroups.
(
1 + k2 if f is cuspidal and ∗ = 0, 1,
Corollary 8.7. L(s, f ) converges for Re(s) >
1 + k if f ∈ Mk (Γ).
We can consider the Euler product for L(s, f ), just like for the Riemann zeta-function.
By definition, we say f is normalized if a1 = 1.
Theorem 8.8. For f ∈ Sk (SL2 (Z)), the following are equivalent:
(1) f is a normalized eigenform;
(2) L(s, f ) has an Euler product
Y
L(s, f ) =
(1 − ap p−s + pk−1−2s )−1 .
p prime
The property of having an Euler product is obviously not stable under addition. Being an
eigenform means having a simple spectrum. Euler product thus reflects the spectrum.
Proof. (1) ⇒ (2). Suppose f is an eigenform with a1 = 1. For Tm where m ∈ Z≥1 , recall
that
X
an (Tm (f )) =
dk−1 anm/d2 .
d|(m,n)
This implies that the eigenvalue of Tm is am . Since Tm Tn = Tmn for (m, n) = 1 and
Tpi+1 = Tp Tpi − pk−1 Tpi−1 , we have the same relations
(
am an = amn
if (m, n) = 1,
(2)
k−1
api+1 = ap api − p api−1 if i ≥ 1.
Hence,
L(s, f ) =
Y
∞
X
p prime
k=1
!
apk p−ks .
pi
The sequence bi = a has recursive relation bi+1 = ap bi − pk−1 bi−1 , hence characteristic
polynomial T 2 − ap T + pk−1 . Thus
∞
X
apk p−ks = (1 − ap p−s + pk−1−2s )−1 .
k=1
28
(2) ⇒ (1). The same proof goes backwards. The Euler product implies that the Fourier
coefficients an satisfy the relations (2). It suffices to verify Tp f = ap f for primes p (since Tp
generate the Hecke algebra), which is equivalent to
an (Tp f ) = an ap
for all n, i.e.
X
dk−1 anp/d2 = an ap .
d|(n,p)
(
anp
But the left hand side is just
anp + pk−1 an/p
if (n, p) = 1,
which is equal to an ap by (2). if p | n,
9. Lecture 9 (February 20, 2013)
9.1. Hecke operators. We have studied the Hecke theory for the level 1 case Γ = SL2 (Z)
using double cosets. For all n ≥ 1, we defined the Hecke operator Tn . Fix k ≥ 1 and consider
the space of cusp forms Sk (Γ). Then Tn ∈ End(Sk (Γ)).
Consider the subalgebra T = C[Tn ]n≥1 ⊂ End(Sk (Γ)), which is a commutative C-algebra.
Since each operator Tn commutes with its adjoint, i.e. is normal, T is semisimple and hence
isomorphic to C × · · · × C.
We can extend this to level N . Let Γ = Γ∗ (N ), where ∗ = 0, 1. We start by defining Tp
for all primes (p, N ) = 1. Tp corresponds to the double coset
a
1 0
1 j
a b
p 0
Γ
Γ=
Γ
tΓ
0 p
0 p
N p
0 1
0≤j≤p−1
a b
∈ Γ0 (N ).
N p
We can define hdi for (d, N ) = 1.
We can extend the definition to Tpi by using the recursive relation
where
Tpi := Tp Tpi−1 − pk−1 hpiTpi−2
and to Tn for (n, N ) = 1 by
Tm Tn = Tmn
whenever (m, n) = 1.
It remains to define Tp for p dividing N . This operator is often called Up , but still Tp in
Diamond-Shurman. In this case, we have the double coset decomposition
a
1 0
1 j
Γ
Γ=
Γ
.
0 p
0 p
0≤j≤p−1
We use the same recursive relations as above to define Tn , and define hdi = 0 if (d, N ) > 1.
Remark. All the above relations can be combined into
∞
X
Y
Y
Tn n−s =
(1 − Tp p−s + hpipk−1−2s )−1 ·
(1 − Tp p−s )−1 .
n=1
p-N
p|N
29
Therefore, we have defined Tn and hni for all n ≥ 1, which are endomorphisms of Sk (Γ).
The Hecke algebra T generated by them is still a commutative C-algebra, but not necessarily
semisimple.
Lemma 9.1. Let (n, N ) = 1. Then the adjoint of Tn with respect to the Petersson inner
product is
Tn∗ = hni−1 Tn .
In particular, Tn is a normal operator if (n, N ) = 1.
This5 suggests we should look at the subalgebra T0 ⊂ T given by
T0 = C[Tn , hni](n,N )=1 ,
which is semisimple by the lemma.
What happens to Tp = Up when p | N ? We
can
still find a formula for its adjoint. Recall
1 0
that Tp is associated to the double coset Γ
Γ, so Tp∗ is associated to the double coset
0 p
p 0
Γ
Γ. These two are in general not the same double coset. Let us introduce one more
0 1
operator.
For Γ = Γ∗ (N ), define the normalizer
N (Γ) = NGL2 (R)+ (Γ) = {γ ∈ GL2 (R)+ : γ −1 Γγ = Γ}.
The quotient N (Γ)/Γ is interesting because it induces automorphisms on the modular curve,
i.e. we have a map
N (Γ)/Γ → Aut(Γ\H∗ ).
It is non-trivial to find a normalizer, but here is one. Let
0 −1
1 0
0 −1
wN =
=
N 0
0 N
1 0
which is contained in N (Γ), so that the double coset ΓwN Γ is just a coset ΓwN . Then we
can check that f 7→ f [wN ]k maps into the same space of cusp forms, so we can consider
wN = [wN ]k ∈ End(Sk (Γ)).
2
= −N , we see that wN is an involution on the modular curve Γ\H∗ , but it is
Since wN
not quite an involution on Sk (Γ).
Lemma 9.2.
2
(1) wN
acts by (−1)k N k−2 on Sk (Γ).
(2) For all n, we have
(
−1
Tn∗ = wN Tn wN
,
−1
∗
hni = wN hniwN
.
Proof.
2
2
k k−2
(1) wN
= [wN
]k = (N 2 )k−1 (−N )−k = (−1)
N . 1 0
p 0
and
differ by conjugation by wN .
(2) The key point is that the matrices
0 p
0 1
5We
can also check that hni∗ = hni−1 for (n, N ) = 1, so hni is indeed normal.
30
Corollary 9.3. For Γ = Γ0 (N ) and (n, N ) = 1, we have
−1
Tn∗ = Tn = wN Tn wN
.
Thus wN commutes with Tn for (n, N ) = 1.
9.2. Atkin-Lehner theory. To understand the relationship between T0 and T, we need the
Atkin-Lehner theory (also known as the newform theory), and later we will give an example
where T is not semisimple. The idea is to study the newforms of Sk (Γ1 (N )), which are the
“primitive” forms in some sense.
When M | N , we
always
have the inclusion Sk (Γ1 (M )) ,→ Sk (Γ1 (N )). Consider the span
d 0
N
. In fact it is enough
of f (dτ ) = d1−k f
where f ∈ Sk (Γ1 (M )), M | N and d | M
0 1 k
to consider forms of level dividing N/p where p | N is a prime:
Skold (Γ1 (N )) = span{f (dτ ) : f ∈ Sk (Γ1 (N/p)), d | p, p | N } ⊂ Sk (Γ1 (N ))
and define the newspace
Sknew (Γ1 (N )) = Skold (Γ1 (N ))⊥ .
Note that we are not calling this the “space of newforms” yet. Newform will have a specific
meaning – it is an eigenform.
Atkin and Lehner proved that the newspace behaves like the level 1 case, where we just
need to look at T0 . Before we make this more precise, let us prove a few lemmas.
Lemma 9.4. The spaces Skold and Sknew are T-stable, where T is the full Hecke algebra.
Proof (Sketch). Introduce T∗ , the algebra generated by the adjoints Tn∗ and hni∗ . It suffices
−1
to prove that Skold is both T- and T∗ -stable. But T∗ = wN TwN
, so it is enough to prove Skold
is T-stable and wN -stable.
Define the operator Vp : Sk (Γ1 (N/p)) →P
Sk (Γ1 (N )) by f 7→ f (pτ ). This is a one-sided
n
inverse of the Up operator. Indeed, for f = ∞
n=0 an q ,
X
X
X
τ +j
1 j
−1
=p
f
Up (f ) =
f
=
an q n/p
0 p k
p
0≤j≤p−1
0≤j≤p−1
p|n
whereas
Vp (f ) =
∞
X
an q pn .
n=0
It is clear that Up Vp = Id.
It remains to show that the sum of the images of Sk (Γ1 (N/p)) ,→ Sk (Γ1 (N )) and Vp :
Sk (Γ1 (N/p)) → Sk (Γ1 (N )) is stable under the operators Tq , hqi and w for all primes q.
For example, consider the diagram (note the two Tp ’s are different!)
Sk (Γ1 (N/p))
Vp
/
Tp
Sk (Γ1 (N ))
Sk (Γ1 (N/p))
Vp
/
Tp
Sk (Γ1 (N ))
which is not quite commutative. Since p | N , the Tp on the right is equal to Up and so
Tp Vp (f ) = f for all f ∈ Sk (Γ1 (N/p)). This shows Tp preserves Skold .
31
The other operators for other primes can be checked similarly6.
Example 9.5. For p - N , Up acts on Sk (Γ0 (p3 N )). Let f ∈ Sk (Γ0 (N )) be an eigenform
of Tp . Consider the space spanned by fi = f (pi τ ) for 0 ≤ i ≤ 3. We claim that Up acts
non-semisimply on it.
Since Vp fi = fi+1 for 0 ≤ i ≤ 2, we have Up fi = fi−1 for 1 ≤ i ≤ 3. Using the formula
X
Tp (f ) =
an q n/p + pk−1 hpif (pτ )
p|N
and the fact that f = f0 is an eigenform, we see that there exists λ ∈ C such that7
λf0 = Tp f0 = Up f0 + pk−1 f1 ,
i.e.

Up f0 = λf0 − pk−1 f1 .

λ
1
−pk−1 0 1 
, which is not semisimple8.
Thus Up has matrix 

0 1
0
Theorem 9.6 (Atkin-Lehner). If f ∈ Sknew (Γ1 (N )) is an eigenform for T0 , then f is also
an eigenform for the full Hecke algebra T.
We are not going to prove this, since the proof is just some combinatorics and not inspirational. There is a more conceptual proof using representation theory.
Definition 9.7. f ∈ Sk (Γ1 (N )) is a newform if it is contained in Sknew (Γ1 (N )), an eigenform
for T and normalized such that a1 (f ) = 1.
Newforms have a rigid meaning. In particular, newforms do not form a vector space, just
a set!
Corollary 9.8 (Multiplicity one). If f is a newform with eigenvalue λ : T → C, then
dim{g ∈ Sk (Γ1 (N )) : T g = λ(T )g for all T ∈ T} = 1.
10. Lecture 10 (February 25, 2013)
10.1. Atkin-Lehner theory. We were discussing the Aktin-Lehner theory about newforms,
and trying to explain the failure of semisimplicity of Hecke algebras, as well as to study the
structure of the Hecke module Sk (Γ) for Γ = Γ∗ (N ) where ∗ = 0, 1. There is no need to
consider the principal congruence subgroup Γ(N ) because it is conjugate to Γ1 (N 2 ).
Consider T = C[Tn , hdi : n ≥ 1, (d, N ) = 1] ⊂ EndC (Sk (Γ)) (recall that the operators
depend on the level N and weight k, although the notation doesn’t suggest so!). Literature
also considers EndC (Mk (Γ)), but we restrict ourselves to the simpler case of cusp forms.
Consider the subalgebra T0 = C[Tn , hdi : (n, N ) = 1, (d, N ) = 1] ⊂ T. Last time we saw
that T0 acts semisimply, but T does not.
6Refer
to Proposition 5.6.2 of Diamond-Shurman for details.
(d, N ) = 1, the diamond operator hdi acts trivially on Sk (Γ0 (N )), so hpif0 (pτ ) = f0 (pτ ) = f1 . Also,
note that Up ∈ End(Sk (Γ0 (p3 N )) but Tp ∈ End(Sk (Γ0 (N )).
8This matrix has characteristic polynomial X 2 (X 2 − λX + pk−1 ), but its kernel is only 1-dimensional.
7For
32
an q n ,
X
Up f =
anp q n
Write Tp = Up if p | N . Then for f =
P
n≥1
n≥1
and
Vp f = p
1−k
X
p 0
f
=
an q np
0 1 k
n≥1
so that Up is a left inverse to Vp .
The guiding example is as follows. For p - N , let f 6= 0 be an eigenform for Tp of level N ,
with Tp f = λf . Consider the span of ei = (Vp )i f where i = 0, 1, · · · , d, which are linearly
independent. Then Up , as a Hecke operator of level N pd , is given by
(
Up ei = ei−1
for 1 ≤ i ≤ d,
k−1
Up e0 = λe0 − p e1 ,
so its matrix is of the form

λ
1
−pk−1 0 1 
.
Up,d = 

0 1
0
If d ≥ 3, then this matrix is not diagonalizable, i.e. the action of Up is not semisimple. This
follows from an easy exercise in linear algebra.

Exercise. The minimal polynomial of Up,d is X d−1 (X 2 − λX + pk−1 ).
The Atkin-Lehner theory takes into account these operators Up .
Lemma 10.1. The map T × Sk (Γ) → C defined by (T, f ) 7→ a1 (T f ) is a perfect pairing.
Proof. Recall that a1 (Tm f ) = am (f ).
If (T, f ) = 0 for all T , then in particular (Tn , f ) = 0, so an (f ) = a1 (Tn f ) = 0 for all n and
f = 0.
If (T0 , f ) = 0 for all f ∈ Sk (Γ), we want to show T0 = 0, i.e. T0 f = 0. Indeed, for all n,
an (T0 f ) = a1 (Tn T0 f ) = a1 (T0 Tn f ) = (T0 , Tn f ) = 0.
If f ∈ Sk (Γ) is an eigenform of T, then T f = λf (T )f satisfies λf (Tn Tm ) = λf (Tn )λf (Tm )
and so defines a character λf : T → C. Define Sk (Γ)[λf ] to be the λf -eigenspace, i.e.
Sk (Γ)[λf ] = {g ∈ Sk (Γ) : T g = λf (T )g for all T ∈ T}.
Lemma 10.2 (Multiplicity one). If f ∈ Sk (Γ) is an eigenform of T, then dim Sk (Γ)[λf ] = 1.
Proof. For g ∈ Sk (Γ)[λf ], we have an (g) = a1 (Tn g) = λf (Tn )a1 (g) for all n ≥ 1.
Theorem 10.3 (Atkin-Lehner).
(1) If f ∈ Sknew (Γ) is an eigenform of T0 , then it is an eigenform of T.
(2) Let f be a newform of level Nf dividing N , and Sf be the span of the linearly independent elements Vd f = f (dτ ) where d | N/Nf . Then
Sf = {g ∈ Sk (Γ∗ (N )) : T g = λf (T )g for all T ∈ T0 }.
In particular, Sf is stable under T.
33
(3) There is a decomposition
Sk (Γ∗ (N )) =
M
M
Sf
M |N f newform
of level M
as Hecke modules.
Remark.
By Lemma 10.2, if M = N , then dim Sf = 1. In general, Sf has dimension
P
1.
d|N/Nf
Remark (Strong multiplicity one9). The association
{f newform of level dividing N } → {non-zero algebra homomorphism λ : T0 → C}
is injective (in fact bijective).
Corollary 10.4.
(1) If f ∈ Sk (Γ∗ (N )) with Fourier coefficients an (f ) = 0 for all (n, N ) = 1, then
X
f∈
Vp (Sk (Γ∗ (N/p))).
p|N
(2) If f, g ∈ Sknew (Γ) with an (f ) = an (g) for all (n, N ) = 1, then f = g.
We will not prove any of these theorems. In fact, Corollary 10.4 is known as the Main
Lemma and proved first in Diamond-Shurman. There are better proofs, e.g. in Casselman.
Example 10.5. Recall that
dim S2 (Γ0 (N )) = g(X0 (N )).
For example, using the genus formula, we get dim S2 (Γ0 (11)) = 1 and dim S2 (Γ0 (22)) = 2.
For any non-zero f ∈ S2 (Γ0 (11)) (necessarily an eigenform on level 11), we have
S2 (Γ0 (22)) = Cf (τ ) ⊕ Cf (2τ ).
λ2 1
With respect to this basis, U2 has matrix
, where T2 f = λ2 f in S2 (Γ0 (11)). We
−2 0
can check that10 λ2 = −2, so U2 is semisimple. Thus, T0N =22 ∼
= C and T = T0 [U2 ] = C × C.
10.2. Rationality and Integrality. We move the story over C to Q. Define
TQ = Q[Tn , hdi : n ≥ 1, (d, N ) = 1] ⊂ EndC (Sk (Γ))
and similarly for TZ . We do not know if they are even finitely generated.
Example 10.6. Let V be a 1-dimensional vector space over C, and set A = 2, B = π. Then
Q[A, B] ⊂ EndC (V ) ∼
= C is not a finitely generated Q-algebra.
We want to give a rational or integral structure to the space of modular forms.
Theorem 10.7. For R = Q, Z, there exists Sk,R (Γ) ⊂ Sk,C (Γ) := Sk (Γ) which is TR -stable,
and such that Sk,C (Γ) = Sk,R (Γ) ⊗R C.
9This
statement is stronger than “multiplicity one” above, but not as strong as the one in automorphic
representations. Perhaps we should only call this “stronger multiplicity one”.
10Or look up Cremona’s table!
34
In fact we can describe Sk,R (Γ) explicitly in terms of Fourier coefficients. For any subring
R ⊂ C, define
Sk,R (Γ) = {f ∈ Sk,C (Γ) : an (f ) ∈ R for all n}
and similarly
Mk,R (Γ) = {f ∈ Mk,C (Γ) : an (f ) ∈ R for all n}.
Today let us only consider the level 1 case Γ = SL2 (Z). Recall the graded algebra
M
Mk,C = C[E4 , E6 ]
k∈Z
where Ek = 1 −
immediately get
k
Bk
P∞
n=1
σk−1 (n)q n for k ≥ 4 even. Since Ek has rational coefficients, we
Corollary 10.8. Mk,Q ⊗ C ∼
= Mk,C .
The basis element E4a E6b is contained in Mk,Q , and this space is stable under TQ since
an (Tm f ) =
X
dk−1 anm/d2 (f ).
d|(n,m)
Thus TQ is a subalgebra of EndQ (Mk,Q ), hence finitely generated over Q, i.e. a finitedimensional Q-vector space.
Similarly, we have Sk,Q ⊗ C ∼
= Sk,C , and TQ × Sk,Q → Q is a perfect pairing.
10.3. Plan. The plan before spring break is to study the rational and integral structures of
spaces of modular forms, and prove a result of Shimura on the algebraicity of special values
of L-functions. We will start discussing p-adic modular forms after the break.
11. Lecture 11 (February 27, 2013)
12. Lecture 12 (March 4, 2013)
13. Lecture 13 (March 6, 2013)
14. Lecture 14 (March 11, 2013)
I was away for the Arizona Winter School.
15. Lecture 15 (March 13, 2013)
I was away for the Arizona Winter School.
35
16. Lecture 16 (March 25, 2013)
17. Lecture 17 (March 27, 2013)
18. Lecture 18 (April 1, 2013)
19. Lecture 19 (April 3, 2013)
20. Lecture 20 (April 8, 2013)
21. Lecture 21 (April 10, 2013)
22. Lecture 22 (April 15, 2013)
23. Lecture 23 (April 17, 2013)
24. Lecture 24 (April 22, 2013)
25. Lecture 25 (April 24, 2013)
26. Lecture 26 (April 29, 2013)
27. Lecture 27 (May 1, 2013)
36