= −ieπig/NEg,0(N τ ) = −ieπig/NEg(τ ).
2.4 Petersson Inner Product and Hecke Operators
2.4.1 Petersson inner product
Let Γ be a subgroup of P SL2(Z) of finite index. The vector space Sk(Γ) = {f : f is a cusp form of wight k on Γ}
is equipped with an inner product, called the Petersson inner product.
Definition 2.4.1 Let D be a fundamental domain for Γ. Then the Petersson inner
The Petersson inner product is defined only on Sk(Γ) because if f and g are not cusp forms then the integral may not be finite. However, the integral will converge whenever at least one of f and g is a cusp form. We remark that
(1) The hyperbolic measure dxdy/y2 is invariant under the substitution τ 7→ γτ for any γ ∈ GL+2(R).
(2) The factor 1/[P SL2(Z) : Γ] is inserted so that the inner product hf, gi will remain the same if we consider f and g as modular forms on a smaller subgroup Γ0.
(3) The Petersson inner product is independent of the choice of the fundamental domain D.
2.4.2 Hecke Operators on modular forms on Γ
0(N )
To define the Hecke operators Tn on Γ0(N ) it is necessary to distinguish the cases gcd(n, N ) = 1 from the cases gcd(n, N ) 6= 1. When n is a positive integer relatively prime to N , we consider the sets
M(N )n =a b
instead. Then the modular group Γ0(N ) acts on M(N )n by multiplication from left. We denote the set of orbits (equivalence classes) by Sn(N ).
Proposition 2.4.2 Let σ ∈ Γ0(N ). Then φσ : Sn(N ) → Sn(N ) given by φσ([γ]) = [γσ] is a well-defined permutation on the elements of Sn(N ).
Proposition 2.4.3 Let n be a positive integer relatively prime to N . A complete set of representatives of orbits in Sn(N ) is
a b
Definition 2.4.4 Let k be an even integer and n a positive integer. Write n in the form mpe11. . . perr such that gcd(m, N ) = 1 and pi|N are prime divisors of N . Then the nth Hecke operator Tn on the space of modular forms of weight k is defined by
Tn= TmTpe11. . . Tperr, where
Thf = hk/2−1 X
[γ]∈ Sh(N )
f [γ]k.
Proposition 2.4.5 The Hecke operators Tn are linear transformations on the space of modular forms on Γ0(N ). Moreover, if f is a cusp form, then so is Tnf .
Proposition 2.4.6 For all positive integers m and n with gcd(m, n) = 1 we have Tmn= TmTn.
Furthermore, suppose that m and n are positive integers such that both of them are rela-tively prime to N . Then we have
TmTn = X
d| gcd(m,n)
dk−1Tmn/d2 = TnTm.
In the following contents we will determine the adjoints Tn∗ of the Hecke operators Tn with respect to the Petersson inner product. For convenience, throughout this section the notation h·, ·i without any subscript denotes the inner product on the space of modular forms on P SL2(Z), while h·, ·iΓ carries the usual meaning.
Proposition 2.4.7 Let n be a positive integer relatively prime to N . The Hecke operators Tn are self-adjoint (also called hermitian) with respect to the Petersson inner product.
That is, we have
hTnf, gi = hf, Tngi.
for all modular forms f and g of weight k on Γ0(N ).
Since Tn are self-adjoint, an elementary result in linear algebra asserts that the eigen-values of Tn are all real.
Proposition 2.4.8 For all positive integers n with gcd(n, N ) = 1, the eigenvalues of Tn are all real.
Now we have a family of self-adjoint linear operators Tnthat are commuting with each other on an inner product space. By a well-known theorem in linear algebra, These linear operators Tn are simultaneously diagonalizable. In other words, the vector space has a basis consisting entirely of simultaneous eigenvectors.
Proposition 2.4.9 There is a decomposition of the vector space Sk(Γ0(N )) into a direct sum
Sk(Γ0(N )) = ⊕ Vi
of orthogonal subspaces Vi such that each Vi is a simultaneous eigenspace for all Tn with gcd(n, N ) = 1.
Moreover, if f is an eigenform in Vi, then so is Tpf for p|N . Therefore, each Tp, p|N , stabilizes Vi. However, in general, Vi may not have a basis consisting of simultaneous eigenforms for all Tn. There does not exist a basis whose elements are all simultaneous eigenforms for all Tn. Nevertheless, if f =P∞
n=1cnqn is a simultaneous eigenforms for all Tn, then f still enjoys the property that Tnf = cnf .
In general, let f be a non-vanishing modular form of weight k on P SL2(Z).
Definition 2.4.10 If f is a simultaneous eigenform for all Hecke operators Tnon P SL2(Z), then we say f is a simultaneous eigenform or a Hecke eigenform. If the Fourier expansion of f has leading coefficient 1, then f is normalized.
Proposition 2.4.11 The space Sk(P SL2(Z)) of cusp forms of weight k on P SL2(Z) is spanned by simultaneous eigenforms.
2.4.3 Properties of Hecke eigenforms
Throughout this section we let k be a positive even integer and d be the dimension of Sk(P SL2(Z)). We assume that {f1, . . . , fd} is a basis of Sk(P SL2(Z)) consisting of Hecke eigenforms. In this section we will study properties of fi.
Proposition 2.4.12 Let f be a Hecke eigenform, and assume that the Fourier expansion of f is c1q + c2q2+ · · · . Then c1 6= 0.
Recall that we say a Hecke eigenform is normalized if the leading Fourier coefficient is 1, the Fourier coefficients of a normalized Hecke eigenform are multiplicative.
Proposition 2.4.13 Let f be a normalized Hecke eigenform with a Fourier expansion q + c2q2+ c3q3+ · · · . Then Tnf = cnf for all positive integers n.
Proposition 2.4.14 If f is a normalized Hecke eigenform with a Fourier expansion q + c2q2+ c3q3· · · , then we have
cmcn= X
d| gcd(m,n)
dk−1cmn/d2
for all positive integers m and n. In particular, if gcd(m, n) = 1, then cmn = cmcn. Proposition 2.4.15 Let f 6= g be two normalized Hecke eigenforms. Then f and g are orthogonal with respect to the Petersson inner product.
2.4.4 Newforms and oldforms
Some of the cusp forms in Sk(Γ0(N )) actually have level smaller than N . Namely, if M is an integer dividing N , then any cusp forms on Γ0(M ) is also a cusp form on Γ0(N ).
Proposition 2.4.16 Let M be a positive integer dividing N , and f (τ ) be a cusp form on Γ0(M ). Then for any h|(N/M ), the function f (hτ ) is a cusp form on Γ0(N ). Moreover, if f (τ ) is a simultaneous eigenform for all Tn with gcd(n, N ) = 1, then so is f (hτ ).
Definition 2.4.17 If f (τ ) ∈ Sk(Γ0(N )) satisfies f (τ ) = g(hτ ) for some simultaneous eigen-form g(τ ) ∈ Sk(Γ0(M )) with M |N , M < N , and h|(N/M ), then f (τ ) is called an oldform.
The subspace spanned by all oldforms is called the space of oldforms and is denoted by Soldk (Γ0(N )). The orthogonal complement is called the space of newforms and is denoted by Snewk (Γ0(N )).
Proposition 2.4.18 Each of Soldk (Γ0(N )) and Snewk (Γ0(N )) is stable under Tn for all n with gcd(n, N ) = 1.
Proposition 2.4.19 The subspace Snewk (Γ0(N )) has a basis consisting of simultaneous eigenforms for all Tn with gcd(n, N ) = 1.
Definition 2.4.20 Let f ∈ Snewk (Γ0(N )). If f is a non-vanishing simultaneous eigenform for all Tn with gcd(n, N ) = 1, then f is a newform.
In fact, more about Snewk (Γ0(N )) is true. To facilitate further discussion. Let us recall a result of Atkin and Lehner.
Proposition 2.4.21 (Atkin-Lehner) Let f (τ ) =P∞
n=1cnqn be a cusp form on Γ0(N ).
If there exists a positive integer M such that whenever gcd(n, M ) = 1 the coefficient an
vanishes, then f lies in the space of oldforms.
Proposition 2.4.22 Let f (τ ) =P∞
n=1cnqn be a newform on Γ0(N ). Then c1 6= 0.
Proposition 2.4.23 Let Snewk (Γ0(N )) = ⊕ Vi be the decomposition of Snewk (Γ0(N )) into a direct sum of simultaneous eigenspaces for all Tn with gcd(n, N ) = 1. Then each Vi has dimension 1.
Before we state our last results in this chapter, let us recall that the Atkin-Lehner involutions
wn=
1
√n
an b cN dn
, n|N, gcd(n, N/n) = 1, adn2− bcN = n
, normalize Γ0(N ). Thus if f is a modular form of weight k on Γ0(N ), then so is f
[wn]k. For the ease of notation, given a modular form on Γ0(N ), we write
Wnf = f [wn].
We remark that since wn2 ∈ Γ0(N ), the coset Γ0(N )wn is all the same for a fixed n, regardless of which wnwe choose. Therefore, the definition of Wn does not depend on the representative wn.
Proposition 2.4.24 If m is a positive integer such that gcd(m, N ) = 1, then TmWn = WnTm.
Proposition 2.4.25 The space of newforms is spanned by cusp forms that are simulta-neous eigenforms for all Atkin-Lehner involutions and all Hecke operators.
Remark. Since w2n ∈ Γ0(N ), we have Wn2f = f . Thus, the eigenvalues of Wn must be 1 or −1.
Finally we study the eigenvalue of Tp, p|N , associated with a newform.
Proposition 2.4.26 Let f =P∞
n=1cnqn be a normalized newform on Γ0(N ). Let p be a prime divisor of N . If p2|N , then cp = 0. If p2 does not divide N , then cp = −pk/2−1, where is the eigenvalue of Wp associated with f .
Chapter 3
Elliptic Curves
An elliptic curve is a pair (E, O), where E is a smooth projective curve of genus 1 and O is a basepoint of E.
Let K be any perfect field. The elliptic curve (E, O) is said to be defined over K if the curve is defined over K, which is also denoted by E/K, and O is a rational point on E defined over K. Every such curve can be written as the locus in P2 of a cubic equation with only one point on the line at infinity. That is, after scaling X and Y , as an equation of the form
Y2Z + a1XY Z + a3Y Z2 = X3+ a2X2Z + a4XZ2+ a6Z3. Here O = [0, 1, 0] and ai ∈ K. And there exists a morphism
+ : E × E → E defined by (P1, P2) 7→ P1+ P2
giving the group law such that the set of rational points on E with identity O (point at infinity) forms an abelian group.
3.1 Definitions
In this chapter, we consider the special case K = Q.
A cubic curve in normal form looks like
y2 = f (x) = x3+ ax2+ bx + c.
Assuming that the roots of f (x) are distinct, that is, this curve is nonsingular. Then such a curve is an elliptic curve, and every point on an elliptic curve has a well-defined tangent line. The reason is as follows. If we write the equation as F (x, y) = y2− f (x) = 0 and take the partial derivatives,
∂F
∂x = −f0(x), and ∂F
∂y = 2y.
We can see that there is no point on the curve at which the partial derivatives vanish simultaneously, since the curve is nonsingular. The details can be found in Rational Points on Elliptic Curves by J.H. Silverman [13].
If this curve is singular, we classify the singular points depending on its tangent di-rections.
Definition 3.1.1 Let P be a singular point on the curve F (x, y) = 0. We say that P is a node if there are two distinct tangent directions at P . P is a cusp if there is one tangent direction at P .
3.1.1 Minimal Weierstrass Equation
We may assume that each elliptic curve have a Weierstrass equation of the form E : y2+ a1xy + a3y = x3+ a2x2+ a4x + a6.
And this equation has an associated discriminant
∆ = − (a21+ 4a2)2(a21a6+ 4a2a6− a1a3a4+ a2a23 − a24) − 8(2a4+ a1a3)3
− 27(a23+ 4a6)2+ 9(a21+ 4a2)(2a4 + a1a3)(a23+ 4a6).
If we replace the variables (x, y) by (x/u2, y/u3), then each ai in the Weierstrass equation E : y2 + a1xy + a3y = x3+ a2x2+ a4x + a6 becomes uiai and the discriminant is u12∆. In this way, we can choose u such that uiai are all integers.
Let E0 be a new equation
E0 : y2+ b1xy + b3y = x3+ b2x2+ b4x + b6
of this elliptic curve E form changing variables such that bi are all integers. Let ∆0 be the discriminant of E0. For each prime p, define vp(∆0) as the power of p such that pvp(∆0) | ∆0 but pk - ∆0 if k > vp(∆0).
Definition 3.1.2 A Weierstrass equation is called a minimal Weierstrass equation E0 for E at p if vp(∆0) is minimized subject to the condition bi ∈ Z.
If we define
vp(∆) = min
E0 vp(∆0), then the minimal discriminant of E is defined by
D =Y
p
pvp(∆).
Definition 3.1.3 A Weierstrass equation is called a global minimal Weierstrass equation for E if E is simultaneously minimal at all primes of Q. The discriminant ∆ of this global minimal Weierstrass equation is equal to the minimal discriminant D of E/Q.
Proposition 3.1.4 ([12], Corollary 8.3) Every elliptic curve E/Q has a global mini-mal Weierstrass equation.
We remark that we can find a global minimal Weierstrass equation for E/Q by finding local minimal equations(e.g. by using Tate’s algorithm).
3.1.2 Reduction
The reduction of E modulo p, denoted eE, is then the curve over Zp defined by the equation E : ye 2+ae1xy +ae3y = x3+ae2x2+ae4x +ae6,
whereaei denotes reduction modulo p. The curve eE may be singular; its non-singular part is denoted eEns.
Definition 3.1.5 We say that
(1) E has good (stable) reduction if eE is non-singular.
(2) E has multiplicative (semi-stable) reduction if eE has a node. And the reduction is called split if the tangent directions are defined over Zp, otherwise it is non-split.
(3) E has additive (unstable) reduction if eE has a cusp.
In cases (2) and (3), E is naturally said to have bad reduction.
3.1.3 Conductor
The minimal discriminant is a measure of the bad reduction of E. Another such measure is the conductor of E/Q.
Definition 3.1.6 The conductor of E/Q is defined by N (E/Q) =Y
p
pfp(E/Q),
where the exponents fp(E/Q) are given by
fp(E/Q) =
0, if E has good reduction at p,
1, if E has multiplicicative reduction at p, 2 + δp, if E has additive reduction at p,
where δp = 0 if p - 6.
Further, fp may be computed by using Ogg’s formula[11].
3.1.4 L-Series
The L-series of an elliptic curve is a generating function which records information about the reduction of the curve modulo every prime.
Let E be an elliptic curve. Set q = pk, for some prime p.
Definition 3.1.7 Let E(Fqr) be the set of points on E with coordinates in Fqr. The zeta function of E over Fqr is given by the formal power series
Z(E/Fq; T ) = exp Proposition 3.1.8 [12] There is an integer a so that
Z(E/Fq; T ) = 1 − aT + qT2
For each prime p, if E has good reduction at p, let ap = p + 1 − # eE(Fp). The local
1 − T, if E has split multiplicicative reduction at p, 1 + T, if E has non-split multiplicicative reduction at p, 1, if E has additive reduction at p.
In all cases, the relation Lp(1/p) = # eE(Fp)/p holds.
Definition 3.1.9 We make substitution T = p−s in Z(E/Fp; T ), and define Hasse-Weil L-series L(E, s) by
L(E, s) = ζ(s)ζ(s − 1) ΠpZ(E/Fp; p−s), where ζ(s) is the Riemann zeta function defined by
ζ(s) =X
n∈N
1
ns, for Re(s) > 1 and we can express ζ(s) as
ζ(s) = Y
2. For each prime p - N , let T (p) be the corresponding Hecke operator; and let W be the operator (W f )(τ ) = f (−1/N τ ). Then
T (p)f = cpf and W f = wf, where w = ±1 is the sign of the functional equation
ξE(s) = wξE(2 − s), where ξE(s) = (2π)−sNs/2Γ(s)L(E, s), for s ∈ C, with the well-know Gamma-function Γ(s).
3. Let ω be an invariant differential on E/Q. There exists a morphism φ : X0(N ) → E, defined over Q, such that φ∗(ω) is a multiple of the differential form on X0(N ) represented by f (τ )dτ .