Modular Forms

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Modular Forms

Andrew Kobin Fall 2019 Contents Contents

Contents

0 The Web of Modularity 1 0.1 The Partition Function ...... 2 0.2 Quadratic Forms ...... 3 0.3 Elliptic Curves ...... 4 0.4 Galois Theory and Quantum Gravity ...... 4 0.5 Some Perspectives on Modular Forms ...... 5

1 Theta Functions 6 1.1 Poisson Summation ...... 6 1.2 The Jacobi Theta Function ...... 8

2 Modular Groups 12 2.1 The Modular Group SL2(Z)...... 12 2.2 Congruence Subgroups ...... 12 2.3 Modular Curves ...... 15

3 Modular Forms 17 3.1 Deﬁnitions and Examples ...... 17 3.2 Modular Forms for SL2(Z)...... 32 3.3 Petersson Inner Product ...... 35

4 Hecke Operators 37 4.1 Hecke Operators on SL2(Z)...... 37 4.2 The Trace Form ...... 41 4.3 Atkin–Lehner Theory ...... 43 4.4 Oldforms and Newforms ...... 48

5 Connections to L-Functions 51 5.1 L-Functions for Hecke Eigenforms ...... 51 5.2 Modular Forms with Complex Multiplication ...... 53 5.3 Connection to Elliptic Curves with CM ...... 55 5.4 Periods and Critical L-Values ...... 57 5.5 Shimura Correspondence ...... 60

i 0 The Web of Modularity

0 The Web of Modularity

These notes were taken during a course / seminar on modular forms taught by Dr. Ken Ono at the University of Virginia in fall 2019. The reading list includes: Koblitz’s Introduction to Elliptic Curves and Modular Forms, Cohen-Stromberg’s Modular Forms, Chapter VII in Serre’s A Course in Arithmetic, Diamond-Shurman’s A First Course in Modular Forms and Stein’s Modular Forms: A Computational Approach among many others. The course centered around a list of special topics directly tied to the theory of modular forms. Some of these topics include:

(1) The theory of elliptic functions and harmonic Maas forms

(2) Classical theory of complex multiplication and its connections to abelian class ﬁeld theory

(3) Exact combinatorial formulas, e.g. the partition function p(n), values of the Riemann zeta function like ζ(2n), L-functions of an elliptic curve.

(4) Quadratic forms – here’s an interesting question that turns out to have a natural modular interpretation: which integers can be represented by an arbitrary quadratic form? For example, if the quadratic form is q = x2, the answer is “all perfect squares” of course. For q = x2 + y2, Legendre and Fermat showed that this comes down to whether the integer is divisible by primes congruent to 3 (mod 4), and this can alternatively be phrased in terms of class ﬁeld theory. For q = x2 + y2 + 10z2, the question is wide open. None of the odd integers n = 3, 7 and 2719 are representable by q, and in fact 2719 is the largest known non-example. Further:

Theorem 0.0.1 (Soundararajan). Assuming the Generalized Riemann Hypothesis, 2719 is the last odd integer not of the form x2 + y2 + 10z2.

(5) Modularity of elliptic curves

(6) Modular forms mod p

(7) Brauer’s Problem 19: given a ﬁnite simple group G and a ﬁxed prime p, is there a complex irreducible representation ρ : G → GL(V ) which remains irreducible mod p?

(8) Thompson’s Conjecture: let G be a finite group and K(G) the number field obtained by adjoining the entries of the character table for G to Q. Thompson proved that if n > 24 and m∗ := (−1)(m−1)/2m for m odd, then √ ∗ K(An) = Q({ p | odd primes p < n, p 6= n − 2}). He then conjectured that if Π is a finite set of odd primes, then for n sufficiently large (depending on Π),

KΠ(An) = Q({χ | χ in the char. table for An restricted to g ∈ An with order p, p ∈ Π}) √ is equal to Q({ p∗ | p ∈ Π, p 6= n − 2}).

1 0.1 The Partition Function 0 The Web of Modularity

Loosely speaking, a modular form is a function f : h → C satisfying az + b a b An invariance property: f = (cz + d)kf(z) for all lying in a ﬁxed cz + d c d subgroup Γ ≤ SL2(Z) and z ∈ h. The number k is called the weight of f, while (cz + d)k is called the factor of automorphy.

Analytic, meromorphic or real analytic conditions.

In the next few sections, we introduce some classical number theory problems that can be elucidated with the theory of modular forms. In some sense these are the foundations of the theory, as many modern techniques were developed with one of these applications in mind.

0.1 The Partition Function

Recall that a partition of an integer n is any nonincreasing sequence of positive integers which sum to n. The partition function p : N → N is deﬁned by p(n) = the number of partitions of n. It turns out that all of the irreducible representations of every ﬁnite group can be explicitly constructed from the combinatorial properties of p(n). For example, if n = 4 there are are 5 partitions: (4), (3, 1), (2, 2), (2, 1, 1), (1, 1, 1, 1). Modular forms enter when one tries to compute the generating function for p(n).

Lemma 0.1.1 (Euler). The generating function for p(n) is

∞ ∞ X Y 1 p(n)qn = . 1 − qn n=0 n=1 The right side of this equation is what is known as an ‘Euler product’. This simple, elegant formula has been a testing ground for the theory of modular forms. For example:

The ﬁrst Hecke operators were deﬁned for this function.

The circle method was originally developed with this function in mind.

Early examples of Galois representations were computed for this example.

Many of the earliest theorems about p(n) are due to Ramanujan, including:

Theorem 0.1.2 (Ramanujan). For all n ≥ 0,

p(5n + 4) ≡ 0 (mod 5) p(7n + 5) ≡ 0 (mod 7) p(11n + 6) ≡ 0 (mod 11).

These were later proven to be the only examples of such congruences.

Theorem 0.1.3 (Ahlgren-Boylan). If ` is prime and p(`n + b) ≡ 0 (mod `) for some b ∈ Z and all n ≥ 0, then (b, `) = (4, 5), (5, 7) or (6, 11).

2 0.2 Quadratic Forms 0 The Web of Modularity

On the other hand: Theorem 0.1.4 (Ono). For any (Q, 6) = 1, there are inﬁnitely many non-nested arithmetic progressions An + B such that

p(An + B) ≡ 0 (mod Q) for all n ≥ 0.

For example, p(4063467631n + 30064597) ≡ 0 (mod 31) for all n ≥ 0. The asymptotic size of p(n) has also been known for decades: 1 √ Theorem 0.1.5 (Hardy-Ramanujan). p(n) ∼ √ eπ 2n/3. 4n 3 1 All of these results can be accessed by studying the meromorphic, weight − 2 modular ∞ Y 1 function P (z) = q−1/24 , where q = q(z) = e2πiz. Unfortunately, this function has 1 − qn n=1 a a pole at “i∞” and at every rational number b ∈ Q ⊆ C.

0.2 Quadratic Forms

Let v ∈ N and deﬁne

v 2 2 rv(n) = #{(x1, . . . , xv) ∈ Z | x1 + ... + xv = n}. This is a classical function with many old results known, including: Theorem 0.2.1 (Gauss). Every nonnegative integer is the sum of 3 triangular numbers, i.e. m(m+1) those of the form 2 for some m.

This√ statement reduces to studying r3(8n + 3), namely Gauss showed that r3(8n + 3) = αh(Q( −8n − 3)) for some α ≤ 1, where h is the class number. Theorem 0.2.2 (Lagrange). Every nonnegative integer is the sum of four squares. Theorem 0.2.3 (Landau). For all n ≥ 1, X r4(n) = 8 d. 1≤d|n 4-d These results about quadratic forms can be interpreted as statements about the holomorphic weight v/2 modular form

∞ X ||x¯||2 X n θv(z) = q = rv(n)q . x¯∈Zv n=0 Unlike P (z) in the last section, this function has no poles at i∞ or any rational number, a and in fact the values of θv(z) as z → b ∈ Q encode important data about values of ζ- and L-functions.

3 0.3 Elliptic Curves 0 The Web of Modularity

0.3 Elliptic Curves

An elliptic curve is an algebraic curve E given by an aﬃne model (or Weierstrass equation) y2 = x3 + Ax + B where x3 + Ax + B has no repeated roots. A standard problem in number theory involves counting the rational (or integral) points of E.

Theorem 0.3.1 (Mordell-Weil). If K is a number ﬁeld and E is an elliptic curve, then E(K) is a ﬁnitely generated abelian group.

∼ r(E,K) In particular, E(K) = E(K)tors × Z where E(K)tors is a ﬁnite abelian group and r(E,K) ≥ 0 is the rank of E(K). The fundamental question in the theory of elliptic curves is: what values of r(E,K) are possible?

Example 0.3.2. Consider the elliptic curve E : y2 = x3 − x over Q. Then (0, 0), (1, 0) and (−1, 0) are all order 2 points of E, so E(Q) contains a subgroup isomorphic to Z/2Z×Z/2Z. Deﬁne a sequence a(n) = aE(n) by

∞ ∞ X Y a(n)qn = q (1 − q4n)2(1 − q8n)2. n=1 n=1

If p is an odd prime, then a(p) = p − #E(Fp) and in fact, a general theorem of Coates-Wiles implies that r(E, Q) = 0 in this case since the ﬁrst moment of the generating function above is nonzero. The last condition can be expressed as the nonvanishing of an integral of a certain cusp form associated to the elliptic curve. This is a special case of Wiles-Taylor’s Modularity Theorem:

Theorem 0.3.3 (Modularity). If E/Q is an elliptic curve, then there is a sequence a(n) = aE(n) such that if p is a prime of good reduction for E, then a(p) = p − #E(Fp), and the function ∞ X n fE(z) = a(n)q n=1 is a weight 2 cusp form.

0.4 Galois Theory and Quantum Gravity

Klein’s j-function is deﬁned on the upper half-plane by

P∞ P 3 n 1 + 240 n=1 d|n d q j(z) = = q−1 + 744 + 196884q + ... Q∞ n 24 q n=1(1 − q ) This has important connections to algebraic number theory and Galois theory which can be summarized in the ﬁrst fundamental theorem of complex multiplication:

Theorem 0.4.1. j(z) is a weakly holomorphic function of weight 0 such that:

4 0.5 Some Perspectives on Modular Forms 0 The Web of Modularity

2 2 (1) If az + bz + c = 0 where a, b, c ∈ Z such that√ b − 4ac < 0 and z ∈ h, then j(z) is an 2 algebraic√ integer with the property that Q( b − 4ac, j(z)) is the Hilbert class ﬁeld of Q( b2 − 4ac). (2) Further, if im z is large, then diophantine approximation exists for j(z) and it can be used to give bounds on Gauss sums.

(3) The regular representation theory of the monster group M encodes certain information about counting states in the theory of 3-dimensional quantum gravity, by way of the j-function.

The content of (3) goes by the name Monstrous Moonshine, originally developed by Borcherds, who won the Fields Medal in 1998 in part for this work.

0.5 Some Perspectives on Modular Forms

Modular forms can be viewed from several diﬀerent perspectives in math, including:

P n (1) as generating functions n a(n)q , whose coeﬃcients a(n) turn out to be important sequences in number theory and combinatorics (e.g. the partition function, the representation numbers for quadratic forms);

(2) as complex functions f : h → C, whose analytic properties encode deep number-theoretic information otherwise inaccessible by elementary methods. For example, the modular forms η(z), ∆(z) and j(z) know a lot about algebraic number theory, by way of class ﬁeld theory, elliptic curves with complex multiplication, etc.;

(3) in relation to L-functions and zeta functions, speciﬁcally by encoding their analytic continuations;

(4) as holomorphic sections of certain line bundles on the modular curves X(Γ) for congruence subgroups Γ ≤ SL2(Z);

5 1 Theta Functions

1 Theta Functions

Theta functions are some of the simplest examples of modular forms. In fact, we already saw an example of a theta function in Section 0.2. They naturally arise by summing over a lattice, so they are easy to construct and understand as complex functions. In the next section, we recall the Poisson summation formula from complex analysis.

1.1 Poisson Summation

2 R 2 Deﬁnition. Suppose f : >0 → is an L -function, i.e. |f| < ∞. Then f is a R C R Schwartz function if it decays rapidly as x → ±∞, i.e. f and any of its derivatives f (n) decay to 0 as x → ±∞ faster than any inverse power of x.

Definition. For a complex-valued function f ∈ L1(R), the Fourier transform of f is defined by Z fˆ(y) = f(x)e−2πixy dx. R Proposition 1.1.1 (Poisson Summation). Let f be a Schwartz function. Then X X f(n) = fˆ(n). n∈Z n∈Z Proof. Set F (x) = P f(x+n) which converges since f(x) decays rapidly as |x| gets large. n∈Z Then F (x) is 1-periodic, so it has a Fourier series with kth Fourier coefficient given by

Z 1 X −2πikx ak = f(x + n)e dx. 0 n∈Z Since f is Schwartz, Fubini’s theorem allows us to swap the order of integration and summation: Z 1 Z n+1 Z X −2πikx X −2πikx −2πikx ak = f(x + n)e dx = f(x)e dx = f(x)e dx. 0 n n∈Z n∈Z R ˆ ˆ (In the last step we use periodicity.) Thus ak = f(k) where f is the Fourier transform of f. Now since F is analytic (it is even Schwartz), it equals its Fourier series on R:

X 2πikx X ˆ 2πikx F (x) = ake = f(k)e . k∈Z k∈Z Plugging in x = 0 gives the result.

Proposition 1.1.2. If f(x) = e−πx2 then fˆ(y) = f(y).

6 1.1 Poisson Summation 1 Theta Functions

Proof. For any y, Z Z fˆ(y) = e−πx2 e−2πixy dx = e−π(x2+2ixy) dx ZR R = e−π(x+iy)2 e−πy2 dx by completing the square R Z = e−πy2 e−π(x+iy)2 dx. R So it’s enough to show that R e−π(x+iy)2 dx = 1. Now the change of variables u = x + iy R gives us Z Z e−π(x+iy)2 dx = e−πu2 du. R iy+R Since e−πu2 is an entire function and decays rapidly as |Re(u)| gets large, the contour integral along the vertical pieces in the contour

iy + R

tend to 0 as they move outward, and thus the integrals along R and along iy + R are equal. Then by a standard computation, Z Z e−πu2 du = e−πu2 du = 1. iy+R R

In other words, the function f(x) = e−πx2 is a ﬁxed point of the Fourier transform operator. By the same proof, we also have:

2 −πx a ˆ √1 1 Proposition 1.1.3. For any a > 0, fa(x) = e satisﬁes f(y) = a f a . Example 1.1.4. Let θ : → by the function θ(x) = P e−πn2x. Then the Poisson R>0 C n∈Z summation formula shows that for all x > 0, 1 1 θ(x) = √ θ . x x

2 \−πn2x √1 −πn x This follows from Proposition 1.1.3: e = x e . The formula above is a basic example of a modularity property for θ, and in fact is useful in the solutions to the sphere packing problem in dimensions 16 and 24.

7 1.2 The Jacobi Theta Function 1 Theta Functions

Example 1.1.5. Let h = {z ∈ C | im z > 0} be the complex upper half-plane. Deﬁne a function θe : h → C by X 2 θe(z) = eπin z. n∈Z Then θe satisﬁes: (1) θe(z + 2) = θe(z) for any z ∈ h. That is, θe is 2-periodic.

√ 1 1 (2) θe(z) = −iz θe − z for any z ∈ h.

This mean θe is our ﬁrst example of a modular form on h.

1.2 The Jacobi Theta Function

Deﬁnition. The Jacobi theta function is the function

X 2 θ(z; τ) = eπin τ+2πin(z+1/2) 1 n∈ 2 Z for any z ∈ C and τ ∈ h. We call z the elliptic variable and τ the modular variable. We can think of θ(z; τ) as a ‘master theta function’ since we know a lot about the transformation properties of θ(z; τ) for any inputs, and plugging in certain values for z or τ recovers more concrete theta functions.

Theorem 1.2.1. For any (z, τ) ∈ C × h, (1) θ(z + 1; τ) = −θ(z; τ). (2) θ(z + τ; τ) = −e−πiτ−2πizθ(z; τ). (3) θ(z; τ + 1) = eπi/4θ(z; τ). √ z 1 (4) θ τ ; − τ = −i −iτθ(z; τ). Proof. (1) follows from Euler’s identity eπi = −1. (2) Distribute τ in the second part of the exponent. (3) Factor out the eπi/4. (4) follows from√ Poisson summation (Proposition 1.1.1), with a bit of√ careful manipula- tion. Here, we take z to be the unique complex function deﬁned by + x on the real axis and extended by analytic continuation. The Jacobi theta function has many applications in number theory, such as the following.

Theorem 1.2.2 (Jacobi Triple Product). For all (z, τ) ∈ C × h, ∞ Y θ(z; τ) = −iq1/8ζ−1/2 (1 − qn)(1 − ζqn)(1 − ζ−1qn−1) n=1 where q = q(τ) = e2πiτ and ζ = ζ(z) = e2πiz.

8 1.2 The Jacobi Theta Function 1 Theta Functions

Remark. The Jacobi triple product takes many diﬀerent forms in math. For example, in combinatorics, it can be written

∞ ∞ X 2 Y (−1)nanqn = (1 − qn)(1 − aqn)(1 − a−1qn−1). n=−∞ n=1

This has an elegant proof, but it is not easy using just combinatorics. In another direction, the individual factors of the triple product can be interpreted in terms of heights on elliptic curves; for example, the product of the (1 − ζqn)(1 − ζ−1qn−1) terms comes from the Weier- Q∞ n strass σ-function. On the other hand, n=1(1−q ) is tied to the fact that all modular forms on h of level SL2(Z) are generated algebraically by the Dedekind eta function:

∞ Y η(τ) = q1/24 (1 − qn). n=1 Euler’s product formula for the partition function (Lemma 0.1.1) can then be interpreted as

∞ X 1 p(n)qn−1/24 = . η(τ) n=0 Proposition 1.2.3. For any τ ∈ h,

1 1 η(τ + 1) = eπi/12η(τ) and η(τ) = √ η − . −iτ τ

Proof. This can be shown using Poisson summation (Proposition 1.1.1), but it follows more easily from Theorem 1.2.1(4), replacing q with q3 and ζ with q.

Corollary 1.2.4. The Dedekind eta function η(τ) satisﬁes the following identities.

(1) For all τ ∈ h, ∞ ∞ Y X 12 2 η(24τ) = q (1 − q24n) = qn n n=1 n=1 12 holds, where n is the Jacobi symbol deﬁned by  0, (12, n) 6= 1 12  = 1, n ≡ ±1 (mod 12) n −1, n ≡ ±5 (mod 12).

(2) For all τ ∈ h,

∞ ∞ Y X 2 η(8τ)3 = q (1 − q8n)3 = (−1)k(2n + 1)q(2n+1) . n=1 n=0

9 1.2 The Jacobi Theta Function 1 Theta Functions

(3) For all τ ∈ h,

5 η(2τ) X 2 = qn = 1 + 2q + 2q4 + 2q9 + 2q16 + ... η(τ)2η(4τ)2 n∈Z

η(2τ)5 In particular, the quotient η(τ)2η(4τ)2 is always congruent to 1 mod 2. (4) For all τ ∈ h,

2 η(τ) X 2 = (−1)nqn = 1 − 2q + 2q4 − 2q9 + 2q16 − ... η(2τ) n∈Z which is also always congruent to 1 mod 2.

(5) For all τ ∈ h, 2 ∞ η(16τ) X 2 = q(2n+1) . η(8τ) n=0

Example 1.2.5. Suppose λ = (λ1, . . . , λt) is a partition of n ≥ 1. From representation theory, we know there is an irreducible representation ρλ : Sn → GL(Vλ) of the symmetric group Sn corresponding to λ (which is unique up to conjugation). A classical result of Frame-Robinson-Thrall says that n! dimC Vλ = Q i,j hi,j

where hi,j are the hook numbers for λ. For a prime p, the partition λ is called a p-core if p does not divide any hook number hi,j for λ. By Brauer’s theorem in representation theory,

ρλ is irreducible mod p if and only if p divides both dimC Vλ and n! = |Sn| to the same power. As a consequence, the representation ρλ of Sn is irreducible mod p if and only if λ is a p-core.

Theorem 1.2.6. If p is prime and cp(n) is the number of p-core partitions of n for each n ≥ 1, then ∞ ∞ X Y (1 − qpn)p c (n)qn = . p (1 − qn) n=0 n=1 Moreover, there is an exact formula

( p−1 p−1 ) X p X c (n) = # (x , . . . , x ) ∈ p x = 0, (x2 + ... + x2 ) + ix = n . p 0 p−1 Z i 2 0 p−1 i i=0 i=0 Note that the formula for p = 2 was already given in Corollary 1.2.4(5) but this combinatorial formula proves that ρλ is irreducible mod 2 if and only if λ is of the form (1, 2, . . . , k). Theorem 1.2.7 (Brauer’s Problem 19; Granville-Ono). If G is a ﬁnite simple group then for all primes p ≥ 5, there exist a representation ρ : G → GL(V ) which is irreducible mod p.

10 1.2 The Jacobi Theta Function 1 Theta Functions

Example 1.2.8. For p = 5, the formula in Theorem 1.2.6 becomes

∞ ∞ X η(5τ)5 X X d n2 c (n)qn+1 = = qn. 5 η(τ) 5 d2 n=0 n=1 d|n

For p ≥ 7, the formulas become harder and proving Brauer’s Problem 19 requires the Weil Conjectures.

Here’s another important application of theta functions.

Theorem 1.2.9. Let Q(x1, . . . , xk) be a Z-positive deﬁnite quadratic form. Then the theta function for Q, deﬁned by X Q(¯x) θQ(τ) := q , x¯∈Zk is a weight k/2 modular form.

We will explore this interpretation of theta functions further when we look at L-functions.

11 2 Modular Groups

2 Modular Groups

2.1 The Modular Group SL2(Z)

Deﬁnition. The group Γ = SL2(Z) is called the modular group.

Lemma 2.1.1. Let Γ = SL2(Z) be the modular group. Then (a) Γ is generated by two matrices

0 −1 1 1 S = and T = . 1 0 0 1

(b) S has order 4 and T has inﬁnite order in Γ.

Deﬁnition. The upper half-plane in the complex plane C is the open half-plane

h = {z ∈ C | Im(z) > 0} equipped with the subspace topology. The completed upper half-plane is the set

∗ h = h ∪ {∞} ∪ Q equipped with the topology coming from taking open sets about ∞ (identiﬁed as i∞) to be half-planes {z ∈ C | Im(z) > y0 > 0} and viewing Q as a subset of the real axis in C.

The group SL2(Z) acts on C by fractional linear transformations: a b az + b z = . c d cz + d Note that a b Im(z) Im z = . c d |cz + d|2

This shows that SL2(Z) acts on h.

2.2 Congruence Subgroups

We distinguish several important subgroups of Γ = SL2(Z). Deﬁnition. Fix an integer N ≥ 1. Then the level N modular group is the subgroup Γ(N) ≤ SL2(Z) deﬁned by a b a b 1 0 Γ(N) = ∈ SL ( ): ≡ mod N . c d 2 Z c d 0 1

∗ ∗ A subgroup Γ ≤ SL2(Z) is a congruence subgroup of level N if Γ(N) ≤ Γ . Lemma 2.2.1. For N ≥ 1,

12 2.2 Congruence Subgroups 2 Modular Groups

(a) Γ(N) is a normal subgroup of SL2(Z). Y 1 (b) |SL ( /N )| = [SL ( ) : Γ(N)] = N 3 1 − where the product is over all 2 Z Z 2 Z p2 p|N primes p | N.

Proof. The reduction homomorphism ϕN : SL2(Z) → SL2(Z/NZ) is surjective with ker- ∼ nel Γ(N), so Γ(N) is a normal subgroup. Further, this also shows that SL2(Z/NZ) = Qk mi SL2(Z)/Γ(N). Then to prove (2), write N = i=1 pi for distinct primes pi. Then by the Chinese remainder theorem,

k ∼ Y mi SL2(Z/NZ) = SL2(Z/pi Z). i=1

m 3m 1 Therefore it suﬃces to observe that |SL2(Z/p Z)| = p 1 − p2 for any prime p and m ≥ 1.

Example 2.2.2. When N = 1, Γ(1) = Γ0(1) = Γ1(1) = SL2(Z). Example 2.2.3. For each N ≥ 1, we distinguish two Hecke subgroups of level N:

a b a b a b Γ (N) = ∈ SL ( ): ≡ mod N 0 c d 2 Z c d 0 d a b a b 1 b and Γ (N) = ∈ SL ( ): ≡ mod N . 1 c d 2 Z c d 0 1

Note that Γ1(N) ≤ Γ0(N). One can think of Γ0(N) as the subgroup of “upper triangular matrices mod N” and Γ1(N) as the “unipotent matrices mod N”.

The arithmetic captured by Γ(N) ⊂ Γ1(N) ⊂ Γ0(N) ⊂ SL2(Z) as N varies is incredibly rich. For example, the p-core problem from Theorem 1.2.6 can be studied using Γ0(p). On the other hand, modular elliptic curves can be parametrized (they form a moduli space) using Γ0(NE) where NE is the conductor of an elliptic curve. Lemma 2.2.4. For N ≥ 1,

(a) Γ1(N) ⊂ Γ0(N) are subgroups of SL2(Z). Y 1 (b) [SL ( ):Γ (N)] = N 1 + . 2 Z 0 p p|N

Y 1 (c) [SL ( ):Γ (N)] = N 2 1 − . 2 Z 1 p2 p|N

(d) The action of SL2(Z) on h restricts to well-deﬁned, faithful actions of each Γ(N), Γ0(N) and Γ1(N) on h.

13 2.2 Congruence Subgroups 2 Modular Groups

Proof. (a) and (d) are routine. a b (b) Deﬁne a homomorphism ψ :Γ (N) → ( /N )×, 7→ d (mod N). Then 0,N 0 Z Z c d ψ0,N is surjective with kernel Γ(N) so

[SL2(Z) : Γ(N)] [SL2(Z) : Γ(N)] [SL2(Z):Γ0(N)] = = [Γ0(N) : Γ(N)] φ(N) where φ is Euler’s totient function. This implies the formula in (2). a b (c) Deﬁne a homomorphism ψ :Γ (N) → /N , 7→ b (mod N). Then ψ 1,N 1 Z Z c d 1,N is surjective with kernel Γ(N) so

[SL2(Z) : Γ(N)] [SL2(Z) : Γ(N)] [SL2(Z):Γ1(N)] = = . [Γ1(N) : Γ(N)] N

∗ For the rest of the chapter, let Γ be a level N congruence subgroup of Γ = SL2(Z).

∗ Deﬁnition. A fundamental domain of Γ is a connected set DΓ∗ satisfying:

0 ∗ 0 (1) For every τ ∈ h there exist τ ∈ DΓ∗ and γ ∈ Γ such that γτ = τ.

∗ (2) If τ1 6= τ2 lie in DΓ∗ then for any γ ∈ Γ , γτ1 = τ2.

1 Example 2.2.5. We claim the region D = z ∈ h : |z| ≥ 1, |Re(z)| ≤ 2 is a fundamental domain for Γ = SL2(Z).

ρ i −ρ¯

Re(z) 1 1 −1 − 2 2 1

Im(z)

We specify three points on the boundary of D: the fourth root of unity i = eiπ, the third root of unity ρ = e2πi/3 and its negative conjugate, the sixth root of unity −ρ¯ = eπi/3. The 1 generators S,T ∈ SL2(Z) act on z ∈ h by S(z) = − z and T (z) = z + 1. To show D is a fundamental domain, we must show that for any z ∈ D and each nontrivial element g ∈ Γ,

14 2.3 Modular Curves 2 Modular Groups

a b gz 6∈ D. Suppose g = and without loss of generality assume Im(gz) ≥ Im(z). Then c d 1 b |cz + d| ≤ 1 which means either c = 0 or c = ±1. If c = 0, g = and gz = z + b with 0 1 b ∈ Z, so gz 6∈ D. If c = −1, we may multiply by −I to get to the c = 1 case. Finally, for c = 1, |z + d| > 1 holds unless d = 0 or z = ρ, −ρ¯, in which case z ∈ ∂D. Hence D is a fundamental domain for Γ.

2.3 Modular Curves

The quotient space Y (Γ∗) := h/Γ∗, called the incomplete modular curve associated to Γ∗, admits the structure of an open Riemann surface, that is, a complex curve of genus g with some positive number of punctures. To obtain a complete modular curve (i.e. a compact Riemann surface), we deﬁne the following points.

∗ ∗ Deﬁnition. The set of cusps for a congruence subgroup Γ ≤ SL2(Z) is the set of Γ -orbits of P1(Q) = Q ∪ ∞ in h∗ = h ∪ Q ∪ {∞}. Lemma 2.3.1. For any congruence subgroup Γ∗, the number of cusps of Γ∗ is ﬁnite.

∗ Proof. This follows from the fact that [SL2(Z):Γ ] < ∞.

Example 2.3.2. The equivalence class of ∞ is the only cusp for Γ = SL2(Z).

Example 2.3.3. If p is prime, then Γ0(p) has two cusps corresponding to the orbits of 0 and ∞ in h∗.

Deﬁnition. The complete modular curve associated to a congruence subgroup Γ∗ ≤ ∗ ∗ ∗ ∗ SL2(Z) is the quotient space X(Γ ) := h /Γ = h/Γ ∪ {cusps}. It can be shown that X(Γ∗) is a compact Riemann surface, so it has a genus g = g(X(Γ∗)), a hyperbolic metric (if g ≥ 2), a well-understood cohomology theory, etc. For the principal congruence subgroups Γ(N), Γ0(N) and Γ1(N), we write their corresponding incomplete and complete modular curves by Y (N),Y0(N),Y1(N) and X(N),X0(N),X1(N), respectively.

Example 2.3.4. For Γ = SL ( ), X(Γ) = X(1) = X (1) = X (1) ∼ 1 , the Riemann 2 Z 0 1 = PC sphere. It has genus 0 and one cusp corresponding to the point at ∞.

Example 2.3.5. For Γ0(4), the complete modular curve X0(4) has genus 0 with 3 cusps.

Example 2.3.6. The modular curve X0(11) has genus 1 and 2 cusps, so it is an elliptic curve. In a precise sense it is the ‘simplest’ modular elliptic curve over C.

Example 2.3.7. The modular curve X0(32) has genus 1 and can be given by the aﬃne model y2 = x3 − x.

15 2.3 Modular Curves 2 Modular Groups

As N → ∞, the genus of X0(N) tends to get large, but the Modularity Theorem says all elliptic curves are modular curves so where do they come from? It turns out that many modular elliptic curves can be obtained as factors of the Jacobian of X0(N), an abelian variety denoted by J0(N).

∗ Lemma 2.3.8. For Γ = PSL2(Z), the only nontrivial stabilizers of the action on h are Γ∗(i) = hSi and Γ∗(ρ) = hST i which are ﬁnite groups of respective orders 2 and 3.

Alternatively, one can describe all the stabilizer subgroups of the action of Γ = SL2(Z) on h by:  h±Si, z = i  Γ(z) = h±ST i, z = ρ h−Ii, z 6∈ {i, ρ}.

Definition. A point in h which is in the SL2(Z)-orbit of i or ρ is called an elliptic point. 1 The period of an elliptic point z is defined to be 2 |Γ(z)|. It turns out that the topology of the upper half-plane h is insufficient to study modular forms. The correct topology comes from studying the following PSL2(Z)-invariant metric. Recall the hyperbolic trig functions

ex − e−x ex + e−x cosh(x) = and sinh(x) = . 2 2

1 Up to normalization, these are the generalized Bessel functions of the first kind of index 2 3 and 2 , respectively. Theorem 2.3.9 (PoincaréMetric). Let z, w ∈ h. Then |z − w|2 (a) The formula d(z, w) := cosh−1 1 + defines a metric on h. 2 im z im w

(b) For any γ ∈ SL2(R), d(γz, γw) = d(z, w).

∗ (c) For any congruence subgroup Γ ≤ SL2(Z), the SL2(R)-invariant measure induced by ∗ dx dy d on h/Γ is given by dµ = y2 where z = x + iy. This allows us to endow spaces of modular forms with inner products, e.g. if f and g are modular forms of weight k then Z dx dy hf, gi := f(z)g(z) y2−k DΓ∗ is an inner product, called the Petersson inner product.

16 3 Modular Forms

3 Modular Forms

3.1 Deﬁnitions and Examples

∗ Deﬁnition. Let k ∈ Z and suppose Γ ≤ SL2(Z) is a congruence subgroup. A holomorphic function f : h → C is a weakly modular function of weight k of level Γ∗ if for all a b g = ∈ Γ∗, c d f(z) = (cz + d)−kf(gz). ∗ When Γ = SL2(Z), these will just be called weakly modular functions (of weight k).

Lemma 3.1.1. A holomorphic function f : h → C is weakly modular of weight k if and only if 1 f(z + 1) = f(z) and f − = zkf(z) z for all z ∈ h.

As a consequence of the ﬁrst relation, a weakly modular function f(z) has a Fourier series expansion in the variable q = e2πiz:

∞ X n f(q) = anq . n=−∞

Identifying z = i∞ with q = 0, we can think of this as a power series expansion of f about the point at inﬁnity.

P n Deﬁnition. Let f(z) be a holomorphic function on C with q-expansion f = anq . If an = 0 for all n << 0, then f is said to be meromorphic at ∞. If an = 0 for all n < 0, then f is said to be holomorphic at ∞.

Deﬁnition. A weakly modular function f (of weight k) is a modular function (of weight k) if it is meromorphic at ∞ and a modular form (of weight k) if it is holomorphic at ∞. Further, if a0 = 0 in its q-expansion, then f is called a cusp form.

For each k ∈ Z, let Mk be the complex vector space of modular forms of weight k. Then if f ∈ Mk and g ∈ M`, it is easy to see that fg ∈ Mk+`, so these spaces form a graded ring of modular forms M M = Mk. k∈Z

For each k, let Sk be the space of cusp forms of weight k. We will see that Mk and Sk are ﬁnite dimensional complex vector spaces. However, the space of all weakly modular ! functions of weight k, sometimes written Mk, is inﬁnite dimensional in general. Remark. A further generalization of modular functions and forms are called Maass forms, which are real analytic functions f : h∗ → C that are eigenfunctions of a Laplacian operator.

17 3.1 Deﬁnitions and Examples 3 Modular Forms

The spaces Mk and Sk have a rich algebraic structure. For example, they are spanned by many diﬀerent objects, including:

theta series;

Eisenstein series;

Poincar´eseries;

for Sk, the collection of cusp forms coming from Galois representations.

∗ To deﬁne modular and cusp forms for diﬀerent congruence subgroups Γ ≤ SL2(Z), we need a notion of Fourier expansion at the cusps of Γ∗. Recall that for z ∈ h∗, the subgroup ∗ ∗ Γ (z) ≤ Γ is called the isotropy subgroup of z. Because SL2(Z) has only one cusp, ∞, the set of cusps of any congruence subgroup Γ∗ is in one-to-one correspondence with the ∗ coset space SL2(Z)/Γ . As a consequence, every Fourier expansion can be exhibited as an ∗ expansion at ∞ after shifting by some [γ] ∈ SL2(Z)/Γ . This is made precise in the following lemma.

∗ Lemma 3.1.2. Let Γ ≤ SL2(Z) be a congruence subgroup. Then

(a) Γ∗(∞) = {T n | n ∈ Z}.

(b) If f is a weakly modular function for SL2(Z), then f has a Fourier expansion

∞ X f(z) = a(n)qn n≥−N

2πiz where q = q(z) = e and N = ord∞(f) is the order of vanishing of f at the cusp ∞.

∗ ∗ ∗ (c) If f is a weakly modular function for Γ 6≤ SL2(Z) and z0 ∈ h is a cusp for Γ , then f has a Fourier expansion about z0 given by

∞ X f(z) = a(n)qn/a n≥−N

where γ ∈ SL2(Z) such that γz0 = ∞, N = ord∞(f|γ) and a is the minimal positive integer such that T a ∈ γ−1Γ∗γ.

∗ Deﬁnition. Let f : h → C be a holomorphic function and Γ ≤ SL2(Z) a congruence subgroup. Then f is a modular form of weight k for Γ∗ if

(1) f(z) is weakly modular for Γ∗.

(2) f(z) is holomorphic at the cusps of Γ∗, i.e. for all γ ∈ Γ∗ taking ∞ to a cusp ∗ z0 = γ∞ ∈ h , f(γz) is holomorphic at ∞. This is equivalent to saying a(n) = 0 for all n < 0 in the Fourier expansion above.

18 3.1 Deﬁnitions and Examples 3 Modular Forms

∗ A cusp form of weight k for Γ is a modular form which vanishes at every cusp z0 = γ∞ of ∗ ∗ ∗ Γ . Write Mk(Γ ) and Sk(Γ ) for the spaces of modular forms and cusp forms, respectively, of weight k for Γ∗. ∗ Deﬁnition. When Γ = Γ0(N), we write Mk(N) and Sk(N) for the spaces of modular forms and cusp forms, respectively, of weight k for Γ0(N). Such an f(z) ∈ Mk(N) (resp. Sk(N)) is called a modular form (resp. cusp form) of level N. 1/24 Q∞ n Example 3.1.3. Given Dedekind’s eta function η(z) = q n=1(1 − q ), deﬁne a function η(z)5 f(z) = . η(5z)

Using Proposition 1.2.3, one can show that f ∈ M2(5). Moreover, it is easy to see from the product forms of η(z)5 and η(5z) that f(q) = 1 + 5F (q) where F ∈ Z[[q]]. In particular, f(q) ≡ 1 (mod 5), so f cannot be a cusp form of level 5. On the other hand, a similar analysis of the Fourier expansion of f about the cusp 0 shows that f(0) = 0. One can even deduce an exact formula for all the coeﬃcients of f at these cusps using formulas like the one in Example 1.2.8. Lemma 3.1.4. Let k ∈ Z. Then ! (a) If k is odd, then Mk = 0. ! (b) If k ≡ 2 (mod 4), then for every f ∈ Mk, f(i) = 0. ! (c) If k 6≡ 0 (mod 3), then for every f ∈ Mk, f(ρ) = 0. Remark. One can interpret (b) and (c) in the theory of elliptic curves as saying that √ j(OQ(i)) = 1728 and j(OQ( −3)) = 0. We will come back to this interpretation later. Example 3.1.5. (Eisenstein series) From complex analysis, we know that if Λ ⊂ C is a lattice then the sum X 1 |γ|σ γ∈Λr{0} converges for all σ ∈ C with Re(σ) > 2. Using this, for each even k ≥ 2 we can deﬁne a function Gk : h → C by X 1 G (z) = . k (mz + n)2k 2 (m,n)∈Z (m,n)6=(0,0)

By construction, Gk is a weakly modular function of weight k. (To see that Gk converges uniformly on h, ﬁrst observe that it converges uniformly on the fundamental domain D since for any z ∈ D, |mz + n| is bounded below by |mρ − n|. Now extend this convergence to all of h by applying the action of SL2(Z) and the modular condition.) What happens at inﬁnity? Viewing ∞ = i∞, it is enough to consider the limit of Gk(z) as z → ∞ within D, but since the series Gk converges uniformly on D, we may take the limit term-by-term to get X 1 Gk(∞) = lim Gk(z) = = 2ζ(k). Im(z)→∞ nk n∈Zr{0}

19 3.1 Deﬁnitions and Examples 3 Modular Forms

This shows that Gk is holomorphic at ∞, so Gk is in fact a holomorphic form of weight k. Note that when k = 2, the sum

X 1 G (z) = 2 (mz + n)2 (m,n)6=(0,0)

converges conditionally but not uniformly on h, so G2 is not a modular form. The Eisenstein series have important q-expansions:

∞ 2(−2πi)k X G (z) = 2ζ(k) + σ (n)qn k (k − 1)! k−1 n=1

P k where σk(n) = d|n d is the generalized divisor sum function. To prove this, consider the well-known formula ∞ Y z2 sin z = z 1 − . n2π2 n=1 Taking the logarithmic derivative yields

∞ X z2 z cot z = 1 + 2 . z2 − n2π2 n=1 Evaluating at πz and dividing out by z, we have two equivalent expressions for π cot(πz):

∞ 1 X 1 1 π cot(πz) = + + z z + m z − m m=1 ∞ 2πi X and π cot(πz) = πi − = πi − 2πi qn. 1 − e2πiz n=0 Equating these two expressions and taking the kth derivative with respect to z yields the following formula: ∞ X 1 (−2πi)k X = dk−1qd. (m + z)k (k − 1)! m∈Z d=1

This implies the q-expansion of Gk(z) stated above. We also deﬁne the normalized Eisenstein series Ek(z) by

∞ 1 2k X E (z) := G (z) = 1 − σ (n)qn k 2ζ(k) k B k−1 k n=1

20 3.1 Deﬁnitions and Examples 3 Modular Forms

where Bi is the ith Bernoulli number. The ﬁrst few normalized Eisenstein series are: ∞ X n 2 3 E4(z) = 1 + 240 σ3(n)q = 1 + 240q + 2160q + 67200q + ... n=1 ∞ X n 2 3 E6(z) = 1 − 504 σ5(n)q = 1 − 504q − 16632q − 122976q − ... n=1 ∞ X n 2 3 E8(z) = 1 + 480 σ7(n)q = 1 + 480q + 61920q + 1030240q + ... n=1 ∞ X n 2 3 E10(z) = 1 − 264 σ9(n)q = 1 − 264q − 135432q − 5196576q − .... n=1 One can prove the following surprising identities:

2 E4 = E8,E4E6 = E10,E4E10 = E7,E6E8 = E14. Comparing the q-expansions of these identities, we obtain the following interesting relations among the generalized divisor sum functions:

n−1 X σ7(n) = σ3(n) + 120 σ3(n)σ3(n − m) m=1 n−1 X 11σ9(n) = 21σ5(n) − 10σ3(n) + 5040 σ3(n)σ5(n − m). m=1 While these are small miracles, it turns out there are only ﬁnitely many pairs k, ` satisfying Ek(z)E`(z) = Ek+`(z) so one should not expect formulas like these in general.

Example 3.1.6. Although E2(z) is not a modular form, it does arise as a p-adic modular form for all primes p (see below). On the other hand, there is a “completed” Eisenstein series ∗ of weight 2, given by G2(z) = G2(z) + h(z) for some non-holomorphic h, which satisﬁes 1 G∗ − = z2G∗(z) and G∗(z + 1) = G∗(z). 2 z 2 2 2

3 In fact, h can be deﬁned by h(z) = − π im z . Proposition 3.1.7. Let k ≥ 4 be even. Then

(a) Ek(z) ≡ 1 (mod 24).

(b) If p is prime and p − 1 divides k, then Ek(z) ≡ 1 (mod p). Moreover, if k ≡ ` m m+1 (mod (p − 1)p ) for some m ≥ 0, then Ek(z) ≡ E`(z) (mod p ). Proof. (a) follows from Von Staudt’s theorem that 6 divides the denominator of every even Bernoulli number Bk. Q (b) follows from the formula for the denominator of Bk, which is: p||k(p − 1) where p || k denotes any prime that divides k exactly (or a prime p | k such that p2 - k).

21 3.1 Deﬁnitions and Examples 3 Modular Forms

Although it’s not a modular form, E2(z) satisﬁes the congruence E2(z) ≡ Ep+1(z) (mod p) for any odd prime p. This follows from Proposition 3.1.7(b) and Fermat’s Lit- tle Theorem. This again hints at the fact that E2(z) is a p-adic modular form for all odd primes p.

∗ Example 3.1.8. Suppose Γ is one of the congruence subgroups Γ(N), Γ0(N) or Γ1(N) and ∗ let τ be a cusp for Γ . Then for every k ≥ 2, there is an Eisenstein series Gk(τ; z) deﬁned as follows. If τ = ∞, then X 1 G (∞; z) := k j(γ; z)k γ∈Γ∗/Γ∗(∞) a b where j(γ; z) = cz + d if γ = . Then when k ≥ 4, G (∞; z) is a modular form of c d k weight k for Γ∗. Moreover, there is a Fourier expansion

∞ X ∗ n Gk(∞; z) = c(∞) + σk−1(n)q n=1

∗ where c(∞) is a constant depending on a value of a ζ-function and σi is a divisor sum ∗ ∗ function depending on Γ . For an arbitrary cusp τ of Γ , we deﬁne Gk(τ; z) by the Fourier −1 ∗ expansion of Gk(∞; z)|kγ for the subgroup γ Γ γ, where γ ∈ SL2(Z) satisﬁes γτ = ∞. Theorem 3.1.9. Fix k ≥ 4 even. Then for any congruence subgroup Γ∗, the set of Eisenstein ∗ series {Gk(τ; z) | τ is a cusp for Γ } is linearly independent.

∗ ∗ Corollary 3.1.10. For any congruence subgroup Γ and k ≥ 4 even, dim Mk(Γ ) is at least the number of cusps of Γ∗.

Example 3.1.11. (Theta series) Recall the function θ(z) = P qn2z/2 from Example 1.1.5. e n∈Z 1 The properties (1) and (2) from that example show that θe(z) is an example of a weight 2 holomorphic modular form. In fact, θ(z) := θe(2z) ∈ M1/2(Γ0(4)). Further, if a ≥ 1, then

∞ 4a X n θ(z) = r4a(n)q ∈ M2a(Γ0(4)) n=0

so θ(z)4a is a modular form of even integer weight. Odd integer weight modular forms also 2a 4 exist, e.g. θ(z) . When a = 1 for example, θ(z) = E2(z), the Eisenstein series of weight 2 from Example 3.1.5. Though when a ≥ 3, θ(z)4a is not purely Eisenstein – there are “error terms” coming from coefficients of cusp forms in S2a(Γ0(4)). Example 3.1.12. (Poincaréseries) In the definition of the Eisenstein series, the factor (cz+d)k that appears in the denominator of the summation can be regarded as an automorphy a b factor for g = . Then G (z) is really an average of the automorphy factors (cz +d)−k c d k over the coset space SL2(Z)/Γ(∞), where Γ(∞) is the isotropy subgroup of the cusp ∞. More general automorphy factors lead to other examples of modular forms.

22 3.1 Deﬁnitions and Examples 3 Modular Forms

α Let k, m ∈ Z, N ≥ 1 and let ϕm : R>0 → C be a function such that ϕm(y) = O(y ) for ∗ mx some fixed α > 0. Set ϕm(z) = ϕm(y)q where z = x + iy. Then the mth Poincaréseries of weight k for the automorphy factor ϕm (for the congruence subgroup Γ0(N)) is defined as

X ∗ P (m, k, N, ϕm; z) := (ϕm|g)(z)

g∈Γ0(N)/Γ(∞) where az + b a b (f|g)(z) := (cz + d)−kf if g = . cz + d c d 2πimy When ϕm = 1, we get the Eisenstein series Ek(z) for Γ0(N). More generally, ϕm(y) = e gives an important class of Poincar´eseries:

X e2πim P (k, N; z) := P (m, k, N, e2πimy; z) = . m (cz + d)k 2 (c,d)∈Z c≥0,(c,d)=1

Theorem 3.1.13. For any k, m ∈ Z, N ≥ 1 and ϕm : R>0 → C as above, 2πimy ! (1) If k is even and m ≤ −1, then P (m, k, N, e ; z) ∈ Mk(Γ0(N)).

2πimy (2) If k is even and m ≥ 1, then P (m, k, N, e ; z) ∈ Sk(Γ0(N)).

2πimy (3) In fact, Sk(Γ0(N)) = SpanC{P (m, k, N, e ; z) | m ≥ 1}. Next, we determine the q-expansions of the Poincar´eseries.

Deﬁnition. For a, b, c ∈ N, the Kloosterman sum for (a, b, c) is the sum X ax + bx−1 K(a, b, c) := exp c x∈(Z/cZ)×

where x−1 denotes the unique element of (Z/cZ)× such that xx−1 ≡ 1 and exp(b) = e2πib. Remark. In other words, a Kloosterman sum is a sum of φ(c) roots of unity. One can show that |K(a, b, c)| ≤ φ(c) << c but this bound is not precise in general. A deeper result is the Weil bound:

|K(a, b, c)| << c1/2+ε.

Theorem 3.1.14. Let k, m ∈ Z and N ≥ 1. Then m P∞ n (1) If m ≥ 1, then Pm(k, N; z) = q + n=1 a(n)q where √ n(k−1)/2 X K(m, n, c) 4π mn a(n) = (2πi)k J 2 c k−1 c c>0 c≡0 (mod N)

where Jk−1 is the (k − 1)st J-Bessel function.

23 3.1 Deﬁnitions and Examples 3 Modular Forms

m P∞ n (2) If m ≤ −1, then Pm(k, N; z) = q + n=1 b(n)q where p ! n(k−1)/2 X K(m, n, c) 4π |m|n b(n) = (2πi)k I 2 c k−1 c c>0 c≡0 (mod N)

where Ik−1 is the (k − 1)st I-Bessel function.

Remark. The I-Bessel functions Iα(x) grow rapidly, though they are sub-exponential in x. Meanwhile, the J-Bessel functions Jα(x) are asymptotically polynomial in x.

Deﬁnition. Let χ be a Dirichlet character mod N, i.e. a homomorphism χ :(Z/NZ)× → C. A modular form f for Γ0(N) has nebentypus χ if a b f(γz) = χ(d)(cz + d)kf(z) for all γ = ∈ Γ (N). c d 0

! Let Mk(N, χ), Mk(N, χ) and Sk(N, χ) for the spaces of modular functions, modular forms and cusp forms, resp., of weight k and level Γ0(N) with nebentypus χ.

2 Q∞ 12n 2 Example 3.1.15. The modular form f(z) = η(12z) = q n=1(1 − q ) is a cusp form of weight 1 for Γ0(144) with nebentypus χ−3, where χ−3 is the Kronecker character:  0, if (n, 12) > 1   −3 −3  , if n = p 6= 2, 3 is prime χ−3(n) := = p n   r Y −3 r  , if n = Q pai for primes p 6= 2, 3.  pai i=1 i i i=1 i

−3 Here, p is the Legendre symbol for primes p.

In certain circumstances, one√ can make sense of ‘half-integral weight’ modular forms using a suitable interpretation of cz + d. To do this rigorously, Shimura defined an extended a Legendre symbol as follows. If a ∈ Z and p is an odd prime, p is just the ordinary Legendre Qr ai a Qr a symbol. If n ∈ N is odd, write n = i=1 pi for primes pi and take n := i=1 ai . For pi an integer n < 0, define   a , if a > 0 a  |n| = n  a − |n| , if a < 0. 0 0 a Setting 1 = −1 = 1, this defines n for all a, n ∈ Z with n odd. Next, we define an ε-factor for each odd n: ( 1 if n ≡ 1 (mod 4) εn = i if n ≡ 3 (mod 4).

24 3.1 Deﬁnitions and Examples 3 Modular Forms

√ Taking z to be the branch of the square root function on C that is positive on the positive π π real axis (meaning the argument is always taken to be between − 2 and 2 ), we deﬁne half- integral weight modular forms as follows.

Deﬁnition. Let λ be a nonnegative integer. A meromorphic function f : h∗ → C is a 1 modular function of half-integral weight λ + 2 for Γ0(4N), with nebentypus χ, if c 2λ+1 f(γz) = χ(d) ε−1−2λ(cz + d)λ+1/2f(z) d d a b for some λ ∈ and for all γ = ∈ Γ (4N). N0 c d 0 Example 3.1.16. By this deﬁnition, the theta series θ(z) = 1 + 2q + 2q4 + ... is a weight 1 2 modular form for Γ0(4). Further, for any γ ∈ Γ0(4), put θ(γz) j(γ; z) = . θ(z)

! 2λ+1 Then saying f ∈ Mλ+1/2(Γ0(4N)), χ) is equivalent to saying f(γz) = χ(d)j(γ; z) f(z) for all γ ∈ Γ0(4N).

Deﬁnition. Let χ : Z/nZ → C be a Dirichlet character and deﬁne the standard univariate theta function θ(χ, ·; z) by

X 2 θ(χ, 0; z) := χ(n)qn if χ is an even function n∈Z ∞ X 2 θ(χ, 1; z) := nχ(n)qn if χ is an odd function. n=1

Theorem 3.1.17 (Jacobi Transformation Laws). Let χ−1 be the quadratic Dirichlet character given by ( 1, n ≡ 1 (mod 4) χ−1(n) = −1, n ≡ 3 (mod 4). Then for any Dirichlet character χ mod N,

1 2 (1) If χ is even, θ(χ, 0; z) is a weight 2 modular form of level Γ0(4N ) with nebentypus χ. 3 2 (2) If χ is even, θ(χ, 1; z) is a weight 2 modular form of level Γ0(4N ) with nebentypus χχ−1.

2 Theorem 3.1.18 (Serre-Stark). Let N ≥ 1. Then the vector space M1/2(Γ1(4N )) of 1 2 weight 2 modular forms of level Γ1(4N ) is generated by the following collection of standard univariate θ-series: N M (Γ (4N 2)) = θ(χ, 0; δz) : cond(χ) divides N and δ divides 1/2 1 cond(χ) where cond(χ) denotes the conductor of χ.

25 3.1 Deﬁnitions and Examples 3 Modular Forms

2 Remark. Unfortunately, S3/2(Γ0(4N ), χ) in general is not generated by the space of cusp- idal θ-functions.

3 Example 3.1.19. By Corollary 1.2.4(2), η(8z) = θ(χ−1, 1; z) ∈ S3/2(Γ0(64), χtriv) where χ−1 is the conductor 4 character from above and χtriv is the trivial character.

For N ≥ 1 and a sequence of integers ar ∈ Z, for r ≥ 1, deﬁne Y F (z) = η(rz)ar . r|N

Theorem 3.1.20. Assume the sequence (ar) satisﬁes X (1) rar ≡ 0 (mod 24). r|N

X N (2) a ≡ 0 (mod 24). r r r|N

1 X (3) The number k := a is an integer. 2 r r|N Then (cz + d)kD (−1)k F (γz) = F F (z) d a b Y for all γ = ∈ Γ (N) and z ∈ h, where D = rar . c d 0 F r|N

Corollary 3.1.21. For any sequence (ar) satisfying the conditions above,

k ! DF (−1) (1) F (z) ∈ Mk Γ0(N), · .

1 X (2) If k = a is a half-integer, the theorem applies to 2 r r|N

η(2z)5 Fe(z) := F (z) . η(z)2η(4z)2

a1 a2 (3) If P,Q are cusps for Γ0(N) represented by rational numbers d and d , respectively, with (ai, d) = 1, then ordP (F ) = ordQ(F ). Example 3.1.22. Let F (z) = η(4z)2η(8z)2. Then the conditions of Theorem 3.1.20 are 2 DF satisﬁed for any (ar) when N = 32 or 64. In the case that N = 32, DF = 32 , so · is the trivial character and we get F ∈ S2(Γ0(32), χtriv).

26 3.1 Deﬁnitions and Examples 3 Modular Forms

Example 3.1.23. Consider the η-quotient η(5z)5 F (z) = . η(z)

Then the conditions of Theorem 3.1.20 hold for N = 5, in which case DF = 5 and we can 5 5 check that F ∈ M2 Γ0(5), · . This F is not a cusp form since S2 Γ0(5), · = 0. −1 ! 12 Example 3.1.24. Let F (z) = η(24z) ∈ M−1/2 Γ0(576), · . Then since η(24z) is a cusp form which is nonvanishing on h, F is also nonvanishing on h and has a pole at every 1 cusp of Γ0(576). Notice that this F is also the Poincar´eseries with m = −1 and weight − 2 . Deﬁnition. The Ramanujan ∆-function is

∞ ∞ Y X ∆(z) := η(z)24 = q (1 − qn)24 = τ(n)qn. n=1 n=1 The coeﬃcient function τ(n) is called Ramanujan’s τ-function. The ∆-function has very deep properties in number theory: The Ramanunan-Petersson conjecture, proven as part of Deligne’s proofs of the Weil 11/2 conjectures, says that |τ(n)| ≤ σ0(n)n for all n. The τ-function is multiplicative: if (n, m) = 1 then τ(n)τ(m) = τ(nm). This fact is not at all obvious from the deﬁnition of τ and indeed, it is a consequence of a deep conjecture of Mordell (now Mordell’s theorem).

The theory of Hecke operators implies that for all primes p and a ≥ 1, τ(p)τ(pa) = τ(pa+1) + p11τ(pa−1). This property can also be expressed as

∞ X τ(pn) 1 = . pns 1 − τ(p)p−s + p11−2s n=0 More generally, ∞ X τ(n) Y 1 = ns 1 − τ(p)p−s + p11−2s n=0 p prime 13 for Re(s) > 2 , so the ﬁrst expression above can be viewed as a “local Euler factor” of an L-function.

The τ-function satisﬁes certain congruences, including: ( 1 (mod 2), if n = (2m + 1)2 (i) τ(n) ≡ 0 (mod 2), otherwise n (ii) τ(n) ≡ 0 (mod 23) if = −1 23 (iii) τ(n) ≡ σ11(n) (mod 691).

27 3.1 Deﬁnitions and Examples 3 Modular Forms

The ﬁrst two can be shown directly from the identities of Euler and Jacobi about the η-function (see Corollary 1.2.4). The third congruence follows from expressing Θ ∈ S12(SL2(Z)) as a linear combination of Eisenstein series and using the q-expansions of E4,E6 and E12 from Example 3.1.5. Remark. Though these “low-lying examples” are easy to obtain by brute-force linear algebra, using the fact that Mk(Γ) are all ﬁnite dimensional vector spaces, it turns out that all congruences for τ(n) can be explained using algebraic number theory. Let ` be prime and let Z` be the ring of `-adic integers. Then there is a 2-dimensional continuous `-adic Galois representation ρ∆,` : Gal(Q/Q) −→ GL2(Z`) satisfying the following properties:

(i) If p 6= ` is prime, then Tr(ρ∆,`(Frobp)) = τ(p), where Frobp is the Frobenius of p.

11 (ii) If p 6= ` is prime, then det(ρ∆,`(Frobp)) = p .

Reducing mod ` gives us a finite group H∆,` :=ρ ¯∆,`(Gal(Q/Q)) ≤ GL2(F`). By Galois theory, there is a finite Galois extension L∆,`/Q corresponding to H∆,` ≤ GL2(F`) and this number field determines the congruences of τ(p) mod ` for all p 6= `. Even better, Deligne and Serre proved that L∆,`/Q is ramified only at `, so we can deduce certain easy congruences directly. For example, there are no Galois extensions of Q ramified only at ` = 2, so L∆,2 = Q and this implies τ(p) ≡ 0(mod 2) for all odd primes p. Likewise, congruence (ii) is equivalent to the statement that H∆,23 ≤ GL2(F23) corresponds to a dihedral subgroup of P GL2(F23) and thus half of its elements have vanishing trace.

Example 3.1.25. Consider the elliptic curve E/Q given by aﬃne equation y2 = x3 + 1. Consider the modular form

∞ 4 X n FE(z) := η(6z) = a(n)q ∈ S2(Γ0(36)). n=1

We claim that for all primes p ≥ 5, a(p) = p + 1 − #E(Fp), which proves the modularity × theorem for E. To see this, ﬁrst suppose p ≡ 2 (mod 3). Then 3 does not divide Fp which 3 × implies that the map ψ : a 7→ a is an automorphism on Fp . That is,

X a3 + 1 X a = = 0. p p a∈Fp a∈Fp

So when p ≡ 2 (mod 3),

X a3 + 1 X a #E( ) = 1 + 1 + = 1 + p + = p + 1. Fp p p a∈Fp a∈Fp

On the other hand, a(n) = 0 if n 6≡ 1(mod 6) so for p ≡ 2(mod 3), a(p) = 0 and thus a(p) = p + 1 − #E(Fp) as claimed.

28 3.1 Deﬁnitions and Examples 3 Modular Forms

The case when p ≡ 1(mod 3) takes more work. From Corollary 1.2.4, we know

∞ ∞ Y 6n Y 6m 3 FE(z) = q (1 − q ) (1 − q ) . n=1 m=1 Using this, one can show that for p ≡ 1(mod 3), √ √ a(p) = (x + −3y) + (x − −3y) = 2x

for some choice of integers x, y such that p = x2 + 3y2. On the other hand,

X a3 + 1 p a∈Fp can be computed by the law of cubic reciprocity, and combining the results as above proves the claim for these primes p.

Example 3.1.26. The Nekrasov-Okounkov formula states that

∞ ∞ Y n −1−z X n (1 − q ) = Qn(z)q n=1 n=0

for a degree n polynomial Qn ∈ Q[z] given by the hook length formula X Y z Q (z) := 1 + n h2 λ`n h∈H(λ) where H(λ) denotes the hook set of λ. For example:

Q0(z) = 1

Q1(z) = 1 + z 5 1 Q (z) = 2 + z + z2 2 2 2 29 1 Q (z) = 3 + z + 2z2 + z3. 3 6 6

2 It turns out that Qn(−z ) are characters of certain inﬁnite dimensional Lie algebras of A-type (cf. MacDonald’s identities).

Conjecture 3.1.27. The Qn(z) have unimodal coeﬃcients for all n ≥ 0. Example 3.1.28. (Borcherds products) Borcherds won the Fields Medal primarily for two results: his proof of the Monstrous Moonshine Conjecture and a study of certain product formulas which we describe here. Recall (Section 0.4) that Klein’s j-function is deﬁned by

E (z)3 j(z) = 4 = q−1 + 744 + 19688q + ... ∆(z)

29 3.1 Deﬁnitions and Examples 3 Modular Forms

Then j is a weakly modular function of weight 0. By the η-quotient theorem (3.1.20), ∆ can be written as an inﬁnite product of η-quotients. On the other hand, since 3 doesn’t 2πi/3 2πi/3 divide 4, E4(e ) = 0 (the isotropy group of e is larger than just h±Ii) so E4(z) is not an inﬁnite product of η-quotients. One can show that every weakly modular function P n f(z) = n a(n)q coincides with the meromorphic continuation of a product expansion of the form ∞ Y f = qordf (∞) (1 − qn)a(n). n=1 Here’s an example. Set     !,+  ! X n M1/2(Γ0(4)) := f ∈ M1/2(Γ0(4)) : f = a(n)q .  n∈Z   n≡0,1 (mod 4) 

It follows from Serre-Stark (3.1.18) that

!,+ M M1/2(Γ0(4)) = Cfd(z) d≥0 d≡0,3 (mod 4)

!,+ where fd is a particular weakly modular function in M1/2(Γ0(4)) of the form

−d X n fd(z) = q + ad(n)q . n≥1 n≡0,1 (mod 4)

4 9 −3 4 For example, f0(z) = θ0(z) = 1 + 2q + 2q + 2q + ... and f3(z) = q − 248q + 26752q − 5 85995q + .... For d ≥ 4, d ≡ 0, 3(mod 4), set Gd(z) := fd−4(z)j(4z). Then Gd(z) ∈ !,+ −d M1/2(Γ0(4)) and the leading term of Gd(z) is q , so the formulas for fd(z) can be extracted from Gd and lower pole-order terms. 1 X Theorem 3.1.29 (Borcherds). Let He(z) = − + h(−n)qn where h(−n) are the 12 n>1 n≡0,3(mod 4) √ Hurwitz class numbers (e.g. when n is squarefree, h(−n) = h(Q( −n)) is an imaginary !,+ quadratic class number). For any f ∈ M1/2(Γ0(4)), deﬁne hf to be the constant term of f(z)He(z) and ∞ Y 2 ψ(f)(z) := qhf (1 − qn)a(n ). n=1

Then ψ(f)(z) is a weight a(0) modular form on SL2(Z) with zeroes and poles only at CM P 2 points of discriminant D < 0 for n<0 a(Dn ) 6= 0. Moreover, if MH denotes the space of all meromorphic modular forms for SL2(Z) with ﬂat Heegner divisor, then !,+ ψ : M1/2(Γ0(4)) −→ MH , f 7−→ ψ(f) is an isomorphism (taking additive structure to multiplicative structure).

30 3.1 Deﬁnitions and Examples 3 Modular Forms

These product formulas, called Borcherds products or sometimes theta lifts, are a modern alternative to the classical interpretation of modular forms as q-series.

Example 3.1.30. Let f(z) = 12f0(z) = 12θ0(z). Then ψ(f) = ∆ ∈ S12(SL2(Z)).

! Example 3.1.31. Let f(z) = 3f3(z). Then ψ(f) = j(z) ∈ M0(SL2(Z)). What’s amazing is that, using the fact that E (z)3 j(z) = 4 ∆(z) and the previous example, it follows that E4(z) is an inﬁnite product of η-quotients! More precisely, E4(z) = ψ(f) where f(z) = 4θ0(z) + f3(z). Here are some more results about the j-function.

Theorem 3.1.32. Let j(z) be Klein’s j-function. Then

! (1) j(z) ∈ M0(SL2(Z)).

! (2) M0(SL2(Z)) = C[j(z)] as polynomial rings.

(3) Every modular function on SL2(Z) is a rational function in j(z). Proof. (1) It is straightforward to verify j(z) has no poles in h. ∗ (2) Recall that X0(SL2(Z)) = h /SL2(Z) is the Riemann sphere. Note that j(z) has exactly one pole at ∞. For any α ∈ C, the formula Fα(z) = j(z) − α deﬁnes a modular form on h (since the number of zeroes equals the number of poles of Fα). Thus Fα has a unique root in X0(SL2(Z)). By varying α, we get a one-to-one correspondence

j : D ←→ C

! where D ⊆ h is a fundamental domain for SL2(Z). This implies M0(SL2(Z)) is the polynomial ring in j(z) – in fact, it shows that j(z) exactly corresponds to the indeterminate t ∈ C[t]. (3) Now suppose 0 ∞ X n X m f(z) = γnq + a(m)q n=−∞ m=1 ! is a weakly modular function of weight 0, i.e. f ∈ M0(SL2(Z)). Then there is a unique polynomial p(j(z)) whose Fourier expansion has the same principal part as f; that is, ! p(j(z)) − f(z) ∈ M0(SL2(Z)) with Fourier expansion

∞ X a(m)qm. m=1 Since there are no poles of this Fourier expansion, there are no zeroes either so it must be 0. Hence f(z) = p(j(z)).

31 3.2 Modular Forms for SL2(Z) 3 Modular Forms

! Remark. For any N ≥ 1, M0(Γ0(N)) = C(j(z), j(Nz)). Further, there are only finitely many N for which the genus of X0(N) is 0, or equivalently, there are only finitely many N such that j(z) and j(Nz) satisfy an algebraic equation. In these cases, there exists a ! modular function jN (z), called a Hauptmodul, such that M0(Γ0(N)) = C(jN (z)). Example 3.1.33. For any z ∈ h, j(z) turns out to be the j-invariant for the complex elliptic curve E = C/Λz where Λz is the lattice Z ⊕ Zz. In fact, E is given by an affine Weierstrass 2 3 equation y = 4x +g2x+g3 where g2 (resp. g3) corresponds to the Eisenstein series of weight 3 4 (resp. weight 6). One can show that the discriminant of the polynomial 4x + g2x + g3 is precisely ∆(z).

3.2 Modular Forms for SL2(Z)

∗ We know that for any congruence subgroup Γ ≤ SL2(Z) and any k ∈ Z, the spaces ! ∗ ∗ ∗ Mk(Γ ), Mk(Γ ) and Sk(Γ ) are all C-vector spaces. In this section we will describe their ! ! structure. Let Mk = Mk(SL2(Z)) and likewise deﬁne Mk and Sk. ! k Example 3.2.1. If k is odd, then Mk = 0: the modular condition implies f(z) = (−1) f(z) for any z ∈ h, which is only possible if f ≡ 0.

! ! ! Example 3.2.2. If k = 0, we know j(z) ∈ M0 and C[j(z)] ⊆ M0. In particular, M0 is ! ! ! an inﬁnite dimensional C-vector space. Further, since j(z)Mk ⊆ Mk, we see that Mk is inﬁnite dimensional for any even k.

∗ Theorem 3.2.3. Suppose Γ ≤ SL2(Z) is a congruence subgroup. Then ∗ ∗ (1) Mk(Γ ) is a ﬁnite dimensional C-vector space, and thus so is Sk(Γ ).

∗ ∗ ∗ ∗ (2) If k > 2, then dimC Mk(Γ ) = c(Γ )+dimC Sk(Γ ) where c(Γ ) is the number of cusps of Γ∗. Proof. (3) When k > 2, we constructed (Example 3.1.8) an Eisenstein series of weight k for Γ∗ for each cusp of Γ∗, so the dimension formula is equivalent to

∗ M ∗ Mk(Γ ) = CGk(τ; z) ⊕ Sk(Γ ). τ a cusp

Remark. In the nebentypus and half-integral weight cases, formulas for dimC Mk(Γ0(N), χ) and dimC Sk(Γ0(N), χ) can be found in Ono’s Web of Modularity.

Theorem 3.2.4 (Valence formula for SL2(Z)). For the full modular group SL2(Z) and any ! f ∈ Mk where k ∈ Z is even, k 1 1 X = ord (f) + ord (f) + ord (f) + ord (f). 12 ∞ 2 i 3 ρ x x∈D x6=i,ρ

Here, D is the standard fundamental domain for SL2(Z).

32 3.2 Modular Forms for SL2(Z) 3 Modular Forms

Corollary 3.2.5. If k ≥ 2 is even, then

 k  , k ≡ 2 (mod 12)  12 dimC Mk =  k  + 1, k 6≡ 2 (mod 12).  12

Example 3.2.6. For the weights k = 0, 2, 4,..., 14, we have:

dimC Mk = 0, 0, 1, 1, 1, 1, 2, 1. In fact, this allows us to compute bases of these vector spaces in terms of the Eisenstein series Ek:

M0 = M2 = 0 M4 = CE4 M6 = CE6 2 M8 = CE4 = CE8 M10 = CE4E6 = CE10 3 2 M12 = CE4 ⊕ C∆ = CE6 ⊕ C∆ = CE12 ⊕ C∆ 2 M14 = CE4 E6 = CE14.

! Corollary 3.2.7. M0 = C[j(z)]. L Corollary 3.2.8. Let M = k∈ Mk be the ring of modular forms for SL2(Z). Then ∼ Z M = C[E4,E6], the polynomial ring in the Eisenstein series E4 and E6.

! 2 2 Remark. Suppose f ∈ Mk for k ≥ 0 even. Consider the set E = {1,E4,E6,E4 ,E4E6,E4 E6} and for each k, let Eek denote the unique element of E whose weight represents k mod 12. Then there is a unique nk ∈ Z such that f(z) ∈ M! . n 0 ∆(z) k Eek(z)

nk Therefore we can write f(z) as ∆(z) Eek(z) times a polynomial in j(z), called the divisor polynomial of f and written gf (j(z)). Explicitly,

∗ ∗ ord (f) ordρ(f) Y ordα(f) gf (j(z)) = (j(z) − i) i (j(z) − ρ) (j(z) − α) α∈C α6=i,ρ

∗ where for τ = i, ρ, ordτ (f) = ordτ (f) − ordτ (Eek). Theorem 3.2.9. Suppose X(Γ∗) has genus 0. Then the ﬁeld of modular functions for Γ∗ is −1 P∞ n simply generated. In particular, there exists a unique hauptmodul h(z) = q + n=1 a(n)q ∈ ! ∗ ! ∗ M0(Γ ) such that M0(Γ ) = C[h].

33 3.2 Modular Forms for SL2(Z) 3 Modular Forms

! Example 3.2.10. We saw that M0(SL2(Z)) = C[j] where j = j(z) is the j-function. Remark. Theorem 3.2.9 has some notable consequences, including the fact that for every ! 1 0 d ! f(z) ∈ M0 = C[j], the derivative of f is still modular: 2πi f (z) = q dq f(q) ∈ M2. For example, 1 E2E j0(z) = − 4 6 ∈ M! . 2πi ∆ 2 This proves that every weight 2 modular function is a derivative of a weight 0 modular function. That is, we have proven:

2 ! E4 E6 Corollary 3.2.11. M2 = C[j] ∆ . In particular, the constant term of the q-expansion of ! any f ∈ M2 is 0. L Theorem 3.2.12. Let M = k∈2 Mk be the ring of modular forms for SL2(Z). Then ∼ Z M = C[E4,E6].

a b Proof. We will show that Mk is generated by the set {E4 E6 | a, b ≥ 0 and 4a + 6b = k} for all even k ≥ 0. When k ≤ 6, this follows from Example 3.2.6. Let k > 6 and induct. a b For a, b ≥ 0 such that 4a + 6b = k, the modular form E4 E6 is not a cusp form, but for any f ∈ Mk, the form f(∞) a b h = f − a b E4 E6 E4(∞) E6(∞) a0 b0 is a cusp form, so by Example 3.2.6, h = g∆ for some g ∈ Mk−12. By induction, g = E4 E6 0 0 for 4a + 6b = k − 12, but ∆ is also a linear combination of powers of E4 and E6, so f is as well. 6 a b E4 Finally, these E4 E6 form a basis for M since if not, the function 4 would satisfy an E6 algebraic equation over C and hence be a scalar. But this is impossible, since E4(i) 6= a b 0,E4(ρ) = 0,E6(i) = 0 and E6(ρ) 6= 0. Therefore {E4 E6 | a, b ≥ 0, 4a + 6b = k} is a basis for Mk.

On the other hand, one can always create a “diagonal basis” for each Mk:

Lemma 3.2.13 (Miller Basis). If k ≥ 4 is even, then there exists a basis of Mk consisting of forms with q-expansions

∞ X n b0 = 1 + a0(n)q n=1 ∞ X n b2 = q + a2(n)q n=2 . . ∞ d−1 X n bd−1 = q + ad−1(n)q n=d

where d = dimC Mk.

34 3.3 Petersson Inner Product 3 Modular Forms

One can always take b0 = Ek. The Miller basis {br} for Mk is useful for calculations, but is inadequate for proving deep theorems. We will see that a diﬀerent basis, the Hecke basis, allows one to describe L-functions which should satisfy the generalized Riemann hypothesis. Moreover, these Hecke bases are algebraically integral, so there is an interesting Galois structure coming from these diﬀerent choices of bases.

3.3 Petersson Inner Product

dx dy ∗ Recall (Theorem 2.3.9) that dµ = y2 is an SL2(R)-invariant measure on h/Γ for any ∗ ∗ congruence subgroup Γ ≤ SL2(Z). Suppose T ∈ Γ .

∗ ∗ Deﬁnition. The Petersson inner product on Mk(Γ ) is deﬁned for any f, g ∈ Mk(Γ ) ∗ such that fg ∈ S2k(Γ ) by 1 Z dx dy hf, gi ∗ := f(z)g(z) . Γ [SL ( ):Γ∗] y2−k 2 Z DΓ∗ Remark. The Petersson inner product has several useful properties:

(1) h·, ·i is bilinear and satisﬁes several other simple relations, which we will prove once we have developed the theory of Hecke operators.

(2) If either f or g is described in terms of Poincar´eseries, then hf, gi is easy to calculate.

∗ 2 (3) Deﬁne the Petersson norm on Sk(Γ ) by ||f|| := hf, fi . Then one can associate a ∗ Dirichlet series to each f ∈ Sk(Γ ) whose periods are related to ||f||.

2πimy Let Pm(k, N; z) = P (m, k, N, e ; z) be the Poincar´eseries for the automorphy factor 2πimy ϕm = E , which is a modular form of weight k for the congruence subgroup Γ0(N). Then by Theorem 3.1.13(2), if k is even, Pm(k, N; z) is a cusp form for m ≥ 1. Moreover, each of these has a q-expansion of the form

∞ m X n Pm(k, N; z) = q + a(m, k, N; n)q n=1 where the a(m, k, N; n) can be expressed in terms of Kloosterman sums.

Example 3.3.1. If k = 12 and N = 1, then for every m ≥ 1, Pm(12, 1; z) = αm∆(z) for some αm ∈ C. It is possible for αm = 0, but no such example is known! (This is related to Lehmer’s Conjecture.)

P∞ n Theorem 3.3.2. If f = n=1 a(n)q ∈ Sk(Γ0(N)). Then for any m ≥ 1, Γ(k − 1) hf, P (k, N)i = a(m). m (4πm)k−1

Moreover, hf, fi ≥ 0 with equality if and only if f ≡ 0.

35 3.3 Petersson Inner Product 3 Modular Forms

Here, Γ is the complex Γ-function Z ∞ Γ(s) = xs−1e−s dx 0

deﬁned for Re(s) > 0. If n ∈ N, then Γ(n) = (n − 1)!. 2πimy Proof. For simplicity, take N = 1 and set ϕm = e . Then Z dx dy hf, Pm(k, 1)i = f(z)Pm(k, 1; z) 2−k D y ∞ X Z dx dy = a(n) e2πin(x+iy)P (k, 1; z) m y2−k n=1 D ∞ Z X 2πin(x+iy) X ∗ dx dy = a(n) e (ϕm | γ) 2−k D y n=1 γ∈SL2(Z)/Γ(∞) ∞ X Z ∞ Z 1 = a(n) e2πi(n−m)xe−2π(n+m)yyk−2 dx dy n=1 0 0 ∞ Z ∞ X −2π(n+m)y k−2 = a(n)δnm e y dy n=1 0 Z ∞ = a(m) e−4πmyyk−2 dy 0 a(m)Γ(k − 1) = . (4πm)k−1

Note that this gives a proof of Theorem 3.1.13(3): let {br} be a basis of Sk(Γ0(N)). Then hbr,Pm(k, N)i= 6 0 so the Pm(n, K) must span Sk(Γ0(N)). Now the statement about nonde- generacy follows immediately.

Example 3.3.3. Since for m ≥ 1, P (m, k, N; z) ∈ Sk(Γ0(N)), there must be many linear relations among the Poincar´eseries. For example, S4 = 0 so Pm(4, 1; z) = 0 for all m ≥ 1. 2 3 On the other hand, S24 = C∆ ⊕ C∆E4 has dimension 2, so there are nontrivial Poincar´e series, but for all distinct m1, m2, m3 ≥ 1, there are some α1, α2, α3 ∈ C such that α1Pm1 +

α2Pm2 + α3Pm3 = 0. Thus an interesting problem is to ﬁnd a subset of the set of Poincar´e series that forms a basis. This was solved by Rhoades in 2008 using harmonic Maass forms.

Another important question is: what characterizes the orthonormal bases of Sk(Γ0(N)) with respect to the Petersson inner product? It turns out that Poincar´eseries are not the answer, but instead the Hecke bases.

36 4 Hecke Operators

4 Hecke Operators

The main question we will explore in this chapter is: are there natural linear endomorphisms ! ∗ ∗ ∗ on the vector spaces Mk(Γ ), Mk(Γ ) and Sk(Γ ) that encode a deeper structure of modular forms? This has a positive answer, and it is given by the theory of Hecke operators.

4.1 Hecke Operators on SL2(Z)

Lemma 4.1.1. Suppose m ≥ 1 and deﬁne

Dm = {γ ∈ M2(Z) | det(γ) = m}.

Then SL2(Z) acts on Dm by left multiplication and the orbit space is a b D /SL ( ) = : ad = m, 0 ≤ b ≤ d − 1 . m 2 Z 0 d

In particular, the size of each orbit is given by σ1(m). α β Proof. Suppose A = ∈ D . Then there is a unique pair of matrices γ δ m

e f a b ∈ SL ( ) and ∈ D g h 2 Z 0 d m such that α β e f a b A = = . γ δ g h 0 d α γ Indeed, let a = (α, γ), e = a and g = a . This ensures the ﬁrst column of A is attained. Moreover, since (e, g) = 1, there exist h, f ∈ Z with eh − fg = 1, and we can extract some b 1 −1 from this. But by multiplying by it is possible to arrange for 0 ≤ b ≤ d − 1. This 0 1 completes the proof.

Deﬁnition. Suppose k ∈ Z is even. If f is a weight k weakly holomorphic form on h (for SL2(Z)), then for each m ≥ 1, the mth Hecke operator acts on f by

k/2−1 X Tmf(z) = m (f|kγ)(z)

γ∈Dm/SL2(Z) a b where (f| γ) = (det γ)−k/2(cz + d)−kf(γz) if γ = ∈ GL ( ). k c d 2 Z

! ! Lemma 4.1.2. If f ∈ Mk (resp. Mk, Sk) and m ≥ 1, then Tmf ∈ Mk (resp. Mk, Sk).

Proof. Observe that Tm is the trace operator of a faithful, transitive action of SL2(Z) on Dm, so the modularity condition with respect to SL2(Z) is automatically preserved. Further, applying Tm to f cannot produce new poles, and since the poles of f ∈ Mk are only at cusps, which form an SL2(Z)-orbit, we must have Tm(Mk) ⊆ Mk and Tm(Sk) ⊆ Sk.

37 4.1 Hecke Operators on SL2(Z) 4 Hecke Operators

Lemma 4.1.3 (Fourier Expansion Formula). Suppose k ∈ Z even and f ∈ Mk has q- P∞ n expansion f = n=d a(n)q . Then for each m ≥ 1,

∞ X n Tmf = b(n)q n=r where X nm b(n) = dk−1a . d2 d|(n,m) Proof. Assume m = p is prime, in which case the formula says that the coeﬃcients of the k−1 n n q-expansion of Tpf are b(n) = a(np) + p a p , where we formally set a p = 0 if p - n. By deﬁnition, k/2−1 X Tpf(z) = p (f|kγ)(z).

γ∈Dp/SL2(Z)

The orbit space of SL2(Z) on Dp can be decomposed as p 0 1 b D /SL ( ) = ∪ : 0 ≤ b ≤ p − 1 . p 2 Z 0 1 0 p

So we see that

p−1 ! X z + b T (f) = pk/2−1 pk/2f(pz) + p−k/2 f p p b=0 p−1 1 X z + b = pk−1f(pz) + f p p b=0 ∞ ∞ p−1 X 1 X X = pk−1 a(n)qpn + a(n) e2πin(z+b)/p. p n=d n=d b=0

Pp−1 2πinb/p But b=0 e = 0 if p - n and the sum is p if p | n, so the formula reduces to the desired. The general case is proven in a similar fashion.

Corollary 4.1.4. Let m, m1, m2 ≥ 1. Then

(1) If (m1, m2) = 1, then Tm1m2 = Tm1 Tm2 .

k−1 (2) If p is prime and a ≥ 1, then TpTpa = Tpa+1 + p Tpa−1 on Mk.

(3) Suppose f is an eigenform of Tm with constant term 1 in its q-expansion. Then Tmf = a(m)f, that is, the mth coeﬃcient is the eigenvalue of f.

(4) If f is an eigenform of Tm1 and Tm2 , and (m1, m2) = 1, then a(m1m2) = a(m1)a(m2). Deﬁnition. A nonconstant modular form f(z) is an eigenform (for all m ≥ 1) provided it is an eigenvector for each Tm, that is, there exist λ(m) ∈ C such that Tmf = λ(m)f for each m ≥ 1. We say an eigenform f is normalized if c1 = 1.

38 4.1 Hecke Operators on SL2(Z) 4 Hecke Operators

Example 4.1.5. The Eisenstein series Ek(z) are eigenforms of every Hecke operator. In particular, TmEk = σk−1(m)Ek for every m ≥ 1.

24 Example 4.1.6. ∆(z) = η(z) ∈ S12(SL2(Z)) but M12 has dimension 2 by Example 3.2.6, so ∆ is an eigenform. As a consequence, Ramanujan’s τ-function has the following properties since they are the coeﬃcients of the q-expansion of ∆:

(1) τ(mn) = τ(m)τ(n) if (m, n) = 1.

(2) τ(p)τ(pa) = τ(pa+1) + p11τ(pa−1) for any prime p and a ≥ 1.

Deﬁne the L-function (Dirichlet series)

∞ X τ(n) L (s) = . ∆ ns n=1

Then the above properties imply that L∆(s) has a product formula

Y 1 L (s) = . ∆ 1 − τ(p)p−s + p11−2s p prime

It is a famous conjecture of Lehmer that τ(m) 6= 0 for any m ≥ 1, but this remains unsolved. 11/2 However, Deligne showed that |τ(m)| ≤ σ0(m)m and there are inﬁnitely many m for which |τ(m)| is arbitrarily close to the right hand side of the inequality.

Example 4.1.7. For each m ≥ 1, deﬁne Jm = mTm(j − 744) where j is the j-function. Then ∞ −1 −m X n ! Jm = Tm(q + 196884q + ...) = q + b(n)q ∈ M0 = C[j]. n=1

This shows Jm is a monic polynomial of degree m in j(z). For example,

J0 = 1

J1 = j − 744 2 J2 = j − 1488j + 159768.

−n It turns out that for all m ≥ 1, Jm(j(q)) = q + O(q). Moreover, the roots of each Jm are distinct and lie in the interval [0, 1728], and as m → ∞ these roots are equidistributed in [0, 1728].

Theorem 4.1.8. As a power series in Z[x][[q]],

2 ∞ E E6 X 4 = (j(z) − x) J (x)qm. ∆ m m=0

We next describe the action of the Hecke operators on cusp forms for SL2(Z). Let h·, ·i be the Petersson inner product on cusp forms deﬁned in Section 3.3.

39 4.1 Hecke Operators on SL2(Z) 4 Hecke Operators

Lemma 4.1.9. Fix k ≥ 4 even and let Sk = Sk(SL2(Z)). Then h·, ·i is a Hermitian inner product on Sk. Theorem 4.1.10. For any m ≥ 1,

(1) hTmf, gi = hf, Tmgi for all f, g ∈ Sk. (2) Every Hecke eigenvalue is a real algebraic integer.

Proof. (1) follows from the deﬁnition of Tm. (2) Suppose f ∈ Sk is a nonzero eigenfunction of Tm, with Tmf = λmf for some λm ∈ C. Then by Lemma 4.1.9 and (1),

λmhf, fi = hλmf, fi = hTmf, fi ¯ = hf, Tmfi = hf, λmfi = λmhf, fi. ¯ Since hf, fi 6= 0, we get λm = λm, so λm ∈ R. Next, let {f1, . . . , fd} be the Miller basis for Mk (see Lemma 3.2.13). By construction, these lie in Z[[q]], so Tm has a representation in terms of an integral basis:

d X Tmf1 = a1nfn n=1 d X Tmf2 = a2nfn n=1 . . d X Tmfd = adnfn. n=1

Thus Tm can be represented as a matrix (aij) with entries in Z. Since λm is a root of the characteristic polynomial of Tm, this implies λm is an algebraic integer.

Remark. There is also a theory of Hecke operators on Sk(Γ0(N), χ) for any N and Dirichlet character χ.

Let T = Tk be the Hecke algebra, generated as a commutative ring by the Hecke operators Tp on Sk for p prime.

Theorem 4.1.11. For each even k ≥ 2, Sk has a unique orthogonal basis f1, . . . , fdk of normalized Hecke eigenforms with respect to the Petersson inner product.

Corollary 4.1.12. Let f1, . . . , fdk be the orthogonal basis of Sk with respect to h·, ·i. Then for each 1 ≤ m ≤ dk,

∞ X n (1) fm = q + am(n)q . n=2

40 4.2 The Trace Form 4 Hecke Operators

(2) If (n, `) = 1, then am(n`) = am(n)am(`).

s s+1 k−1 s−2 (3) If p is prime, then am(p)am(p ) = am(p ) + p am(p ).

0 (4) hfm, fmi > 0 and hfm, fm0 i = 0 if m 6= m .

(5) If Kk is the number ﬁeld obtained by adjoining the coeﬃcients of fm for all m, then

Kk/Q is a totally real Galois extension with Gal(Kk/Q) ≤ Sdk .

Conjecture 4.1.13 (Maeda). For all k ≥ 12 even, Gal(Kk/Q) = Sdk . Furthermore, for all m ≥ 2, the characteristic polynomial of Tm as an operator on Sk is irreducible in Z[x] with splitting ﬁeld Kk.

Remark. Replacing SL2(Z) by a congruence subgroup Γ0(N) (including with nebentypus) generates number fields of Fourier coefficients with very different behavior. For example, these number fields need not be totally real, and in many cases are “unexpectedly small”. It turns out the unexpected drops in size of such number fields are explained by geometry (e.g. the modularity of elliptic curves, abelian surfaces, Kuga-Sato varieties, etc.).

4.2 The Trace Form

Deﬁnition. For each k ≥ 12 even, let f1, . . . , fm be the basis of eigenforms for Sk. The Eichler-Selberg trace form of weight k (or just the trace cusp form) is deﬁned as

dk ∞ ES X X n Sk (z) = fm(z) = dkq + Tr(Tn|Sk)q m=1 n=2 where dk = dim Sk and Tr is the linear algebraic trace operator.

Remark. Assuming Maeda’s conjecture, each Tr(Tn | Sk) is also equal to a ﬁeld trace of Kk/Q.

ES It turns out there is a closed formla for the coefficients of each Sk . Let k ≥ 12 be even k−2 and define the polynomial Pk(t, N) to be the coefficient of x in the generating function

∞ 1 X = P (t, N)xk−2. 1 − tx + Nx2 k k=2 Also, let H(n) be the class number of positive deﬁnite binary quadratic forms with discriminant −n (and set H(n) = 0 if −n is not a discriminant).

Theorem 4.2.1 (Eichler-Selberg). For k ≥ 12 even and m ≥ 2,

1 X 1 X Tr(T | S ) = − P (t, m)H(4m − t2) − min(d, d0)k−1. m k 2 k 2 t∈Z dd0=m d,d0≥1

41 4.2 The Trace Form 4 Hecke Operators

Example 4.2.2. For m = 5, notice that H(20 − t2) = 0 unless 0 ≤ t ≤ 4. So 1 Tr(T | S ) = − (P (0, 5)H(20)+P (1, 5)H(19)+P (2, 5)H(16)+P (3, 5)H(11)+P (4, 5)H(4)+1). 5 k 2 k k k k k

This shows that Tr(T5 | Sk) only changes with k, and more speciﬁcally, the only things that change with k are the Pk(t, 5). For even integers k ≡ ` (mod 4), one can use the recursive properties of Pk(t, 5) to show that Tr(T5 | Sk) ≡ Tr(T5 | S`) (mod 5). More generally, if pa is a prime power and k ≡ ` (mod pa−1(p − 1)), then a similar ES ES a argument shows that Sk ≡ S` (mod p ).

Definition. Let p ≥ 5 be prime and let E be an elliptic curve over a finite field Fp with good 2 3 3 reduction, i.e. an integral model y = x +Ax+B with A, B ∈ Fp and disc(x +Ax+B) 6≡ 0 (mod p). Then the trace of Frobenius of E/Fp is the sum X x3 + Ax + B a (p) = − E p x∈Fp

· where p denotes the Legendre symbol for p.

Note that #E(Fp) = p+1−aE(p). For R ≥ 1, the 2Rth moment of the traces of Frobenius of elliptic curves over Fp is the sum

p−1 !2R X X x3 + ax + b S (p) = . R p a,b∈Fp x=0 Remark. This isn’t quite the right geometric deﬁnition of the moment of traces of Frobenius – the correct way to sum these traces of Frobenius is by replacing the sum of a, b ∈ Fp with the sum of all j-invariants of elliptic curves over Fp. ∗ 1 Deﬁne SR(p) = p−1 SR(p) − 1. Using the theory of elliptic curves with complex multiplication, in 1967, Birch proved the following theorem. Theorem 4.2.3 (Birch). Let p ≥ 5 be prime and R ≥ 1. Then

R (2R)! X (2R)! S∗ (p) = pR+1 − (2k + 1) pR−k(Tr(T | S ) + 1). R R!(R + 1)! (R − k)!(R + k + 1)! p 2k+2 k=1 (2R)! Notice that the term is the Rth Catalan number. R!(R + 1)! Example 4.2.4. For p ≥ 5, we have

2 S1(p) = p 3 S2(p) = 2p − 3p 4 2 S3(p) = 5p − 9p − 5p 5 3 2 S4(p) = 14p − 28p − 20p − 7p 5 4 3 2 S5(p) = 42p − 90p − 75p − 35p − 9p − τ(p).

42 4.3 Atkin–Lehner Theory 4 Hecke Operators

Moments like S (p) are constructed in order to study the distribution of some random R √ variable. Recall Hasse’s bound: |aE(p)| ≤ 2 p. This can be written a (p) −1 ≤ E√ ≤ 1. 2 p

a (p) E√ Sato and Tate associated to each E and p an angle θE(p) in [0, π] defined by cos θE(p) = 2 p . Birch’s theorem then implies: Corollary 4.2.5 (Birch). Choose 0 < c < d < π. Then as p → ∞, the naive proportion of elliptic curves E/Fp such that θE(p) ∈ [c, d] is 2 Z d sin2 t dt. π c This is a special case of the famous Sato-Tate conjecture on probability measures for elliptic curves over finite fields.

4.3 Atkin–Lehner Theory

In this section we describe a generalization of the theory of Hecke operators to the higher level case. The main discovery of Atkin and Lehner was two kinds of operators extending Hecke theory: Operators U and V which look like “pieces” of the Hecke operators, and

Atkin–Lehner operators. Together, these give a theory of Hecke newforms, which are simultaneous Hecke eigenforms of higher level. Lemma 4.3.1. Suppose k, N, M ≥ 1 with N | M. Then

! ! (1) Mk(Γ0(N)) ⊆ Mk(Γ0(M)), Mk(Γ0(N)) ⊆ Mk(Γ0(M)) and Sk(Γ0(N)) ⊆ Sk(Γ0(M)). M (2) For each M, Sk(Γ1(M)) = Sk(Γ0(M), χ) over all Dirichlet characters χ mod χ mod M ! M, and likewise for Mk(Γ1(M)) and Mk(Γ1(M)).

(3) If f(z) has level M and m ∈ N, then f(mz) has level mM. Proof. (1) is immediate from the deﬁnitions of level subgroups. a b (2) Recall that if f has level Γ (M) and γ = ∈ Γ (M), then 1 c d 1

f(γz) = (cz + d)kχ(d)f(z).

On the other hand, the characters χ mod M generate the 1-dimensional representations of ! Γ0(M)/Γ1(M), so the decomposition holds for each of Mk, Mk, Sk.

43 4.3 Atkin–Lehner Theory 4 Hecke Operators

(3) Set F (z) = f(mz) and suppose

AB AB γ = = ∈ Γ (mM). CD cm D 0

Then Amz + Bm F (γz) = f = f(γz0) Cz + D where z0 = mz ∈ h. On the other hand,

f(γz0) = (cz0 + D)kf(z0) = (Cz + D)kf(mz) = (Cz + D)kF (z).

Therefore F has level mM.

! Remark. This shows that every f ∈ Mk(Γ1(M)) has a unique decomposition X f = αχfχ χ mod M

! for some αχ ∈ C and fχ ∈ Mk(Γ1(M), χ). Moreover, the decomposition restricts to Mk(Γ1(M)) and Sk(Γ1(M)).

Deﬁnition. For each m ≥ 1, deﬁne the Vm-operator by (Vmf)(z) := f(mz).

Lemma 4.3.2. For each m ≥ 1, Vm takes modular forms (resp. cusp forms) of level M (with nebentypus χ) to modular forms (resp. cusp forms) of level mM (with nebentypus χ).

Deﬁnition. For each m ≥ 1, deﬁne the Um-operator by

m−1 1 X z + j (U f)(z) := f . m m m j=0

P∞ n ! Lemma 4.3.3. Let m ≥ 1 and f = n=−b a(n)q ∈ Mk(Γ0(M)). Then

∞ X n Umf = a(mn)q . n=−b

P n Example 4.3.4. Let m = p be prime. Then if p - M and f = a(n)q has level M (and nebentypus χ), we saw that Tpf (resp. Tp,χf) also has level M (and nebentypus χ). Therefore k−1 k−1 Tp = Up + p Vp (resp. Tp,χ = Up + χ(p)p Vp).

In fact, this implies that Up also takes modular forms (resp. cusp forms) of level M (with nebentypus χ) to modular forms (resp. cusp forms) of level mM (with nebentypus χ).

The Vm- and Um-operators give a new proof of Ramanujan’s theorem on congruences of the partition function (Theorem 0.1.2):

44 4.3 Atkin–Lehner Theory 4 Hecke Operators

Proof. By Lemma 0.1.1, ∞ ∞ X Y 1 q1/24 p(n)qn = = . 1 − qn η(z) n=0 n=1 So ∞ 1 X = p(n)q24n−1 ∈ M! (Γ (576), χ ). η(24z) −1/2 0 12 n=0 `2−1 r(`) Let ` ≥ 5 be prime and set r(`) = 24 ∈ Z and F`(z) = ∆(z) . Then ∞ η(`z)` Y (1 − q`n)` F (z) ≡ ≡ qr(`) (mod `). ` η(z) 1 − qn n=1

Fix a residue class b mod ` and define a`(n) to be the Fourier coefficients of F`: ∞ X n F`(z) = a`(n)q . n=r(`) Then we have ∞ ! ∞ ! X n r(`) Y `n ` F`(z) = p(n)q q (1 − q ) n=0 n=1 ∞ ! ∞ ! X Y 2 ≡ p(n)qn qr(`) (1 − q` n) (mod `) n=0 n=1 ∞ ! ! X X 2 2 ≡ p(n)qn (−1)kq` (3k +k)/2+r(`) (mod `) n=0 k∈Z `(3k2+k) where the last congruence is by a formula of Euler. Notice that the exponents 2 + r(`) form an arithmetic progression mod `2 starting with r(`). In particular, for any N ≥ 1, X `2(3k2 + k) a (N) ≡ p(N − r(`)) + (−1)kp N − − r(`) (mod `). ` 2 k6=0 Now the right hand side is a finite sum and all terms have the same arithmetic progression mod `2. So taking r(`) = 1, 2, 5 for ` = 5, 7, 11, we see that

p(5n + 4) ≡ 0 (mod 5) ⇐⇒ a5(5n) ≡ 0 (mod 5)

p(7n + 5) ≡ 0 (mod 7) ⇐⇒ a7(7n) ≡ 0 (mod 7)

p(11n + 6) ≡ 0 (mod 11) ⇐⇒ a11(11n) ≡ 0 (mod 11). for all n ∈ N. Thus it suﬃces to show that U`F` ≡ 0 (mod `) for ` = 5, 7, 11. Notice that on S12, S24 and S60, we have

11 U5 ≡ U5 + 5 V5 = T5 (mod 5) 23 U7 ≡ U7 + 7 V7 = T7 (mod 7) 59 U11 ≡ U11 + 11 V11 = T11 (mod 11).

45 4.3 Atkin–Lehner Theory 4 Hecke Operators

Now it follows from Sturm’s theorem (checking the ﬁrst 1, 2 and 5 coeﬃcients p(n)) that T`F` ≡ 0 (mod `) for ` = 5, 7, 11, proving the theorem.

Remark. It turns out that U`F` 6≡ 0 (mod `) for any primes ` ≥ 13. In particular, for each fixed arithmetic progression `n + b, there are infinitely many n for which p(`n + b) 6≡ 0 (mod `). However, it turns out that there are infinitely many congruences of the form

p(An + B) ≡ 0 (mod `c)

for every prime `. This can be shown by studying the equation

c Tp(U`F`) ≡ 0 (mod ` )

and varying the primes p.

What does it mean geometrically to have T`f ≡ 0 (mod `) for a prime `? An unsatisfying k−1 answer is that this is the same as U`f ≡ 0 (mod `) since T` = U` + ` V` ≡ U` (mod `). So instead, let’s ask: what does it mean geometrically for U`f ≡ 0 (mod `) for a prime `? To provide an answer, we will study the image of U` as a linear map Sk → Sk(Γ0(`)), as (`) well as the mod ` reduction of this setup. Write Sk for the mod ` reduction of the space (`) ∗ (`) (`) of weight k cusp forms, and likewise write Sk (Γ ),Tm ,Um for the mod ` reductions of the appropriate spaces/operators.

Remark. One can show that for any prime `, the Eisenstein series E`−1 satisﬁes E`−1 ≡ 1 (mod `). Thus we get a map

(`) (`) Sk −→ Sk+`−1 f 7−→ fE`−1 (mod `).

P∞ n Lemma 4.3.5. Let f = n=1 a(n)q ∈ Sk be a cusp form. Then for a prime `, the following are equivalent

(a) U`f ≡ 0 (mod `).

(b) P a(n)qn ∈ S(`). `-n k P∞ `−1 n (c) f ≡ n=1 a(n)n q (mod `).

`−1 d (d) θ f ≡ f (mod `) where θ is the operator θ = q dq .

Proof. (a) ⇐⇒ (b) By deﬁnition, U`f ≡ 0 (mod `) if and only if a(`n) ≡ 0 (mod `) for each n ≥ 1. (a) ⇐⇒ (c) For each n ≥ 1, note that ` | n is equivalent to n`−1 ≡ 0 (mod `). Then use Fermat’s little theorem. `−1 P∞ `−1 n (c) ⇐⇒ (d) Note that θ f = n=1 a(n)n q so (d) is just a restatement of (c).

(`) Theorem 4.3.6 (Serre). Let ` ≥ 5 be prime and f ∈ Sk. Then θf ∈ Mk+r(`,k) for some integer r(`, k) depending only on `, k.

46 4.3 Atkin–Lehner Theory 4 Hecke Operators

One obstacle to writing down systems of Hecke eigenforms for level subgroups Γ0(N) is that the operators Um and Vm need not preserve the eigenspaces of Tm. Speciﬁcally, if Tpf = λf, we might not necessarily have TpVmf = λVmf for all (or any) m.

Lemma 4.3.7. Let n, N ≥ 1 and let p be a prime with ordp(N) = m. Then there is a 2 × 2 integral matrix Qpα β W (Qp) = ∈ M2(Z) Nγ Qpdelta m with determinant Qp := p such that the natural action of W (Qp) on Mk(Γ0(N)) (by conjugation) is independent of the choices of α, β, γ, δ.

Deﬁnition. Let n, N, p and m be as above. A matrix W (Qp) is called an Atkin–Lehner m matrix with determinant Qp = p . The corresponding endomorphism of Mk(Γ0(N)) is called the Atkin–Lehner involution.

Deﬁnition. For each N ≥ 1, the Fricke involution is the endomorphism Mk(Γ0(N)) → Mk(Γ0(N)) given by the matrix

0 −1 W (N) = . N 0

Lemma 4.3.8. Let N ≥ 1. Then

(1) W (N) is an involution on Mk(Γ0(N)) and its action on the upper half-plane swaps the points 0 and ∞.

(2) For each prime p, W (Qp) is an involution on Mk(Γ0(N)).

(3) The matrices in the set {W (Qp): p | N} ∪ {W (N)} act faithfully and transitively on the cusps of X0(N).

m (4) When (n, N) = 1, ordp(N) = m and Qp = p , the involutions W (N) and W (Qp) commute with Tn. Here are some applications of Atkin–Lehner theory, which we will explore during the remainder of these notes:

The behavior of L-functions

The Shimura correspondence, which is a collection of Hecke-equivariant maps Sk+1/2 → S2k

Galois representations (for GL2).

47 4.4 Oldforms and Newforms 4 Hecke Operators

4.4 Oldforms and Newforms

Deﬁnition. Suppose k ≥ 2 is even and N | M, N 6= M. A cusp form f ∈ Sk(Γ0(M)) is an oldform if f is a ﬁnite linear combination of cusp forms Vdg for g ∈ Sk(Γ0(N)) M and d dividing N .A newform is an element of the orthogonal complement of the space old of oldforms in Sk(Γ0(M)) with respect to the Petersson inner product. Let Sk (M) (resp. new Sk (M)) denote the subspaces of oldforms (resp. newforms) of level Γ0(M). Theorem 4.4.1. Let N | M. Then

old new (1) Sk(Γ0(M)) = Sk (M) ⊕ Sk (M).

new (2) Sk (M) has an orthogonal basis (with respect to the Petersson inner product) of normalized Hecke eigenforms, each of which is also an eigenform of the Atkin–Lehner involutions.

Remark. There is an extension of Atkin–Lehner theory (due to Winnie Li) to Sk(Γ0(M), χ) for any Dirichlet character χ – here, oldforms must have level divisible by cond(χ), the conductor of χ.

P∞ n Theorem 4.4.2 (Deligne). Suppose f = n=0 a(n)q ∈ Sk(Γ0(M), χ) is a normalized newform. If p - M is prime, then |a(p)| ≤ 2p(k−1)/2. The theory of newforms and, specifically, Deligne’s theorem on the coefficients of newforms, has an application to so-called ‘Brauer theory’ in the modular representation theory of finite groups. In an attempt to attack Artin’s conjecture on modular L-functions, Brauer asked for a classification of all irreducible representations of a finite group G which remain irreducible mod p.

Example 4.4.3. For p = 2, 3, 100% of finite groups have no nontrivial irreducible representations in characteristic 0 whose reductions mod p stay irreducible. Just considering G = Sn, the remaining cases are classified as follows. When p = 2, Sn has an irreducible k(k+1) representation mod 2 if and only if n = 2 is a triangular number. Similarly, when p = 3, 2 2 Sn has an irreducible representation mod 3 if and only if n = x + 3y for x, y ∈ Z. Question 1 (Brauer’s Problem 19). Suppose p ≥ 5 is prime. Does every finite simple group G have at least one irreducible complex representation ρ : G → GL(V ) such that ρ is irreducible mod p?

The answer was shown to be YES, using Deligne’s theorem (4.4.2).

Theorem 4.4.4 (Granville-Ono). Brauer’s Problem 19 has a positive answer. Moreover, for p ≥ 5 prime, there exists a real number αp > 0 such that the number rn,p of irreducible complex representations ρ of Sn such that ρ is irreducible mod p has the following asymptotic behavior: (p−3)/2 rn,p ∼ αpn .

48 4.4 Oldforms and Newforms 4 Hecke Operators

Proof. (Sketch) Recall from Example 1.2.5 that by representation theory (a theorem of Frame-Robinson-Thrall), there is a bijective correspondence between irreducible representations of Sn and conjugacy classes in Sn, and these are in turn in bijection with partitions of n. There is a speciﬁc construction of all irreducible representations ρλ from partitions λ ` n given by the theory of Young tableaux. The hook formula shows that n! dimC ρλ = Q hλi

where hλi are the hook-lengths of λ. On the other hand, for any ﬁnite group G, a representation ρ : G → GL(V ) is irreducible mod p if and only if ordp(|G|) = ordp(dim V ). Applying this to G = Sn and using the hook formula, we are looking for partitions λ ` n such that p does not divide any hook-length of λ. As in Example 1.2.5, let cp(n) denote the number of p-core partitions of n, where p ≥ 5. (p−3)/2 Then we must show cp(n) > 0 for all n and cp(n) ∼ αpn . p2−1 To show the asymptotic formula for cp(n), set δp = 24 and ∞ η(pz)p X F (z) = = c (n)qn+δp . p η(z) p n=0

Since p ≥ 5, δp ∈ N so Fp is a modular form with integer coeﬃcients. By Theorem 3.1.20, ∗ p−1 ∗ Fp ∈ Mk(Γ0(p), χp) where k = 2 and χp is the character

 p  , if p ≡ 1 (mod 4) χ∗(n) = n p −p  n , if p ≡ 3 (mod 4).

Since the modular curve X0(p) has 2 cusps, there are exactly two Eisenstein series in ∗ Mk(Γ0(p), χp), namely Ek,∞ and Ek,0, where Ek,τ is nonvanishing at τ for each τ ∈ {0, ∞}. δp On the other hand, the ﬁrst term of Fp is q so it follows that

∗ Fp = Ek,0 + gp for some gp ∈ Sk(Γ0(p), χp).

The nontrivial part of this proof, which uses the theory of newforms, is that αp is the value of Fp at 0. Write ∞ X n X gp = bp(n)q = cjNp n=0 ∗ for normalized newforms Np ∈ Sk(Γ0(p), χp). Then Deligne’s theorem (4.4.2) shows that the n (p−3)/4+ε (p−3)/2 q coeﬃcient of each Np is O(n ) = o(n ). η(z)p Now, notice WpFp is a scalar multiple of η(pz) . It can be shown that this scalar has to be αp. Once this is established, we can write

∞ X X n/d E = 1 + d(p−3)/2qn, k,0 p n=1 d|n

(p−3)/2 and immediately see that cp(n) ∼ αpn .

49 4.4 Oldforms and Newforms 4 Hecke Operators

Finally, the fact that cp(n) > 0 for all n can be shown as follows. By definition, cp(0) > 1. n If we denote the q coefficient of Ek,0 by sp(n), then by Deligne’s theorem (4.4.2) it suffices to show X (p−3)/4 αp|sp(n)| > |γj| |s0(n)|n .

This boils down to establishing upper bounds on the γj. By Atkin–Lehner theory,

γj ≈ hFp − αpEk,0,Nji = hFp,Nji and this can now be eﬀectively bounded using the coeﬃcients of η(pz)p.

50 5 Connections to L-Functions

5 Connections to L-Functions

Recall that the Riemann zeta function is deﬁned by

∞ X 1 Y 1 ζ(s) = = ns 1 − p−s n=1 p prime for Re(s) > 1. A fundamental result is:

Lemma 5.0.1. ζ(s) has an analytic continuation to C. The proof of analytic continuation for ζ(s) relies crucially on the properties of the theta function θ(z) = P qn2 (see Example 1.1.5). More generally, modular forms are one of n∈Z the main tools for establishing and studying the analytic continuations of zeta functions and L-functions all over number theory.

5.1 L-Functions for Hecke Eigenforms

Assume k is an even integer.

P∞ n Deﬁnition. Let f(z) = q + n=2 a(n)q is a weight k cusp form of level Γ0(N). The Dirichlet series for f is deﬁned by

∞ X a(n) L(f, s) = . ns n=1

We will show that for any cusp form f ∈ Sk(Γ0(N)), L(f, s) has an analytic continuation to C. Lemma 5.1.1. If a, b ∈ R with b 6= 0 and ∞ 1 X = A(n)tn 1 − at + bt2 n=0 then A(0) = 1 and for n ≥ 1, the A(n) satisfy A(n) = aA(n − 1) − bA(n − 2). Proof. Note that ∞ X (1 − at + bt2) A(n)tn = 1. n=0 Then expanding and comparing coeﬃcients, we deduce the claimed formula.

Deﬁnition. For a prime p and a normalized Hecke eigenform f ∈ Sk(Γ0(N)), the function

∞ X a(pn) Z (f, s) = p pns n=0 is called the local zeta function for f at p.

51 5.1 L-Functions for Hecke Eigenforms 5 Connections to L-Functions

Lemma 5.1.2. For any normalized Hecke eigenform f ∈ Sk(Γ0(N)), 1 Z (f, s) = . p 1 − a(p)p−s + pk−1−2s

Proof. Since Tpf = a(p)f for any prime p, we get a(pn+1) = a(p)a(pn) − pk−1a(pn−1) so Lemma 5.1.1 gives the formula by plugging in t = p−s. Remark. The denominator 1 − a(p)t + pk−1t2 in the local zeta function for f has two roots, k−1 say π1(p) and π2(p), and they must satisfy π1(p)π2(p) = p . Then the Riemann hypothesis for these L-functions (as proven by Deligne) says that (k−1)/2 |π1(p) + π2(p)| < 2p .

Theorem 5.1.3. Suppose f ∈ Sk(Γ0(N)) and g(z) = (W (N)f)(z). If ∞ ∞ X af (n) X ag(n) L(f, s) = and L(g, s) = ns ns n=1 n=1 are the L-functions of f, g, then k+1 (1) L(f, s) and L(g, s) are absolutely convergent for Re(s) > 2 . In particular, they are analytic on this domain.

(2) Each of L(f, s) and L(g, s) has an analytic continuation to C. (3) For h = f, g, deﬁne √ !s N Λ(h, s) = Γ(s)L(f, s). 2π Then Λ(f, s) = ikΛ(g, k − s). (k−1)/2+ε Proof. (Sketch) By Deligne’s theorem (4.4.2), |af (n)| and |ag(n)| are O(n ). This implies (1). Assume N = 1 for simplicity. (2) follows from (3), so consider the function Γ(f, s) and the following computation: Z ∞ Z ∞ ∞ s−1 X 2πin(it) s−1 f(it)t dt = af (n)e t dt 0 0 n=1 ∞ Z ∞ X −2πnt s−1 = af (n) e t dt n=1 0 ∞ X af (n) = Γ(s) (2πn)s n=1 1 s = Γ(s)L(f, s) = Λ(f, s). 2π Applying the same calculation to g = W (N)f, after a change of variables we get the functional equation for Λ(f, s).

52 5.2 Modular Forms with Complex Multiplication 5 Connections to L-Functions

Deﬁnition. The function Λ(f, s) is called the completed L-function for the Hecke eigenform f.

new Remark. By Atkin–Lehner theory, Sk (Γ0(N)) has a basis of normalized Hecke eigenforms f satisfying W (N)f = ε(f)f for some root of unity ε(f), called the root number of f.

new Corollary 5.1.4. Suppose f ∈ Sk (Γ0(N)) is a normalized Hecke newform. Then (1) Λ(f, s) = ikε(f)Γ(f, k − s) where ε(f) is the root number of f.

(2) L(f, s) has an Euler product

Y 1 Y 1 L(f, s) = . 1 − a(p)p−s 1 − a(p)p−s + pk−1−2s p|N p-N

Conjecture 5.1.5 (Generalized Riemann Hypothesis for L(f, s)). If 0 < Re(ρ) < k and k L(f, ρ) = 0, then Re(ρ) = 2 . Here’s a glimpse into what is known about this version of the Generalized Riemann Hypothesis (for more, see Section 5.4):

The values L(f, 1),L(f, 2),...,L(f, k − 1) are all integers lying in [0, k], called Deligne periods. They show up in Voevodsky’s proof of the Bloch-Kato Conjecture.

If k = 2, then the value L(f, 1) is related to L-functions for modular elliptic curves.

Morally, the sum k−1 X j Rf (x) = L(f, j)x j=0 k−1 1 is self-reciprocal (by the functional equation), meaning x Rf x should look like Rf (x). Such polynomials, called period polynomials, have a Riemann hypothesis which says that all of the k − 1 zeroes of R (x) lie on the circle |z| = √1 . f N Theorem 5.1.6 (Ono-Soundararajan). The Riemann hypothesis for Deligne periods is true if k ≥ 4.

5.2 Modular Forms with Complex Multiplication

In this section, we construct a class of modular forms coming from algebraic number theory, using the theory of complex multiplication (due to Hecke, Ribet, Serre, et al). √ Let d < 0 be a squarefree integer and consider the imaginary quadratic ﬁeld K = Q( d) with ring of integers OK . Then any given integral basis of OK has a discriminant DK which is congruent√ to 0 or 1 (mod 4). Such a D is called a fundamental discriminant, and we have K = Q( D).

53 5.2 Modular Forms with Complex Multiplication 5 Connections to L-Functions

Definition. For an imaginary quadratic field K/Q, fix an ideal f ⊂ OK and let I(f) be the subgroup of the class group of K consisting of (classes of) ideals that are coprime to f. A map ζ : I(f) → C× is called a Hecke Grössencharakter mod f if ζ is a group homomorphism × × and there exists another group homomorphism ζ∞ : K → C such that ζ([α]) = ζ∞(α) for all α ≡ 1 mod f.

Remark. Here, the notation α ≡ 1 mod f means ordp(α−1) > ordp(f) for all primes p ∈ Z. It does not necessarily imply that α−1 ∈ f, although α−1 ∈ f does imply that α ≡ 1 mod f. √ Theorem 5.2.1. Let K = Q( D) be an imaginary quadratic field with D < 0 a fundamental × discriminant, let OK be its ring of integers and fix an ideal f ⊂ OK . Suppose ζ : I(f) → C is a Hecke Grössencharakter mod f such that for some k ∈ Z, α k−1 ζ (α) = ∞ |α| for all α ∈ K×. Then the function

X (k−1)/2 N (a) K/Q ff(z) = ζ(a)NK/Q(a) q a⊂OK

is a cusp form in Sk(Γ1(N)) where N = 101NK/Q(f). D Remark. In fact, ff ∈ Sk(Γ1(101NK/Q(f)), ζχD) where χD = · is the Kronecker character for D (a generalization of the Legendre symbol).

Corollary 5.2.2. Lf(s) := L(ff, s) has analytic continuation to C and there is a root of unity ε such that the completed L-function √ !s N Λ (s) := Γ(s)L (s) f 2π f satisﬁes the functional equation Λf(s) = εΛf(k − s).

Deﬁnition. A sequence (an) is lacunary if #{n ≤ x | a = 0} lim n = 1. x→∞ x P n Theorem 5.2.3 (Serre). Suppose f = n a(n)q is a weight k > 1 modular form. Then a(n) is lacunary if and only if f is a linear combination of modular forms with complex multiplication.

P δ n2 Example 5.2.4. The theta functions θ = n ψ(n)n q , δ ∈ {0, 1} are lacunary modular 1 3 forms (the coeﬃcients of their q-expansions are lacunary) of weight 2 or 2 (depending on the parity of ψ). More generally, the lacunary property seems to be connected to the existence of an under- lying rank 1 or rank 2 lattice (or collections of such lattices) over which Poisson summation implies modularity.

54 5.3 Connection to Elliptic Curves with CM 5 Connections to L-Functions

Theorem 5.2.5 (Serre). If r = 1, 2, 3, 4, 6, 8, 10, 14 or 26, then η(z)r is lacunary. For any even integer r not on this list, η(z)r is not lacunary.

Proof. (Sketch) For r = 1, 3, this was essentially proven in Chapter 1. For r = 2, 4, 6, this also follows from the results of Chapter 1 applied to η(z)2, η(z)4 and η(z)6. When r = 8, 10, 14, 26, Serre wrote η(z)r as an explicit linear combination of modular forms with CM. To prove the second statement, deﬁne

(−1)r/2 f (z) = η(24z)r ∈ S Γ (576), . r r/2 0 ·

There are only finitely√ many discriminants√ D√ < 0 dividing 576, giving finitely many possible CM fields: Q(i), Q( −2), Q( −3) and Q( −6). By algebraic number theory, if p ≡ 11 (mod 12) is prime then (p) is inert in each of these fields. But the coefficients in the q- expansion of a Hecke eigenform are Hecke eigenvalues and the exponents in a form with CM are norms of ideals, we know that Tpfr = 0 for all p ≡ 11 (mod 12) for any even r not on the list. This implies the second statement.

5.3 Connection to Elliptic Curves with CM

An elliptic curve E has complex multiplication (CM) if End(E) 6= Z. In general, Z ⊆ End(E) ∼ and one can show that for any such E, End(E) = OK where OK is the ring of integers in an imaginary quadratic field K/Q. Example 5.3.1. Consider the elliptic curve E given by affine Weierstrass equation y2 = x3 − √ h 1+ −3 i √ 1. Then End(E) = Z 2 , so E has CM in the imaginary quadratic field K = Q( −3). For any prime p ≥ 5,

p−1 ( X x3 − 1 p + 1, if p ≡ 2 (mod 3) #E( ) = p + 1 + = Fp p x=0 p + 1 ± 2x, if p ≡ 1 (mod 3).

Indeed, if p ≡ 2 (mod 3), then 3 does not divide p − 1, so x 7→ x3 is an automorphism of × Fp . Thus p−1 p−1 X x3 − 1 X x = = 0. p p x=0 x=0 On the other hand, p ≡ 1 (mod 3) is equivalent to x2 + 3y2 = p having an integer solution (x, y). This is in turn equivalent to (p) having a factorization (p) = (π)(¯π) in OK . Then

#E(Fp) = p + 1 + π +π ¯ which implies the formula in the second case. (By a theorem of Deuring, this is true for any elliptic curve E/Q with good reduction at p.)

55 5.3 Connection to Elliptic Curves with CM 5 Connections to L-Functions

Example 5.3.2. The elliptic curve E given by y2 = x3 − x has good reduction at every odd 5 prime p: the conductor of E is NE = 2 (e.g. by Tate’s conductor algorithm). It’s easy to see that E has CM in the ﬁeld K = Q(i). By a similar argument as above, p−1 ( X x3 − x p + 1, if p ≡ 3 (mod 4) #E( ) = p + 1 + = Fp p x=0 p + 1 ± 2x, if p ≡ 1 (mod 4). This is related to Fermat’s theorem that an odd prime p is a sum of two squares if and only if p ≡ 1 (mod 4).

2 2 Theorem 5.3.3. The newform F (z) = η(4z) η(8z) ∈ S2(Γ0(32)) has q-expansion F = P n n a(n)q satisfying p−1 X x3 − x a(p) = − p x=0 for all primes p ≥ 5. Proof. (Gauss) Suppose p ≡ 1 (mod 4) and write p = x2 + y2 where x ≥ 1 is odd and y ≥ 1. Then (x+y+1)/2 #E(Fp) = p + 1 + (−1) 2x. (In particular, this will determine the sign of ±2x in the example above.) Note that

2 ∞ η(16z) X 2 = q(2n+1) η(8z) n=0 ∞ X 2 η(8z)3 = (−1)n(2n + 1)q(2n+1) n=0 2 η(16z) X 2 2 =⇒ η(8z)2η(16z)2 = η(8z)3 = (−1)y(2y + 1)q(2x+1) +(2y+1) . η(8z) x,y≥0 Since F (2z) = η(8z)2η(16z)2, we see that a(p) is the coeﬃcient of q2p in the last expression above. This justiﬁes the formula after expanding p and 2p as sums of two squares.

If E/Q is an elliptic curve with aﬃne Weierstrass equation y2 = f(x), the L-function associated to E is ∞ X aE(n) Y 1 L(E, s) = = ns 1 − a (p)p−s + p1−2s n=1 p E where aE(p) is deﬁned by p−1 X f(x) a (p) = − E p x=0

for primes p - NE, and something similar for primes of bad reduction for E. There is also a modular form associated to E: ∞ X n FE = aE(n)q . n=1

56 5.4 Periods and Critical L-Values 5 Connections to L-Functions

Theorem 5.3.4 (Modularity Theorem – Taylor, Wiles, et al.). If E/Q is an elliptic curve with conductor NE, then

(1) FE is an Atkin–Lehner newform in S2(Γ0(NE)).

(2) L(E, s) has analytic continuation to C. (3) The completed L-function √ N s Λ(E, s) := E Γ(s)L(E, s) 2π

satisﬁes the functional equation Λ(E, s) = εΛ(E, 2 − s) where ε = −λNE is the eigenvalue of FE under W (NE). So far the only known method for proving analytic continuation and the functional equation for more complicated L-functions, like L(E, s), is the one used to prove the modularity theorem, i.e. using modular forms. Take a newform f ∈ Sk(Γ0(N)), where k is even. Then the functional equation Λ(f, s) = εΛ(f, k − s) determines a “critical strip” in C, namely {z ∈ C : 0 < Re(z) < k}. The special values L(f, j) for j = 1, 2, . . . , k − 1 encode important arithmetic data.

5.4 Periods and Critical L-Values

The Mordell–Weil theorem is one of the strongest motivators for studying values of L- functions of arithmetic elliptic curves.

Theorem 5.4.1 (Mordell–Weil). Let K be a number ﬁeld and E/Q an elliptic curve. Then E(K) is a ﬁnitely generated abelian group. A particularly famous unsolved problem in this area is the congruent number problem: recall that a positive integer n is a congruent number if there exists a rational right triangle (with all three side lengths rational) with area n. For example, 6 is congruent (it is the area of the right triangle with side lengths 3, 4, 5) but 1, 2, 3 and 4 are not congruent. One can show that 5 is also congruent. Question 2. Which positive integers are congruent numbers?

Lemma 5.4.2. A positive integer n is congruent if and only if there exist x, y ∈ Q with y 6= 0 such that y2 = x(x − n)(x + n). In other words, the congruent number elliptic curve

2 3 2 En : y = x − n x encodes the congruent number problem in the sense that En has a nontrivial rational point if and only if n is congruent. Furthermore:

Lemma 5.4.3. n is congruent if and only if En has positive rank.

57 5.4 Periods and Critical L-Values 5 Connections to L-Functions

This follows from another famous theorem in arithmetic geometry:

Theorem 5.4.4 (Nagell–Lutz). Given a suitable integral model E/Z for E/Q, for all primes p of good reduction, the reduction map E(Z) → E(Fp) is injective on torsion.

Corollary 5.4.5. If E/Q has an `-torsion point, where ` is prime, then ` divides Np(E) = p + 1 − aE(p) for almost all primes p.

In particular, for almost all primes p, we have aE(p) ≡ 1 + p (mod `). But in particular, for p ≡ 3 (mod 4), aE(p) = 0 so we must have ` = 2. Since any torsion point (x, y) with y = 0 is necessarily of order 2 (and vice versa), any rational point of En with y 6= 0 must have inﬁnite order. Thus the main question we must answer is how to test whether the rank of En(Q) is positive. The following version of Birch and Swinnerton-Dyer’s famous conjecture is sometimes known as the “weak” BSD conjecture.

Conjecture 5.4.6 (Birch–Swinnerton-Dyer). Suppose E/Q is an elliptic curve with conductor NE and consider the Hasse–Weil L-function of E: Y 1 Y 1 L(E, s) = −s −s 1−2s . 1 − aE(p)p 1 − aE(p)p + p p|NE p-NE Then

(1) L(E, s) has an analytic continuation to C.

(2) There exists a number εE ∈ {±1} such that Λ(E, s) = εEΛ(E, 2 − s) where √ N s Λ(E, s) = E Γ(s)L(E, s). 2π

(3) rank(E(Q)) = ords=1(L(E, s)). Remark. Due to its high proﬁle in modern number theory, the BSD conjecture has been well-studied and many things are known, although a full proof of BSD is probably a long way oﬀ.

(1) If εE = −1, ords=1(L(E, s)) is positive and odd. Despite the fact that εE = −1 for 50% of all elliptic curves over Q, we are not good at finding any points of infinite order in these cases. (2) Gross and Zagier’s so-called Heegner point method can often construct a rational point of infinite order from generalized singular moduli in the case ords=1(L(E, s)) = 1. (3) The full Birch–Swinnerton-Dyer conjecture gives meaning to the derivatives L(r)(E, 1) for all r ≥ 1, when ords=1(L(E, s)) = r, namely

(r) L (E, 1) |X(E/Q)|ΩERE Y = 2 cp(E) r! |E(Q)tors| p|NE where

58 5.4 Periods and Critical L-Values 5 Connections to L-Functions

X(E/Q) is the Tate–Shafarevich group of E/Q; Z 1 2 ΩE is the period of E, deﬁned by ΩE = p dx if E : y = f(x); R f(x)

RE is the regulator of E, deﬁned as an r × r determinant of the heights of generators of E(Q)/E(Q)tors;

cp(E) are the Tamagawa numbers of E mod p. (4) It is now known that 50% of numbers are congruent and 50% are not. However, the full BSD conjecture remains open.

The BSD conjecture can be phrased in terms of L-functions of weight 2 newforms, due to work of Manin and Shimura. Suppose f ∈ S2k(Γ0(N)) is a normalized Hecke newform. By Corollary 5.1.4, the completed L-function √ !s N Λ(f, s) = Γ(s)L(f, s) 2π satisfies a functional equation Λ(f, s) = εf Λ(f, 2k − s) where εf is the eigenvalue of f under W (N). The critical L-values L(f, 1),L(f, 2),...,L(f, 2k − 1) are conjectured to be arithmetically significant; in particular, the central L-value L(f, k) is supposed to encode information like the coefficients in the full BSD conjecture (e.g. Tamagawa numbers, |X|). Theorem 5.4.7 (Shimura–Manin). Assuming the hypotheses above,

+ − + − (1) There exist two real periods ωf and ωf such that hf, fi = ωf ωf . Λ(f, j) (2) If 1 ≤ j ≤ k − 1 is even, then + ∈ Q({af (n)}) where {af (n)} are the coeﬃcients ωf in the q-expansion of f. Λ(f, j) (3) If 1 ≤ j ≤ k − 1 is odd, then − ∈ Q({af (n)}. ωf

When k = 2, Eichler–Shimura deﬁne a map ϕ :Γ0(N) −→ C by

ϕ(γ) = Ef (γt) − Ef (t)

where Ef (t) is the function Z i∞ X af (n) n Ef (t) = q dz. t n n∈Z

Theorem 5.4.8 (Eichler–Shimura). If f is a weight 2 newform of level Γ0(N) deﬁned over Z, then ϕ is a group homomorphism whose image is a lattice Λf ⊂ C.

Thus any weight 2 newform f ∈ S2k(Γ0(N)) can be used to construct a modular elliptic curve E = C/Λf . The converse is basically the content of the Modularity Theorem (5.3.4).

59 5.5 Shimura Correspondence 5 Connections to L-Functions

Deﬁnition. For a weight 2 newform f ∈ S2(Γ0(N)), the period polynomial for f is Z i∞ 2k−2 Rf (x) = f(z)(z − x) dz. 0

2k−2 −(2k − 2)! X (2πix)n Lemma 5.4.9. R (x) = L(f, 2k − n − 1). f (2πi)2k−1 n! n=0 Example 5.4.10. The modular form f(z) = η(2z)4η(4z)4 is a weight 4 newform of level Γ0(8) (e.g. by Theorem 3.1.20). Then

2 Rf (x) ≈ −6.9925x + 4.3339ix + 0.87469 with roots approximately ±0.1703767 + 0.309793i. One can show that the square of the 1 norm of each of these roots is 8 (which is related to the level). It is surprising that both roots have the same norm, and something deeper is thought to be true. Conjecture 5.4.11 (Riemann Hypothesis for Period Polynomials). For any weight 2k newform f ∈ S2k(Γ0(N)), k ≥ 2, the roots of the period polynomial Rf (x) all lie on the circle |z| = √1 in the complex plane and are equidistributed. N

Lemma 5.4.12. For any newform f ∈ S2k(Γ0(N)), √ 1 R (x) = −i2k−2ε ( Nx)(k−2)/2R − f f f Nx

where εf is the root number of f.

5.5 Shimura Correspondence

To study critical L-values of newforms f ∈ S2k(Γ0(N)), we might look for these numbers in diﬀerent places in math. One of the best results in this direction says that the L(f, j) and the derivative values L(r)(f, j) are Fourier coeﬃcients of half-integral weight cusp forms (by work of Shimura, Kohnen–Zagier and Waldspurger in the 1980s) and harmonic Maass forms (by work of Bruinier–Ono in the 2010s). Recall that f ∈ Sλ+1/2(Γ0(4N), χ) if a b f(γz) = j(γ, z)2λ+1f(z)χ(d) for any γ = ∈ Γ (4N) c d 0 where j(γ, z) is the factor of automorphy: θ (γz) j(γ, z) = 0 . θ0(z)

1 By Theorem 3.1.18, all holomorphic modular forms of weight 2 are ﬁnite linear combinations of theta series of the form X n2 θr,t(z) = q . n≡r (mod t)

60 5.5 Shimura Correspondence 5 Connections to L-Functions

Moreover, there is even a ﬁnite basis of such theta series given by N θ (δz): t2 divides N, δ divides . r,t t2

This description does not restrict nicely to Sλ+1/2(Γ0(4N), χ) – in fact, the study of these spaces of half-integral weight modular forms is very deep. Example 5.5.1. Consider the quadratic forms

2 2 2 Q1(x, y, z) = x + y + 10z 2 2 2 Q2(x, y, z) = 2x + 2y + 3z − 2xz.

Then det(Q1) − det(Q2) = 10. Deﬁne a modular form

X Q1(x,y,z) Q2(x,y,z) f(q) = θQ1 (q) − θQ2 (q) = (q − q ). (x,y,z)∈Z3

Then f ∈ S3/2(Γ0(40), χ10). Notice that Q1 and Q2 are equivalent over Qp for every prime p, but one can show that they are not globally equivalent (i.e. over Q), so this is a failure of the local-global principle for such cusp forms. This half-integral weight cusp form f has some additional interesting features:

2 If n is odd and p ≥ 5 is prime such that p | n, then r1(n)r2(n) ≥ 1 where ri(n) is the n coeﬃcient of q in the q-expansion of θQi . That is, all such n are represented by each of the two quadratic forms Q1,Q2.

If n > 2719 is odd, then r1(n)r2 ≥ 1. That is, all odd n greater than 2719 are represented by each of the two quadratic forms Q1,Q2. More generally: Proposition 5.5.2. Suppose Q(x, y, z) = ax2 + by2 + cz2 is a positive deﬁnite diagonal ternary quadratic form. If p is a prime not dividing det(Q) and n ≥ 1 such that p2 - n and Q(x, y, z) = n is solvable over Qp, then rQ(n) ≥ 1. What about coeﬃcients for other half-integral weight cusp forms? Shimura proved the following in 1973.

P∞ n Theorem 5.5.3 (Shimura Correspondence). Suppose λ ≥ 1 and g = n=1 b(n)q ∈ Sλ+1/2(Γ0(4N), χ) is a half-integral weight cusp form. For squarefree t ≥ 1, deﬁne

−1λ t ψ (n) = χ(n) t n n

−1 t where · and · are the Legendre and Dirichlet symbols for −1 and t, respectively. Let At(n) be the sequence given by

∞ ∞ 2 X At(n) X b(tn ) = L(s − λ + 1, ψ ) . ns t ns n=1 n=1

61 5.5 Shimura Correspondence 5 Connections to L-Functions

Then the operator St deﬁned by

∞ X n (Stg)(z) := At(n)q n=1 satisﬁes:

2 (1) Stg ∈ M2λ(Γ0(N), χ ).

2 (2) If λ > 1, then Stg ∈ S2λ(Γ0(N), χ ).

3 2 (3) If λ = 1 and g is orthogonal to weight 2 theta functions, then Stg ∈ S2λ(Γ0(N), χ ).

Deﬁnition. The operator St in the theorem is called the tth Shimura correspondence.

Proposition 5.5.4. For every t ≥ 1 squarefree, St is Hecke-equivariant. That is, if p is a prime not dividing 4N, then TpStg = StTp2 g for every g ∈ Sλ+1/2(Γ0(4N), χ). This implies that the coeﬃcient b(n) of Hecke a eigenform g = P b(n)qn can be deter- mined from b(np2) and b(n/p2). Thus the coeﬃcients of such a g break up into two types: b(n) for n squarefree and b(n) for n not squarefree (the latter class containing all Hecke eigenvalues of g).

2 Remark. Suppose Stg ∈ S2λ(Γ0(N), χ ) is a normalized Hecke eigenform. Then TpStg = p λpStg where λp is the coeﬃcient of q in the q-expansion of g. However, this is not enough to determine g: we would still need to understand the coeﬃcients of squarefree n.

Theorem 5.5.5 (Shimura, Kohnen, Waldspurger, et al.). Suppose λ, N ≥ 1 are ﬁxed. Then

(1) For any nebentypus χ, the Shimura correspondences Sλ+1/2(Γ0(4N), χ) → S2λ(Γ0(N)) are all surjective.

+ (2) There is a subspace Sλ+1/2(Γ0(4N), χ) of Sλ+1/2(Γ0(4N), χ) mapping isomorphically onto S2λ(Γ0(N)).

P n (3) Suppose f = a(n)q ∈ S2λ(Γ0(N)) is a normalized Hecke eigenform which is the P n + image of g = b(n)q ∈ Sλ+1/2(Γ0(4N), χ), then

(a) StTp2 g = a(p)f. (b) If n is squarefree and (n, 4N) = 1, then for the L-function

∞ X a(m) L(f ⊗ χ , s) := χ (m), n ms n m=1

2 where χn is the Dirichlet character mod n, we have L(f ⊗ χn, λ) = C|b(n)| for some positive constant C.

62 5.5 Shimura Correspondence 5 Connections to L-Functions

This shows that for any normalized Hecke eigenform f ∈ S2λ(Γ0(N)), the half-integral weight form g corresponding to f under the Shimura correspondence is a generating function for both L(f ⊗χn, λ) and the Hecke eigenvalues of f. Thus in some sense, this g knows more about f than f itself! Coming back to the connections to elliptic curves described in Section 5.3, the modularity theorem (5.3.4) suggests there should be a connection between elliptic curves E/Q with + conductor NE and elements of S3/2(Γ0(4NE)). Theorem 5.5.6 (Kolyvagin). The Birch–Swinnerton-Dyer conjecture for the congruent number elliptic curve En is equivalent to rank(En) = ords=1(L(En, s)) if n is squarefree and ords=1(L(En, s)) ≤ 1. Theorem 5.5.7 (Bruinier–Ono). There is a Hecke-equivariant correspondence

+ S3/2(Γ0(4N)) ←→ H1/2(Γ0(N))

1 where Hλ+1/2 denotes the space of harmonic Maass forms of weight λ + 2 . Thus the following objects are all in correspondence: Elliptic curves E/Q of conductor NE

P∞ aE (n) L-functions L(E, s) = n=1 ns new Hecke newforms fE(z) ∈ S2 (Γ0(NE)) 3 + Weight 2 cusp forms gE(z) ∈ S3/2(Γ0(4NE))

Harmonic Maass forms hE(z) ∈ H1/2(Γ0(NE)).