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 atColumbia University in Spring 2013 (MATH G4657: Algebraic Number Theory). Hopefullythese notes will appear in a more complete form during Fall 2014. I recommend that youvisit my website from time to time for the most updated version.

Due to my own lack of understanding of the materials, I have inevitably introducedboth mathematical and typographical errors in these notes. Please send corrections andcomments to phlee@math.columbia.edu.

Contents

Classical Modular Forms1. Lecture 1 (January 23, 2013) 31.1. Introduction 31.2. Basic Definitions 31.3. Eisenstein series 52. Lecture 2 (January 28, 2013) 62.1. Eisenstein series 63. Lecture 3 (January 30, 2013) 93.1. Modular Curves 94. Lecture 4 (February 4, 2013) 134.1. Modular Curves 134.2. Elliptic curves with level structures 155. Lecture 5 (February 6, 2013) 165.1. Heegner points 165.2. Dimension formulas 176. Lecture 6 (February 11, 2013) 196.1. Hecke operators 197. Lecture 7 (February 13, 2013) 217.1. Hecke operators for level 1 217.2. Hecke operators for level N 237.3. Petersson inner product 258. Lecture 8 (February 18, 2013) 258.1. Petersson inner product 258.2. L-functions 279. Lecture 9 (February 20, 2013) 29

Last updated: September 6, 2014.1

9.1. Hecke operators 299.2. Atkin-Lehner theory 3110. Lecture 10 (February 25, 2013) 3210.1. Atkin-Lehner theory 3210.2. Rationality and Integrality 3410.3. Plan 35

Warning: The following lectures have not been edited.11. Lecture 11 (February 27, 2013) 3512. Lecture 12 (March 4, 2013) 3513. Lecture 13 (March 6, 2013) 3514. Lecture 14 (March 11, 2013) 3515. Lecture 15 (March 13, 2013) 35

p-adic Modular Forms16. Lecture 16 (March 25, 2013) 3617. Lecture 17 (March 27, 2013) 3618. Lecture 18 (April 1, 2013) 3619. Lecture 19 (April 3, 2013) 3620. Lecture 20 (April 8, 2013) 3621. Lecture 21 (April 10, 2013) 3622. Lecture 22 (April 15, 2013) 3623. Lecture 23 (April 17, 2013) 3624. Lecture 24 (April 22, 2013) 3625. Lecture 25 (April 24, 2013) 3626. Lecture 26 (April 29, 2013) 3627. Lecture 27 (May 1, 2013) 36

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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 willfocus on modular forms. The two main topics are:

(1) classical (holomorphic) modular forms for the full modular group SL2(Z) and itscongruence subgroups. We will be interested in both the analytic and arithmetictheory.

(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 beassigned from this book (among other resources).

(2) Serre, A Course in Arithmetic. The last part of this book has a concise explanationof 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 z for any complex number not necessarily contained in H.

Any γ ∈ SL2(R) =

(a bc d

): a, b, c, d ∈ R, det = 1

acts on H via

γ · τ =aτ + b

cτ + d.

Check that this defines a group action. The key point is

Im(γ · τ) =Im(τ)

|cτ + d|2

using the fact that det(γ) = 1. Moreover this action defines a biholomorphic automorphismof 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) forsome positive integer N , where Γ(N) is the principal congruence subgroup of level N definedby

Γ(N) =

(a bc d

)∈ SL2(Z) :

(a bc d

)≡(

1 00 1

)(mod N)

.

We can define various other subgroups that have finite index in SL2(Z), for example, bychanging the condition into

Γ0(N) =

(a bc d

)∈ SL2(Z) :

(a bc d

)≡(∗ ∗0 ∗

)(mod N)

.

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

)= (cτ + d)kf(τ) for all γ =

(a bc d

)∈

SL2(Z).

The group SL2(Z) is generated by two elements, which are the matrices corresponding totranslation τ 7→ τ + 1 and inversion τ 7→ − 1

τ.

Lemma 1.4. SL2(Z) =

⟨(1 10 1

),

(0 −11 0

)⟩.

With this we can show

Lemma 1.5. A fundamental domain of H modulo SL2(Z) is

D =

τ ∈ H : −1

2≤ Re(τ) <

1

2, |τ | > 1

∪τ ∈ H : −1

2≤ Re(τ) ≤ 0, |τ | = 1

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

γ =

(1 10 1

)shows that f satisfies f(τ+1) = f(τ), i.e. f has period 1. We can then consider

its Fourier expansion: for any τ = x+ iy, viewing f as a function of x gives

f(x+ iy) =∑n∈Z

an(y)e2πinx

where the Fourier coefficients are given by

an(y) =

∫ 1

0

f(x+ iy)e−2πinxdx = e−2πny

∫ 1+iy

0+iy

f(τ)e−2πinτdτ

So far we have not used the holomorphicity of f . By contour integration over the rectangle

with vertices iy, 1+iy, iy′, 1+iy′, we have that∫ 1+iy

0+iyf(τ)e−2πinτdτ is independent of y because

the integral over the two vertical edges cancel. Hence we can write

f(τ) =∑n∈Z

ane2πinτ .

It is convenient to introduce q = e2πiτ . Then the Fourier expansion of f is given by

f(τ) =∑n∈Z

anqn.

Geometrically, the map τ 7→ q = e2πiτ sends the upper half plane H into D − 0 whereD = 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 anyFourier expansion.

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Definition 1.6. f is meromorphic at τ =∞ if there exists M ∈ Z such that an = 0 for alln < 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) functionon 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 powersof q);

(2) f(γ · τ) = (cτ + d)kf(τ) for all γ =

(a bc d

)∈ SL2(Z).

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 freeabelian of rank 2 since τ has positive imaginary part. Define

Gk(τ) =∑

w∈Λτ−0

w−k =∑′

(m,n)∈Z2

(mτ + n)−k

where∑′ means we are summing over (m,n) 6= (0, 0).

Gk satisfies the following transformation property: if we replace τ 7→ γ · τ = aτ+bcτ+d

, then

Gk(γ · τ) = (cτ + d)kGk(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 anycompact 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

|w|2 = |mτ + n|2 = m2ττ + 2mnRe(τ) + n2τ∈D≥ m2 −mn+ n2 = |mρ+ n|2

(where ρ is a cube root of unity). Therefore we can bound

|Gk(τ)| ≤∑′

m,n

|w|−k =∑′

m,n

|mρ+ n|−k

which is convergent when k ≥ 3. This proves convergence and uniform convergence overcompact subsets.

By absolute convergence, we can evaluate limits term-wise.

Lemma 1.9. limτ→∞

Gk(τ) =∑′

n∈Z

n−k = 2ζ(k) where ζ is the Riemann zeta function.

Corollary 1.10. For k ≥ 4 even, Gk is a modular form of weight k, called the Eisensteinseries 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

z+∞∑d=1

(1

z + d+

1

z − d

)= π cotπz = π

cos πz

sinπz

which has simple poles at the integers and is holomorphic elsewhere. Be careful not to writethe sum as

∑d∈Z

1z+d

because this is not convergent!For a proof of this identity, Diamond-Shurman gives the hint

sin πz = πz∏n≥1

(1− z2

n2

)but this is like cheating because the two identities are equivalent – the identity we want isjust 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 functionhaving 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

f(z) = f(0) +∞∑i=1

bi

(1

z − ai+

1

ai

)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 theexample of Eisenstein series

G2k(τ) =∑′

ω∈Λτ

ω−2k =∑′

(c,d)∈Z2

(cτ + d)−2k

where Λτ = Z + Zτ . For k ≥ 2, this is absolutely convergent and uniformly convergent oncompact sets. Thus G2k ∈ M2k, the space of modular forms of weight 2k. Also recall thatG2k(∞) = 2ζ(2k).

We want to find the Fourier expansion. We make use of the following identity

π cotπz =1

z+∞∑d=1

(1

z − d+

1

z + d

)=

1

z+∞∑d=1

2z

z2 − 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

πzcos πz

sin πz= 1 + 2

∞∑d=1

z2

z2 − d2= 1− 2

∞∑k=1

ζ(2k)z2k

where we expanded at z = 0 using the geometric seriesz2

z2 − d2= −z

2

d2

1

1− z2

d2

= −∞∑k=1

(z2

d2)k.

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Incidentally we found a way to evalute the Riemann zeta function! This relates to theBernoulli numbers as follows. The left hand side of the above equation can be written asπiz e

πiz+e−πiz

eπiz−e−πiz = t2et+1et−1

where t = 2πiz, so

t

2

(1 +

2

et − 1

)= 1− 2

∞∑k=1

ζ(2k)

(2πi)2kt2k

Recall that

Definition 2.1. The Bernoulli numbers are defined by

t

et − 1=∞∑k=0

Bk

k!tk.

As a corollary,

ζ(2k) = −(2πi)2k

2

B2k

(2k)!∈ π2kQ×

for k ≥ 1.Let us go back to the question of finding the Fourier expansion of the Eisenstein series

G2k(τ) =∑′

(c,d)

(cτ + d)−2k = 2ζ(2k) + 2∞∑c=1

∑d∈Z

1

(cτ + d)2k.

The series expansion of π cotπz becomes 1

∑d∈Z

(z + d)−1 = π cot πz = πi

(1 +

2

e2πiz − 1

)= πi

(−1− 2

∞∑n=1

e2πinz

)which is valid for Im(z) > 0 (so that |e2πiz| < 1). Taking (2k − 1) times derivative withrespect to z, we have

−(2k − 1)!∑d∈Z

(z + d)−2k = −(2πi)2k

∞∑n=1

n2k−1e2πinz.

Taking z = cτ and suming over all positive integers c, we obtain

G2k(τ) = 2ζ(2k) +(2πi)2k

(2k − 1)!

∞∑n=1

σ2k−1(n)qn

where the divisor sum function is defined as σk(n) =∑

1≤m|nmk.

The normalized Eisenstein series of weight 2k is defined to be

E2k(τ) =G2k(τ)

2ζ(2k)= 1− 2k

B2k

∞∑n=1

σ2k−1(n)qn.

For example,

E4 = 1 + 240∞∑n=1

σ3(n)qn ∈ Z[[q]];

1The left hand side is written this way for simplicity. There will not be any convergence issues afterdifferentiation.

7

E6 = 1− 504∞∑n=1

σ5(n)qn ∈ Z[[q]].

In general we only have E2k(τ) ∈ Q[[q]]. The fact that E2k has rational coefficients isimportant in the second half of the course when we discuss p-adic modular forms.

We can generalize modular forms for SL2(Z) to Γ(N), for example∑

c≡0 (mod N)

χ(d)

(cτ + d)k.

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, thisis just the routine definition for complex-valued functions. For τ =∞, consider the Fourierexpansion

f(τ) =∑

n−∞

anqn = qn0

(an0 +

∑n>n0

anqn

)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. ∑τ∈D∪∞

mτvτ (f) =k

12(1)

where

mτ =

12

if τ = i13

if τ = ρ1 if τ =∞ or τ ∈ D\i, ρ

Remark. The weights are present because i and ρ have non-trivial stabilizers. Later we willgive a more geometric proof using modular curves and Riemann-Roch.

Proof (Sketch). Use contour integration along the boundary of D truncated by a horizontalsegment which bounds all the zeroes and poles in D, detouring around ρ, i, ρ + 1 alongsmall circular arcs. We first assume there are no zeroes or poles on the boundary. By theargument principle,

1

2πi

∫f ′(τ)

f(τ)dτ =

∑τ∈D\i,ρ

vτ (f).

Around τ = ρ, ρ+ 1, the contribution from the two arcs is 2(− 12π

π3vρ(f)) = −1

3vρ(f).

Similarly, around τ = i, the contribution is −12vi(f).

Under τ 7→ − 1τ, f(− 1

τ) = τ kf(τ). The arc on the unit circle has contribution k

12.

Under τ 7→ q = e2πiτ , 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 2.3.8

(1) The graded algebra of all modular forms for SL2(Z) is

∞⊕k=0

Mk = C[E4, E6]

i.e. any modular form is a polynomial in E4 and E6.

(2) ∆ =1

1728(E3

4 − E26) is a cusp form of weight 12. ∆(τ) 6= 0 for τ ∈ H and has a

simple zero at ∞.

Proof. If k ≤ 2, there is no integral solution to Equation (1), so there are no non-zeromodular forms of weight ≤ 2 for SL2(Z). The same fact for Γ0(2) is important to the proofof Fermat’s Last Theorem!

If k = 4, the only solution is vρ(E4) = 1 and E4 doesn’t vanish at any other point. Wehave 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 E2

4 and E8 have the same constant term 1,we have E2

4 = E8 (which implies a relation between σ3 and σ7).If k = 10, we have E4E6 = E10.If k ≥ 12, we see that Mk = Sk ⊕ CEk where Sk is the space of cusp forms of weight k.

If f ∈ Sk, by definition v∞(f) ≥ 1. But v∞(∆) = 1 and vτ (∆) = 0 for all τ ∈ H, so f∆

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 byinduction on k.

But the above splitting is not unique. We only required that Ek is not a cusp form. Sincethe equation 4a+ 6b = k always has a solution in non-negative integers if k ≥ 4 is even, werewrite

Mk = ∆Mk−12 ⊕ CEa4E

b6

which finishes the proof of (1). (2) is trivial.

We set

j(τ) =E4(τ)3

∆(τ)=

1

q+ · · · .

Then j is holomorphic for τ ∈ H with v∞(j) = −1. Next time we will see that it defines anisomorphism SL2(Z)\H ∪ ∞ ∼= P1.

3. Lecture 3 (January 30, 2013)

3.1. Modular Curves. We will discuss modular curves, which are Riemann surfaces overC. So far we have studied modular forms over SL2(Z), but the definition works if we replaceit 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

Γ(N) =

(a bc d

)∈ SL2(Z) :

(a bc d

)≡(

1 00 1

)(mod N)

;

Γ1(N) =

(a bc d

)∈ SL2(Z) :

(a bc d

)≡(

1 ∗0 1

)(mod N)

;

Γ0(N) =

(a bc d

)∈ SL2(Z) :

(a bc d

)≡(∗ ∗0 ∗

)(mod N)

.

Note that the principal congruence subgroup Γ(N) is the kernel of the map

SL2(Z)mod N−→ SL2(Z/N).

More generally, if we define the Borel subgroup

B0(Z/N) =

(∗ ∗0 ∗

)⊂ SL2(Z/N)

and denote

B1(Z/N) =

(1 ∗0 1

)⊂ SL2(Z/N);

B(Z/N) =

(1 00 1

)⊂ SL2(Z/N);

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

[SL2(Z/N) : B∗(Z/N)] = [GL2(Z/N) : B∗(Z/N)]

where the latter B∗ is the corresponding subgroup of GL2(Z/N). It is a standard exercise tofind 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 (N

p)4(p2 − 1)(p2 − p) = N4(1− p−1)(1− p−2).

In general, |GL2(Z/N)| = N4∏

p|N(1− p−1)(1− p−2).

On the other hand, for N = pk a prime power, |B0(Z/pk)| = (N(1− p−1))2 ·N = N3(1−p−1)2, so in general |B0(Z/N)| = N3

∏p|N(1− p−1)2.

Therefore, we have

[SL2(Z) : Γ0(N)] = [GL2(Z/N) : B0(Z/N)] = N∏p|N

(1 + p−1).

Let Γ ⊂ SL2(Z) be a congruence subgroup containing ±I. We equip YΓ = Γ\H with thequotient 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 anyτ1, τ2 ∈ H (possibly τ1 = τ2), there exist neighborhoods U1 and U2 of τ1 and τ2 respectivelysuch that for γ ∈ Γ, γU1 ∩ U2 = ∅ if and only if γτ1 = τ2. Thus π(U1) and π(U2) separateany 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 theinterior of the fundamental domain, then it is clearly true. If they are on the boundary, weuse 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 S1).Then Γτ = Γ∩γ−1 SO(2)γ is the intersection of a discrete group and a compact group, henceis finite. But any finite subgroup of S1 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 chartsare 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 ofYΓ is of the form Γτ for τ ∈ H. If τ is not elliptic, applying the proof of Lemma 3.1 toτ1 = τ2 = τ shows that τ has a neighborhood U such that γU ∩U 6= ∅ implies γτ = τ . ThenU gives a local coordinate at τ .

If τ is elliptic, locally its neighborhood is D/µ2 or D/µ3 with local coordinate given byz 7→ z2 or z3.

(Diagram here)

Example 3.6. For Γ = SL2(Z), YΓ = SL2(Z)\H. Recall that the j-function is defined asE3

4

∆. Viewing j as a function on YΓ, the order of vanishing is given by

vπ(ρ)(j) =1

3vρ(j).

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

topological space, consider X = X ∪ ∞ where the open sets of X are the open sets of X

together with ∞∪ cocompact open sets of X. This implies X is compact.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 bc d

)(xy

)=

(ax+ bycx+ 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∗ iscompact. In terms of the fundamental domain, D ∪ ∞ is compact (same proof as one-point 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.Next we consider the local coordinate at ∞. Since the stabilizer SL2(Z)∞ is given by±(

1 ∗0 1

), in general we have Γ∞ = Γ ∩ SL2(Z)∞ ⊃

(1 NZ0 1

)for some positive

integer N . Let h∞ be the minimal positive integer such that

(1 h∞0 1

)∈ Γ∞, called the

width or period. Give local coordinate at ∞ by q = e2πiτ/h∞ . In general, for τ ∈ P1(Q), hτis defined by moving τ to ∞ by SL2(Z).

To summarize, we have shown that Γ\H = YΓ ⊂ XΓ = Γ\H∗ are Riemann surfaces, andXΓ is compact.

We are interested in understanding the invariants of XΓ as a Riemann surface, e.g. itsgenus.

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 anisomorphism.

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\Γ1τ) = [Γ2 :Γ1], and there are only finitely many elliptic points.

To calculate the genus, we use the Riemann-Hurwitz formula. It is enough to calculateramifications, which is where we need to use the local coordinates.

2g1 − 2 = (2g2 − 2) deg(φ) + deg Ram

where deg Ram =∑

x∈X1(ex− 1) =

∑y∈X2

degRy, degRy =∑

x∈φ−1(y)(ex− 1), and ex is theramification degree at x ∈ X.

Note that the ramification points are a subset of the fibers over elliptic points of X2 (butthe 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 ofelliptic 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).12

For h = 2, 3, define eh to be the number of elliptic points in the fiber over the order helliptic point Ph of SL2(Z)\H∗, where P2 = i and P3 = ρ. Then the number of non-ellipticpoints in the fiber over Ph is d−eh

h, so Rh = (h− 1)d−eh

h.

(Diagram here)Define e∞ to be the number of cusps in X1. Then R∞ =

∑(ex − 1) = d− e∞.

Plugging in everything,

2g1 − 2 = −2d+∑

h=2,3,∞

h− 1

h(d− eh) = −2d+

1

2(d− e2) +

2

3(d− e3) + (d− e∞)

So we have proved

Theorem 3.8. g(XΓ) = 1 +d

12− 1

4e2 −

1

3e3 −

1

2e∞.

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” usingelliptic curves.

In general it is hard to calculate eh. It is easy if Γ is a normal subgroup, e.g. Γ(N), becauseevery point in a fiber will either be simultaneously elliptic or non-elliptic. But Γ0(N) is notnormal.

4. Lecture 4 (February 4, 2013)

4.1. Modular Curves. Last time we constructed the modular curve XΓ, which is a compactRiemann surface with genus given by

g(XΓ) = 1 +d

12− e2(Γ)

4− e3(Γ)

3− e∞(Γ)

2

where d is the degree of the natural map XΓ → XSL2(Z), e2, e3 are the number of ellipticpoints 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∗.

Theorem 4.1. For Γ = Γ0(N), we have

e2(Γ0(N)) =

0 if 4 | N,∏

p|N

(1 +

(−4p

))if 4 - N,

=

0 if (Q(

√−1), N) does not satisfy the Heegner condition,

2#odd p|N if (Q(√−1), N) satisfies the Heegner condition,

e3(Γ0(N)) =

0 if 2 | N or 9 | N,∏

p|N

(1 +

(−3p

))otherwise,

=

0 if (Q(

√−3), N) does not satisfy the Heegner condition,

2#odd p|N if (Q(√−3), N) satisfies the Heegner condition,

13

where we define(−4

2

)= 0 and

(−33

)= 0; otherwise

(∗p

)is the Legendre symbol, and

e∞(Γ0(N)) =∑d|N

φ

(gcd

(d,N

d

)),

where φ(n) =∑

c|n 1.

Definition 4.2. The “Heegner condition for (K,N)” means for every p | N , either p is splitin 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 leastas 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 asRiemann surfaces. Equivalently, this is the set of lattices Λ ⊂ C modulo homothety, namelyΛ1 ∼ Λ2 if and only if there exists λ ∈ C× such that λΛ1 = Λ2.

Indeed, we can establish a bijection by sending τ ∈ SL2(Z)\H to Λτ = Z + Zτ . Replacing

τ by γτ where

(a bc d

)∈ SL2(Z), the lattice becomes

Z + Zaτ + b

cτ + d=

1

cτ + d(Z(cτ + d) + Z(aτ + d)) =

1

cτ + d(Z + Zτ),

i.e.

Λγτ =1

cτ + dΛτ

and so the map is well-defined. Bijectivity can then be verified easily.

Remark. A function f on H satisfying the weight k condition

f(γτ) = (cτ + d)kf(τ)

can be interpreted as a function g on the set of lattices of homogeneous degree −kg(λΛ) = λ−kg(Λ).

Given g, we can define f on H by f(τ) = g(Λτ ). Then

f(γτ) = g(Λγτ ) = g(1

cτ + dΛτ ) = (cτ + d)kg(Λτ ) = (cτ + d)kf(τ)

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 ∈ H0(E,Ω1E/C) such that

h(E, λω) = λkh(E,ω).

Given a function h, how do we get f? To any given τ ∈ H we associate the elliptic curveE = C/Λτ and the canonical 1-form dz, which descends to E because it is invariant un-der translation, and define f(τ) = h(E, dz). Then C/ 1

cτ+dΛτ is isomorphic to C/Λτ via

multiplication by cτ + d, so

h(C/Λγτ , dz) = h(C/1

cτ + dΛτ , dz) = h(C/Λτ , (cτ + d)dz) = (cτ + d)kh(C/Λτ , dz),

14

i.e.f(γτ) = (cτ + d)kf(τ).

Note that we have not put any holomorphic conditions on H or at the cusps. We are justtrying to interpret the weight k condition by considering the points of the modular curvesas 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

ι∗ =

cyclic subgroup of order N (of E[N ]) if ∗ = 0,

order N point (of E[N ]) if ∗ = 1,

a Z/N -basis of E[N ]: (α, β) ∈ E[N ]2 with Z/Nα + Z/Nβ = E[N ] if ∗ = nothing,

3

where E[N ] is the subgroup of N -torsion. If E = C/Λ with group structure induced by thecomplex numbers, then E[N ] = 1

NΛ/Λ ∼= (Z/N)2 as abelian groups.

The equivalence relation is given in the obvious way: two pairs (E1, G1) and (E2, G2) are

isomorphic if there is an isomorphism E1φ→∼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

ι∗ =

⟨

1N

+ Λτ

⟩if ∗ = 0,

1N

+ Λτ if ∗ = 1,

( 1N, τN

) if ∗ = nothing.

(For E = C/Λτ , E[N ] = 1N

Λτ/Λτ = ( 1NZ + 1

NZτ)/Λτ .)

Proof (Sketch). Let us check this is well-defined for the Γ0(N)-level structure, which is the

only case we will use. Let γ =

(a bc d

)∈ Γ0(N). We have(

C/1

cτ + dΛτ ,

⟨1

N

⟩)cτ+d−→

(C/Λτ ,

⟨cτ + d

N

⟩)=

(C/Λτ ,

⟨d

N

⟩)=

(C/Λτ ,

⟨1

N

⟩)since N | c and gcd(d,N) = 1. In fact the same calculation proves injectivity as well: ifτ and γτ go to the same pair, then γ must be in Γ0(N). Surjectivity can be proved bychanging basis.

Now we return to the interpretation of elliptic points of X0(N) or Y0(N). The correspon-dence above implies that the stabilizer group of any point in the orbit Γ0(N)τ is isomorphicto

Γ0(N)τ ∼= Aut(E,G)

where G is an order N cyclic subgroup. In the case N = 1, this corresponds to the fact thatAut(E) can only be Z/2,Z/4,Z/6, with the last two cases occurring when E has complexmultiplication 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 haveWeil 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 orderh = 2 and the set of ideals N ⊂ Z[i] such that Z[i]/N ∼= Z/N , and a bijection between theset of elliptic points of X0(N) of order h = 3 and the set of ideals N ⊂ Z[e2πi/3] such thatZ[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!). Assumingthis, we can give a proof of Theorem 4.1.

Proof of Theorem 4.1 (Sketch). It suffices by the Chinese Remainder Theorem to considerN = 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.

If p is inert, then pOK is a prime ideal with OK/(p) ∼= (Z/p)2, 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 ifN = pipj 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 extraautomorphisms 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 thequadratic field K = Q(i). Our goal is to count the number of subgroups G ⊂ C/OK suchthat G is cyclic of order N and the pair (C/OK , G) admits non-trivial automorphisms.

Since the automorphism group of the latticeOK is generated by multiplication by i (iOK =OK = Z[i]), we want [i] : C/OK → C/OK to preserve G. Writing G = M/OK , whereM ⊂ K = Q ⊗ OK is contained in 1

NOK , 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 numberof ideals N =M−1 ⊂ OK such that OK/N =M/OK = Z/N .

Corollary 4.5. If N = p is a prime, then

g(X0(p)) =

bp+1

12c − 1 if p ≡ 1 (mod 12),

bp+112c otherwise.

Note the congruence condition on p precisely depends on its decomposition in Q(i) andQ(√−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 ] = 1N

Λ/Λ. Write G = Λ′/Λ where

Λ′ ⊂ 1N

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 isC/Λ→ C/Λ′.

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/Λ′) where OK ⊂ Λ′ ⊂1NOK and Λ′/OK ∼= Z/N .

Definition 5.1. A Heegner point attached to OK on X0(N) is such a pair (C/OK ,C/Λ′)where both curves have endomorphism ring OK .

This condition can be translated into

Lemma 5.2. EndC(C/Λ′) = OK if and only if Λ′ is an ideal of K, i.e. a fractional ideal ofOK.

Proof. If Λ′ is an ideal, then λΛ′ ⊂ Λ′ ⇔ λ ∈ OK by inverting ideals. The converse is easy,since λ ∈ C : λΛ′ ⊂ Λ′ ⊃ OK means Λ′ is an ideal.

We are reduced to the question of finding ideals Λ′ such that Λ′/OK ∼= Z/N . Last timewe showed

Lemma 5.3. A fractional ideal Λ′ such that Λ′/OK ∼= Z/N exists if and only if p | N impliesp ramifies with p || N or p is split. In particular, if p has an inert factor, then Λ′ does notexist.

This is called the Heegner condition. We summarize our work last time as follows. Let Ehbe 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 bya modular form of weight 2 (using the modularity theorem as a black box!). Mapping theHeegner points into E shows that they are defined over the Hilbert class field of K, andtaking trace pushes them down to Q. This is the only systematic way of producing rationalpoints 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 cuspforms, which is almost trivial after the genus formula, at least for even weights. The idea isto interpret the space of modular forms as the space of global sections of line bundles overRiemann 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 ofmodular forms (resp. cusp forms) of weight k.

17

Lemma 5.5 (Key Lemma).

(1) The map Mk → H0(X,Ω⊗k/2X (εk)) given by

f 7→ ωf = f(τ)(2πidτ)k/2,

where εk := bk4cε2 + bk

3cε3 + k

2ε∞ ∈ Div(X), is an isomorphism. More precisely, this

is the space of meromorphic forms ω of degree k2

with poles div(ω) ≥ −εk.

(2) The same map gives an isomorphism Sk∼→ H0(X,Ω

⊗k/2X (εk − ε∞)).

Proof. For any γ ∈ Γ ⊂ SL2(Z), define the automorphy factor j(γ, τ) = cτ + d. Thend(γτ) = j(γ, τ)−2dτ . Since f(γτ) = j(γ, τ)kf(τ), ωf is invariant under Γ and indeed descendsto a meromorphic form of degree k

2on X.

Let us express the condition of f being holomorphic in terms of ωf . Choose local coordinatez at elliptic points so that τ = z1/h. Then

f(τ)(dτ)k/2 = f(z1/h)(dz1/h)k/2 = z( 1h−1) k

2 f(z1/h)(dz)k/2.

Hence f(τ) is holomorphic on H if and only if vz(ωf ) ≥ −bh−1h

k2c. (Note that invariance

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 h0 1

)⟩. The local coordinate is given by q = e2πiτ/h, so dq/q = 2πi/hdτ . If f has

Fourier expansion f(τ) = g(q) = a0 + a1q + · · · , then

ωf = f(τ)(2πidτ)k/2 = g(q)(hdq

q)k/2 = Cq−k/2g(q)(dq)k/2.

Therefore, f holomorphic (resp. zero) at∞ if and only if vq=0(ωf ) ≥ −k2

(resp. −k2

+ 1).

Corollary 5.6. S2∼= H0(X,ΩX), so dimS2 = 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) = dimH0(X,L).For k ≥ 2, we have

deg Ωk/2(εk) =k

2(2g − 2) + bk

4ce2 + bk

3ce3 +

k

2e∞ ≥ (2g − 2) + e∞ > 2g − 2.

For k ≥ 4 (to study cusp forms), we have

deg Ωk/2(εk − ε∞) =k

2(2g − 2) + bk

4ce2 + bk

3ce3 + (

k

2− 1)e∞ ≥ 2g − 2 + e∞ > 2g − 2.

Applying to L = Ωk/2(εk) and Ωk/2(εk − ε∞) respectively, we conclude

Corollary 5.7. For k ≥ 2, dimMk = (k − 1)(g − 1) + bk4ce2 + bk

3ce3 +

k

2e∞.

For k ≥ 4, dimSk = (k − 1)(g − 1) + bk4ce2 + bk

3ce3 + (

k

2− 1)e∞.

For k odd, the same idea works but the calculations are more complicated. It is mostdifficult 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 modularforms and cusp forms of weight k respectively. We want to study the endomorphisms ofthese vector spaces.

Denote GL2(Q)+ = γ ∈ GL2(Q) : det γ > 0. It defines an action on H and has thefollowing 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−1Hγ and H are “commensurable”, i.e. γ−1Hγ ∩ H is of finite index inH and γ−1Hγ.

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(hgh′) = φ(g) for all h, h′ ∈ H and g ∈ G) and supported on finitely many doublecosets. The last condition is the analogue for φ to have compact support when G is equippedwith a topology, e.g. Example 6.2.

We can define an algebra structure on C[G//H] by

(φ ∗ ϕ)(g) =∑

x∈H\G

φ(gx−1)ϕ(x)

which can be interpreted as the integral

∫G

φ(gx−1)ϕ(x)dx where H\G is given the counting

measure 4. Then (C[G//H], ∗) is an algebra with identity 1H , the characteristic function ofH. 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 = gσxσ, such that everydouble 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).

Lemma 6.4. GL2(Qp) =∐

α≥β∈Z

GL2(Zp)(pα 00 pβ

)GL2(Zp).

This immediately implies the second requirement that every double coset has a represen-tative fixed by σ.

Note σ defines an action on C[G//H] as well by φσ(g) = φ(gσ), 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]. Combiningthese, 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 Lemma5.1.2 of Diamond-Shurman.

19

Remark. (G,H) is called a Gelfand pair, i.e. for all irreducible representations π of G, thespace 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 onlyconsider the subalgebra generated by functions supported in M2(Z) ∩GL2(Q)+. This set iscertainly 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 thedisjoint unions ∐

β|α>0αβ=n

Γ

(α 00 β

)Γ =

∐β|n

0≤x<β

Γ

(n/β x

0 β

)where Γ = SL2(Z). In particular, we have

#Γ\M [n] =∑β|n

β = σ1(n).

(In order to find a double coset representative for any matrix in M [n], note that β will bethe 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 writeM [nm] = M [n] ·M [m]

in an essentially unique way: if γ = γnγm = γ′nγ′m, then γ′−1

n γn = γ′mγ−1m has deter-

minant 1 and is contained in 1nM2(Z) ∩ 1

mM2(Z) = M2(Z), hence is in Γ = SL2(Z).

Thus γ′m ∈ 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+1 − 1

p− 1· p

2 − 1

p− 1=pn+2 − 1

p− 1+ p · p

n − 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−1j(γ, τ)−kf(γτ)

where for γ =

(a bc d

), j(γ, τ) = cτ + d is the automorphy factor. Using this notation, the

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

f [φ]k =∑

Γ\GL2(Q)+

f [γ]kφ(γ)

which is a finite sum. Again this can be interpreted as a formal integral

∫GL2(Q)+

f [γ]kφ(γ)dγ,

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 canalso 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 topk−2Tpn−1 because

f [pγ]k = det(pγ)k−1j(pγ, τ)−kf(γτ) = p2(k−1)p−kf [γ]k

since γ and pγ act on H in the same way. With this, Theorem 6.7 translates into

Theorem 6.10.

(1) TnTm = Tnm for gcd(n,m) = 1.(2) For p prime, TpTpn = Tpn+1 + pk−1Tpn−1.

Lemma 6.11 (Effect on q-expansion). If f =∑∞

n=0 an(f)qn ∈ Mk, then Tmf ∈ Mk withn-th Fourier coefficient given by

an(Tmf) =∑d|(n,m)

dk−1anm/d2(f).

In particular, we have a1(Tmf) = am(f).

Example 6.12. Let ∆ = q∞∏n=1

(1 − qn)24 =∞∑n=1

τ(n)qn = q + · · · , where τ is known as the

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 propertiesin Theorem 6.10, so

τ(mn) = τ(m)τ(n) if (m,n) = 1,

τ(pn+1) = τ(p)τ(pn)− p11τ(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

Tn(f) =∑

β∈Γ\M [n]

f [β]k

(which is a finite sum) for f ∈Mk and n ≥ 1, where

f [β]k(τ) = (det β)k−1j(β, τ)−kf(β(τ))

Lemma 7.2 (Effect on q-expansion). Let f =∑∞

n=0 anqn ∈ Mk(SL2(Z)), and denote

Tm(f) =∑∞

n=0 an(Tmf)qn. Then

an(Tmf) =∑d|(m,n)

dk−1anm/d2 .

In particular, a1(Tmf) = am and a0(Tmf) = σk−1(m)a0.

From this, we can showan(TmTm′f) = an(Tm′Tmf)

which verifies that the subalgebra of End(Mk(SL2(Z)) generated by Tn for all n ≥ 1 iscommutative. 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∐ad=m

0≤b≤d−1

Γ

(a b0 d

).

HenceTm(f)(τ) = mk−1

∑β

j(β, τ)−kf(β(τ))

where β runs through the coset representatives

(a b0 d

). Since β(τ) = aτ+b

d, e2πinβ(τ) =

e2πinaτ+bd and j(β, τ) = d, we have

Tm(f)(τ) = mk−1∑ad=m

∞∑n=0

d−kd−1∑b=0

ane2πin(aτ

d+ bd

)

= mk−1∑ad=m

∑d|n

d−kdane2πin(aτ

d)

=∑ad=m

∞∑n′=0

(md

)k−1

adn′e2πin′aτ

=∞∑n=0

qn

∑n′a=nad=m

ak−1adn′

.

Example 7.3 (Eisenstein series). Recall that the Eisenstein series of weight k is given by

Gk(τ) =∑

w∈Λτ−0

w−k = 2ζ(k) +(2πi)k

(k − 1)!

∞∑n=1

σk−1(n)qn

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) bycomparing the constant terms. In fact we can directly check the formula∑

d|(n,m)

dk−1σk−1

(nmd2

)= σk−1(m)σk−1(n).

Remark. Recall that functions on H satisfying the weight-k condition can be interpretedas functions on lattices of homogeneous of degree −k. With this, we can give a simplerdefinition of Hecke operator

(Tnf)(Λ) = nk−1∑Λ′⊂Λ

[Λ:Λ′]=n

f(Λ′).

For a fixed lattice Λ = Λτ = Z + Zτ ⊂ C, there is a bijection

Λ′ ⊂ Λ of index n ←→ SL2(Z)\M [n]

where we associate to each β =

(a bc d

)∈M [n] the lattice βΛ = span(a+ bτ, c+ dτ), which

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 canextend Hecke operators to Γ∗(N) where ∗ = 0, 1 or nothing. It is enough to study Γ1(N),because Γ1(N) ⊂ Γ0(N) and(

N 00 1

)−1

Γ(N)

(N 00 1

)= Γ1(N2)

(

(a bc d

)is sent to

(a b/NcN d

)). Recall Γ1(N) is normal in Γ0(N): there is an exact

sequence

1→ Γ1(N)→ Γ0(N)→ (Z/N)× → 1

where the last map is given by

(a bc d

)7→ d (mod N).

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) forthe trivial character χ0. It is almost trivial to see that

Mk(Γ1(N)) =⊕

χ:(Z/N)×→C×Mk(Γ1(N), χ).

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(Tnf)(Λ1,Λ2) = nk−1

∑Λ′2⊂Λ2

[Λ2:Λ′2]=n

f((Λ′2 ∩ Λ1),Λ′2).

(2) Equivalently, in the language of elliptic curves, viewing f : (E,G, ω)/ ∼→ C gives

Tn(f)(E,G, ω) = nk−1∑

H⊂E subgroupof order n

f(E/H, (G+H)/H, ωH)

(because (n,N) = 1) where ωH is the pullback of ω under the dual isogeny of E →E/H.

(3) In terms of double cosets, let MN =

(a bc d

)≡(∗ ∗0 ∗

)(mod N)

and MN [n] =

γ ∈MN : det γ = n for (n,N) = 1. Then we have the coset decomposition

MN [n] =∐ad=n

0≤b≤d−1

Γ0(N)

(a b0 d

).

We define Hecke operators for Γ = Γ∗(N) where ∗ = 0, 1 as follows. For p prime, define

Tp = Γ

(1 00 p

)Γ

andTpi = TpTpi−1 − pk−1〈p〉Tpi−2

(the diamond operator 〈p〉 will be defined below). Note when p - N , the second equationreduces to Tpi = (Tp)

i. We then extend the definition to Tn by multiplicativity.We can prove the algebra generated by these is commutative, by using the involution

σ(X) =

(N 00 1

)−1

X t

(N 00 1

)on Γ∗(N).

Lemma 7.5. Let f ∈Mk(Γ1(N), χ) with Nebentypus χ : (Z/N)× → C×. Then

an(Tmf) =∑d|(n,m)

χ(d)dk−1anm/d2

(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) isa normal subgroup of Γ0(N), any coset satisfies Γ1(N)γ = γΓ1(N) for γ ∈ Γ0(N) and henceΓ1(N)γΓ1(N) = Γ1(N)γ = γΓ1(N). This defines a double coset operator which dependsonly on the image of γ under

Γ0(N)→ (Z/N)× → 1

sending γ =

(a bc d

)to d (mod N). This double coset operator is denoted by 〈d〉 and called

the diamond operator.

Theorem 7.6. The operators Tn and 〈d〉, where n ≥ 1 and d ∈ (Z/N)×, are commutativeon 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) inthe following sense.

Definition 7.7. φ : H→ C has weight (m,n) if

φ(γτ) = j(γ, τ)mj(γ, τ)nφ(τ).

Example 7.8. Im(τ) = y = 12i

(τ − 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

〈f, g〉Γ =

∫Γ\H

f(τ)g(τ)ykdτdτ

y2

Claim. If one of f or g is a cusp form, then limy→∞ f(τ)g(τ)yk = 0.

Claim. 〈Gk, Gk〉 diverges.

Note that the Eisenstein series can be given by

Gk =∑′

c,d

(cτ + d)−k =∑

γ∈Γ∞\Γ

j(γ, τ)−k

where Γ = SL2(Z) and Γ∞ =

(1 ∗0 1

). Then for any g ∈Mk(Γ),∫

Γ\HGk(τ)g(τ)yk

dτdτ

y2=

∫Γ\H

∑γ∈Γ∞\Γ

j(γ, τ)−kj(γ, τ)−kg(γτ) Im(τ)kdτdτ

y2

=

∫Γ\H

∑γ∈Γ∞\Γ

g(γτ) Im(γτ)kdτdτ

y2

=

∫Γ∞\H

g(τ) Im(τ)kdτdτ

y2

=

∫Γ∞\H

g(τ)ykdτdτ

y2

is divergent unless g(τ)→ 0 as y →∞, i.e. unless g is a cusp form. This implies the secondclaim.

Next time we will study the inner product space (Sk(Γ∗(N)), 〈·, ·〉), and show that Heckeoperators are normal, i.e. they commute with their adjoints X∗X = XX∗. This implies X isdiagonalizable. Therefore we can simultaneously diagonalize the family of Hecke operatorsand decompose the space as direct sum of Hecke eigenforms.

8. Lecture 8 (February 18, 2013)

8.1. Petersson inner product. Consider the hyperbolic measuredxdy

y2on H which is

invariant under SL2(R). If f and g are modular forms of weight k for a congruence subgroup25

Γ, then the product f(τ)g(τ) Im(τ)k is Γ-invariant. Define

〈f, g〉Γ =

∫Γ\H

f(τ)g(τ)ykdxdy

y2.

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 thefundamental domain will be finitely many copies of D). Write f(τ) = a0 + a1q + · · · andg(τ) = b0 + b1q + · · · . Since a0b0 = 0, there exists some constant C > 0 such that

|f(τ)g(τ)yk| < Ce−2πyyk.

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∫y0

ykdxdy

y2does.

Therefore, 〈f, g〉Γ is a well-defined inner product on Sk(Γ), the space of cusp forms, so itmakes sense to talk about adjoint operators. For Γ = Γ∗(N) where ∗ = 0, 1, consider theHecke operators Tn for (n,N) = 1.

(Morally speaking, they already determine Tp for primes p dividing N . Also we don’t losetoo much just by considering Γ0(N), since Γ1(N) is normal in Γ0(N) with abelian quotient.Last time we introduced the diamond operators 〈d〉 for (d,N) = 1. In fact we don’t lose toomuch just by considering SL2(Z). The same proof techniques work. Anyway we’ll assumeΓ = Γ∗(N) where ∗ = 0, 1.)

Let T ∗n be the adjoint operator of Tn on Sk(Γ), characterized by

〈Tnf, g〉Γ = 〈f, T ∗ng〉Γ.

Theorem 8.2. Let Tα be the operator associated to ΓαΓ, where α ∈M2(Z) with det(α) 6= 0.Then

T ∗α = Tα∗

where α∗ = det(α)α−1 ∈M2(Z).

Corollary 8.3. For Γ = Γ∗(N) and (n,N) = 1, we have

T ∗n = 〈n〉−1Tn.

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,N) = 1, Tp corresponds to the double coset

ΓαΓ and T ∗p corresponds to Γα∗Γ, where α =

(1 00 p

)and α∗ =

(p 00 1

). We claim that(

p 00 1

)∈ Γ1(N)

(1 00 p

)Γ0(N).

(Note: In SL2(Z) we could have simply conjugated by

(0 −11 0

).)

26

If β ∈ Γ1(N) and γ ∈ Γ0(N) are such that α∗ = β−1αγ, then αγ(α∗)−1 = β. Writing

γ =

(a bc d

)∈ Γ0(N), we want(

1 00 p

)(a bc d

)(p−1 00 1

)=

(a/p bc pd

)to be contained in Γ1(N). We can pick a = p, and solve for N | c with pd = bc+ 1.

Therefore, T ∗p corresponds to the double coset

Γ

(1 00 p

)γΓ = Γ

(1 00 p

)Γγ = Γ

(1 00 p

)ΓγΓ

where γ ∈ Γ0(N), i.e.

T ∗p = Tp〈γ〉 = Tp〈d〉 = Tp〈p−1〉.

We will not prove the theorem, which amounts to changing basis. The complexity of thetheory 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 operatorTn and 〈d〉, where (n,N) = 1 and (d,N) = 1. (For Γ0(N), we just need to consider Tn since〈d〉 is trivial.)

Remark. For Γ = SL2(Z), Mk(SL2(Z)) = CEk ⊕ Sk(SL2(Z)). Thus Ek is an eigenform witheigenvalue σk−1(n) for Tn. The same conclusion in Corollary 8.4 holds for Mk(SL2(Z)), with“orthogonal” suitably interpreted (since 〈Ek, Ek〉Γ diverges).

8.2. L-functions. Next we will discussion an application to L-functions. Let f ∈ Mk(Γ)have Fourier expansion f(q) =

∑∞n=1 anq

n. Define

L(s, f) =∞∑n=1

ann−s.

To address convergence issues, note that we have the following trivial bounds on Fouriercoefficients.

Lemma 8.6. If f is a cusp form, then |an| ≤ Cnk/2 for some constant C. In general, wehave |an| ≤ Cnk.

Proof. For simplicity, assume Γ = Γ∗(N) where ∗ = 0, 1, so the local parameter is given byq = e2πiτ (for general congruence subgroups we need to take into account the width of Γ at∞). Then

an =1

2πi

∫f(q)q−n

dq

q=

∫ 1

0

f(x+ iy)e−2πin(x+iy)dx

is independent of y > 0 by holomorphicity. Choosing y = 1n, we have

an = e2π

∫ 1

0

f

(x+

i

n

)e−2πinxdx

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+ i

n)| ≤ C( 1

n)−k/2 = Cnk/2 and so

|an| ≤ Cnk/2.

For the Eisenstein series Ek for SL2(Z), we have

an = σk−1(n) =∑d|n

dk−1 ≤ Cnk.

Since any modular form for SL2(Z) can be written as a sum of an Eisenstein series and acusp form, we obtain the trivial bound.

We omit the proof for general congruence subgroups.

Corollary 8.7. L(s, f) converges for Re(s) >

1 + k

2if f is cuspidal and ∗ = 0, 1,

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

L(s, f) =∏

p prime

(1− app−s + pk−1−2s)−1.

The property of having an Euler product is obviously not stable under addition. Being aneigenform 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, recallthat

an(Tm(f)) =∑d|(m,n)

dk−1anm/d2 .

This implies that the eigenvalue of Tm is am. Since TmTn = Tmn for (m,n) = 1 andTpi+1 = TpTpi − pk−1Tpi−1 , we have the same relations

aman = amn if (m,n) = 1,

api+1 = apapi − pk−1api−1 if i ≥ 1.(2)

Hence,

L(s, f) =∏

p prime

(∞∑k=1

apkp−ks

).

The sequence bi = api

has recursive relation bi+1 = apbi − pk−1bi−1, hence characteristicpolynomial T 2 − apT + pk−1. Thus

∞∑k=1

apkp−ks = (1− app−s + pk−1−2s)−1.

28

(2) ⇒ (1). The same proof goes backwards. The Euler product implies that the Fouriercoefficients an satisfy the relations (2). It suffices to verify Tpf = apf for primes p (since Tpgenerate the Hecke algebra), which is equivalent to

an(Tpf) = anap

for all n, i.e. ∑d|(n,p)

dk−1anp/d2 = anap.

But the left hand side is just

anp if (n, p) = 1,

anp + pk−1an/p if p | n,which is equal to anap by (2).

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 considerthe 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 henceisomorphic to C× · · · × C.

We can extend this to level N . Let Γ = Γ∗(N), where ∗ = 0, 1. We start by defining Tpfor all primes (p,N) = 1. Tp corresponds to the double coset

Γ

(1 00 p

)Γ =

∐0≤j≤p−1

Γ

(1 j0 p

)t Γ

(a bN p

)(p 00 1

)

where

(a bN p

)∈ Γ0(N).

We can define 〈d〉 for (d,N) = 1.We can extend the definition to Tpi by using the recursive relation

Tpi := TpTpi−1 − pk−1〈p〉Tpi−2

and to Tn for (n,N) = 1 by

TmTn = 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

Γ

(1 00 p

)Γ =

∐0≤j≤p−1

Γ

(1 j0 p

).

We use the same recursive relations as above to define Tn, and define 〈d〉 = 0 if (d,N) > 1.

Remark. All the above relations can be combined into∞∑n=1

Tnn−s =

∏p-N

(1− Tpp−s + 〈p〉pk−1−2s)−1 ·∏p|N

(1− Tpp−s)−1.

29

Therefore, we have defined Tn and 〈n〉 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 necessarilysemisimple.

Lemma 9.1. Let (n,N) = 1. Then the adjoint of Tn with respect to the Petersson innerproduct is

T ∗n = 〈n〉−1Tn.

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, 〈n〉](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

that Tp is associated to the double coset Γ

(1 00 p

)Γ, so T ∗p is associated to the double coset

Γ

(p 00 1

)Γ. These two are in general not the same double coset. Let us introduce one more

operator.For Γ = Γ∗(N), define the normalizer

N(Γ) = NGL2(R)+(Γ) = γ ∈ GL2(R)+ : γ−1Γγ = 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

wN =

(0 −1N 0

)=

(1 00 N

)(0 −11 0

)which is contained in N(Γ), so that the double coset ΓwNΓ is just a coset ΓwN . Then wecan check that f 7→ f [wN ]k maps into the same space of cusp forms, so we can considerwN = [wN ]k ∈ End(Sk(Γ)).

Since w2N = −N , we see that wN is an involution on the modular curve Γ\H∗, but it is

not quite an involution on Sk(Γ).

Lemma 9.2.

(1) w2N acts by (−1)kNk−2 on Sk(Γ).

(2) For all n, we have T ∗n = wNTnw

−1N ,

〈n〉∗ = wN〈n〉w−1N .

Proof.

(1) w2N = [w2

N ]k = (N2)k−1(−N)−k = (−1)kNk−2.

(2) The key point is that the matrices

(1 00 p

)and

(p 00 1

)differ by conjugation by wN .

5We can also check that 〈n〉∗ = 〈n〉−1 for (n,N) = 1, so 〈n〉 is indeed normal.

30

Corollary 9.3. For Γ = Γ0(N) and (n,N) = 1, we have

T ∗n = Tn = wNTnw−1N .

Thus wN commutes with Tn for (n,N) = 1.

9.2. Atkin-Lehner theory. To understand the relationship between T0 and T, we need theAtkin-Lehner theory (also known as the newform theory), and later we will give an examplewhere 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

of f(dτ) = d1−kf

[(d 00 1

)]k

where f ∈ Sk(Γ1(M)), M | N and d | NM

. In fact it is enough

to consider forms of level dividing N/p where p | N is a prime:

Soldk (Γ1(N)) = spanf(dτ) : f ∈ Sk(Γ1(N/p)), d | p, p | N ⊂ Sk(Γ1(N))

and define the newspaceSnewk (Γ1(N)) = Sold

k (Γ1(N))⊥.

Note that we are not calling this the “space of newforms” yet. Newform will have a specificmeaning – it is an eigenform.

Atkin and Lehner proved that the newspace behaves like the level 1 case, where we justneed to look at T0. Before we make this more precise, let us prove a few lemmas.

Lemma 9.4. The spaces Soldk and Snew

k are T-stable, where T is the full Hecke algebra.

Proof (Sketch). Introduce T∗, the algebra generated by the adjoints T ∗n and 〈n〉∗. It sufficesto prove that Sold

k is both T- and T∗-stable. But T∗ = wNTw−1N , so it is enough to prove Sold

k

is T-stable and wN -stable.Define the operator Vp : Sk(Γ1(N/p)) → Sk(Γ1(N)) by f 7→ f(pτ). This is a one-sided

inverse of the Up operator. Indeed, for f =∑∞

n=0 anqn,

Up(f) =∑

0≤j≤p−1

f

[(1 j0 p

)]k

= p−1∑

0≤j≤p−1

f

(τ + j

p

)=∑p|n

anqn/p

whereas

Vp(f) =∞∑n=0

anqpn.

It is clear that UpVp = 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, 〈q〉 and w for all primes q.For example, consider the diagram (note the two Tp’s are different!)

Sk(Γ1(N/p))

Tp

Vp// Sk(Γ1(N))

Tp

Sk(Γ1(N/p))Vp// Sk(Γ1(N))

which is not quite commutative. Since p | N , the Tp on the right is equal to Up and soTpVp(f) = f for all f ∈ Sk(Γ1(N/p)). This shows Tp preserves Sold

k .31

The other operators for other primes can be checked similarly6.

Example 9.5. For p - N , Up acts on Sk(Γ0(p3N)). Let f ∈ Sk(Γ0(N)) be an eigenformof Tp. Consider the space spanned by fi = f(piτ) for 0 ≤ i ≤ 3. We claim that Up actsnon-semisimply on it.

Since Vpfi = fi+1 for 0 ≤ i ≤ 2, we have Upfi = fi−1 for 1 ≤ i ≤ 3. Using the formula

Tp(f) =∑p|N

anqn/p + pk−1〈p〉f(pτ)

and the fact that f = f0 is an eigenform, we see that there exists λ ∈ C such that7

λf0 = Tpf0 = Upf0 + pk−1f1,

i.e.Upf0 = λf0 − pk−1f1.

Thus Up has matrix

λ 1

−pk−1 0 10 1

0

, which is not semisimple8.

Theorem 9.6 (Atkin-Lehner). If f ∈ Snewk (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 inspi-rational. There is a more conceptual proof using representation theory.

Definition 9.7. f ∈ Sk(Γ1(N)) is a newform if it is contained in Snewk (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, justa set!

Corollary 9.8 (Multiplicity one). If f is a newform with eigenvalue λ : T→ C, then

dimg ∈ Sk(Γ1(N)) : Tg = 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 thestructure of the Hecke module Sk(Γ) for Γ = Γ∗(N) where ∗ = 0, 1. There is no need toconsider the principal congruence subgroup Γ(N) because it is conjugate to Γ1(N2).

Consider T = C[Tn, 〈d〉 : n ≥ 1, (d,N) = 1] ⊂ EndC(Sk(Γ)) (recall that the operatorsdepend on the level N and weight k, although the notation doesn’t suggest so!). Literaturealso considers EndC(Mk(Γ)), but we restrict ourselves to the simpler case of cusp forms.Consider the subalgebra T0 = C[Tn, 〈d〉 : (n,N) = 1, (d,N) = 1] ⊂ T. Last time we sawthat T0 acts semisimply, but T does not.

6Refer to Proposition 5.6.2 of Diamond-Shurman for details.7For (d,N) = 1, the diamond operator 〈d〉 acts trivially on Sk(Γ0(N)), so 〈p〉f0(pτ) = f0(pτ) = f1. Also,

note that Up ∈ End(Sk(Γ0(p3N)) but Tp ∈ End(Sk(Γ0(N)).8This matrix has characteristic polynomial X2(X2 − λX + pk−1), but its kernel is only 1-dimensional.

32

Write Tp = Up if p | N . Then for f =∑

n≥1 anqn,

Upf =∑n≥1

anpqn

and

Vpf = p1−kf

[(p 00 1

)]k

=∑n≥1

anqnp

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 Tpf = λf . Consider the span of ei = (Vp)if where i = 0, 1, · · · , d, which are linearly

independent. Then Up, as a Hecke operator of level Npd, is given byUpei = ei−1 for 1 ≤ i ≤ d,

Upe0 = λe0 − pk−1e1,

so its matrix is of the form

Up,d =

λ 1

−pk−1 0 10 1

0

.

If d ≥ 3, then this matrix is not diagonalizable, i.e. the action of Up is not semisimple. Thisfollows from an easy exercise in linear algebra.

Exercise. The minimal polynomial of Up,d is Xd−1(X2 − 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(Tf) is a perfect pairing.

Proof. Recall that a1(Tmf) = am(f).If (T, f) = 0 for all T , then in particular (Tn, f) = 0, so an(f) = a1(Tnf) = 0 for all n and

f = 0.If (T0, f) = 0 for all f ∈ Sk(Γ), we want to show T0 = 0, i.e. T0f = 0. Indeed, for all n,

an(T0f) = a1(TnT0f) = a1(T0Tnf) = (T0, Tnf) = 0.

If f ∈ Sk(Γ) is an eigenform of T, then Tf = λf (T )f satisfies λf (TnTm) = 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) : Tg = λf (T )g for all T ∈ T.

Lemma 10.2 (Multiplicity one). If f ∈ Sk(Γ) is an eigenform of T, then dimSk(Γ)[λf ] = 1.

Proof. For g ∈ Sk(Γ)[λf ], we have an(g) = a1(Tng) = λf (Tn)a1(g) for all n ≥ 1.

Theorem 10.3 (Atkin-Lehner).

(1) If f ∈ Snewk (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 inde-pendent elements Vdf = f(dτ) where d | N/Nf . Then

Sf = g ∈ Sk(Γ∗(N)) : Tg = λf (T )g for all T ∈ T0.In particular, Sf is stable under T.

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(3) There is a decomposition

Sk(Γ∗(N)) =⊕M |N

⊕f newformof level M

Sf

as Hecke modules.

Remark. By Lemma 10.2, if M = N , then dimSf = 1. In general, Sf has dimension∑d|N/Nf 1.

Remark (Strong multiplicity one9). The association

f newform of level dividing N → non-zero algebra homomorphism λ : T0 → Cis 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

f ∈∑p|N

Vp(Sk(Γ∗(N/p))).

(2) If f, g ∈ Snewk (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 MainLemma and proved first in Diamond-Shurman. There are better proofs, e.g. in Casselman.

Example 10.5. Recall that

dimS2(Γ0(N)) = g(X0(N)).

For example, using the genus formula, we get dimS2(Γ0(11)) = 1 and dimS2(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).

With respect to this basis, U2 has matrix

(λ2 1−2 0

), where T2f = λ2f in S2(Γ0(11)). We

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, 〈d〉 : 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. ThenQ[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 automorphicrepresentations. 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 subringR ⊂ 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⊕k∈Z

Mk,C = C[E4, E6]

where Ek = 1 − kBk

∑∞n=1 σk−1(n)qn for k ≥ 4 even. Since Ek has rational coefficients, we

immediately get

Corollary 10.8. Mk,Q ⊗ C ∼= Mk,C.

The basis element Ea4E

b6 is contained in Mk,Q, and this space is stable under TQ since

an(Tmf) =∑d|(n,m)

dk−1anm/d2(f).

Thus TQ is a subalgebra of EndQ(Mk,Q), hence finitely generated over Q, i.e. a finite-dimensional 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 ofspaces of modular forms, and prove a result of Shimura on the algebraicity of special valuesof 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