A weak modular form of weight k for a subgroup Γ ≤ GL(2, ℝ): a slash-invariant
function on the upper half plane that is holomorphic (MDiff), but without any
condition on growth at the cusps.
- toFun : UpperHalfPlane → ℂ
- holo' : MDiff ⇑self.toSlashInvariantForm
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Modular forms of weight k for the full modular group SL(2, ℤ) = Γ(1).
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Forgetful map: every modular form for SL(2, ℤ) is in particular a weak modular
form (we drop the boundedness-at-cusps condition).
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Modular forms of weight k for the congruence subgroup Γ₀(N) ≤ SL(2, ℤ).
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Weak modular forms of weight k for Γ₀(N) (holomorphic and slash-invariant,
without cuspidal growth condition).
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Cusp forms of weight k for the full modular group SL(2, ℤ) = Γ(1) (modular
forms vanishing at every cusp).
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Cusp forms of weight k for the congruence subgroup Γ₀(N) ≤ SL(2, ℤ).
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A cusp form for SL(2, ℤ) is in particular a modular form (vanishing at cusps
implies boundedness at cusps).
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A cusp form for SL(2, ℤ) is in particular a weak modular form (drop both the
boundedness and vanishing conditions at cusps).
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The Taylor–Wiles theorem: every semistable elliptic curve over ℚ is modular.
The Breuil–Conrad–Diamond–Taylor theorem extending Taylor–Wiles: every elliptic
curve over ℚ is modular.
The Modularity Theorem for elliptic curves over ℚ: every elliptic curve over
the rationals is modular.
Predicate stating that an integral Weierstrass model W is a model for the
rational Weierstrass curve E, i.e. there is a ℚ-rational change of variables
taking W (viewed over ℚ) to E.
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Structure encoding that W is a minimal integral model for E: it is a model
for E and its discriminant divides that of every other integral model of E.
- is_model_for : IsModelFor E W
- discriminant_dvd (W' : WeierstrassCurve.Affine ℤ) : IsModelFor E W' → WeierstrassCurve.Δ W ∣ WeierstrassCurve.Δ W'
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The minimal discriminant of a rational Weierstrass curve: the discriminant of its minimal integral model.
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The minimal discriminant of a rational Weierstrass curve is nonzero.
The minimal discriminant of E divides the discriminant of any integral
Weierstrass model of E.
The (chosen) minimal integral Weierstrass model of a rational Weierstrass curve.
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The four possible reduction types of an elliptic curve at a prime: good reduction, split multiplicative reduction, nonsplit multiplicative reduction, or additive reduction.
- good : ReductionType
- splitMultiplicative : ReductionType
- nonsplitMultiplicative : ReductionType
- additive : ReductionType
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The reduction type of an elliptic curve E over ℚ at the prime p, computed
from the reduction of its minimal model modulo p.
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The local conductor exponent of an elliptic curve at a prime, giving the power of that prime appearing in the global conductor.
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At a prime of good reduction, the conductor exponent is 0.
At a prime of split multiplicative reduction, the conductor exponent is 1.
At a prime of nonsplit multiplicative reduction, the conductor exponent is 1.
At a prime p > 3 of additive reduction, the conductor exponent is 2.
The conductor of an elliptic curve over ℚ: the product over primes dividing
the minimal discriminant of p raised to the local conductor exponent.
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The conductor of an elliptic curve over ℚ is positive.
An elliptic curve over ℚ is semistable if it has good or multiplicative
(but never additive) reduction at every prime.
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If at every bad prime the reduction of E is split or nonsplit multiplicative
(so that the local exponent is 1), the conductor is simply the squarefree
product of those primes.
A semistable elliptic curve has squarefree conductor.
An elliptic curve with squarefree conductor is semistable.
Characterisation: an elliptic curve over ℚ is semistable iff its conductor
is squarefree.
The trace of Frobenius of E at the prime p: at primes of good reduction
it equals p + 1 − #E(𝔽_p), and at primes of bad reduction it takes the
conventional values 0 (additive), 1 (split multiplicative), or −1
(nonsplit multiplicative).
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The local character χ(p): equals 1 at primes of good reduction and 0
at primes of bad reduction.
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The local L-polynomial at p: 1 − a_p X + p X² at primes of good reduction,
and 1, 1 − X, or 1 + X at primes of additive, split multiplicative, or
nonsplit multiplicative reduction respectively.
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The local Euler factor at p and complex variable s: the reciprocal of the
local L-polynomial evaluated at p^{-s}.
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The (Hasse–Weil) L-function of an elliptic curve over ℚ, defined as the
Euler product of local factors over all primes.
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At a prime of additive reduction, the trace of Frobenius is 0.
At a prime of split multiplicative reduction, the trace of Frobenius is 1.
At a prime of nonsplit multiplicative reduction, the trace of Frobenius is
-1.
At any prime of bad reduction, the trace of Frobenius lies in {0, 1, -1}.
The Euler product defining the L-function converges (as an unordered product)
for Re(s) > 3/2.
The local zeta function of E at p as a rational function in T: the
local L-polynomial divided by (1 − T)(1 − pT).
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The number of 𝔽_{p^n}-points on the reduction of E modulo p.
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At a prime of good reduction, the local zeta function admits an exponential
expression in terms of the point counts over extensions of 𝔽_p.
Hasse bound: at a prime of good reduction, a_p² ≤ 4p, equivalently
|a_p| ≤ 2√p.
The integer Fourier coefficients of the (conjectural) weight-2 newform
attached to the elliptic curve E.
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At every prime p, the p-th q-expansion coefficient of E agrees with the
trace of Frobenius a_p(E).
Eichler–Shimura–Carayol: every integral normalized weight-2 newform f arises
as the q-expansion of an elliptic curve E over ℚ of conductor f.level.
Variant of Eichler–Shimura–Carayol at primes: every integral normalized weight-2 newform comes from an elliptic curve whose traces of Frobenius match the newform's prime coefficients.
An elliptic curve E over ℚ is modular if there exists an integral
weight-2 newform whose Fourier coefficients agree with the q-expansion
coefficients of E.
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A modular elliptic curve E is associated with a newform whose level equals
the conductor of E.
The L-function of a newform: the Dirichlet L-series with coefficients
a_n ∈ ℂ.
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An elliptic curve E has a modular L-function if its L-function equals
that of some newform whose level matches the conductor of E.
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The L-function of an elliptic curve agrees with the Dirichlet L-series formed from its q-expansion coefficients.
If the q-expansion coefficients of E agree with those of a newform f,
then their L-functions agree.
Converse: if the L-functions of E and a newform f agree, then so do
their coefficients.
The two notions of modularity (coefficient-level and L-function-level) coincide.
Modularity theorem (coefficient form): every elliptic curve over ℚ is
modular at the level of Fourier coefficients.
Modularity theorem (L-function form): every elliptic curve over ℚ has a
newform with matching L-function.
Global statement of the modularity theorem in the namespace's preferred form:
every elliptic curve over ℚ is modular.
Predicate stating that p is a prime of good reduction for both E₁ and
E₂.
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Faltings–Tate isogeny theorem: if E₁ and E₂ have matching traces of
Frobenius at all but finitely many common good primes, then they are
isogenous.
Isogenous elliptic curves have the same trace of Frobenius at every prime.
Isogenous elliptic curves have the same reduction type at every prime.
Isogenous elliptic curves have equal L-functions.
If two elliptic curves have the same L-function, then their traces of Frobenius agree at every prime.
Two elliptic curves over ℚ are isogenous iff they have the same
L-function.
Isogenous elliptic curves have equal conductor.
Two elliptic curves are isogenous iff there are only finitely many common good primes at which their traces of Frobenius differ.
Combined characterisation: isogeny is equivalent to equality of L-functions together with matching traces at almost all common good primes.
Global re-statement: two elliptic curves over ℚ are isogenous iff they
have the same L-function.
The free abelian group on the set of complex lattices, viewed as the divisor group of lattices.
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Coercion of a positive natural number n to a nonzero complex unit.
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The action of a nonzero complex scalar l on a complex lattice, scaling
both periods.
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The Hecke operator T_n acting on the free abelian group of lattices: sends
each lattice to the formal sum of its index-n sublattices.
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The Hecke operator at n = 1 is the identity.
The homothety operator R_l: the ℤ-linear extension of the scaling action
on lattices by the scalar l ∈ ℂˣ.
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Scalings compose by multiplication of scalars.
The Hecke operator T_n commutes with every homothety R_l.
Homotheties multiply: R_l ∘ R_μ = R_{lμ}.
The finite set of index-n sublattices of a given lattice L.
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Defining identity for the Hecke operator: T_n sends a basis lattice to the
formal sum of its index-n sublattices.
The composition T_{p^{r+1}} ∘ T_p applied to a lattice unfolds to a double
sum over sublattices counting pairs (L', L'').
The double sum decomposition: pairs (L', L'') of nested sublattices split
into index-p^{r+2} sublattices of L plus a p-multiplied correction term
from scaled index-p^r sublattices.
Recurrence for Hecke operators at prime powers:
T_{p^{r+2}} = T_{p^{r+1}} T_p − p · T_{p^r} R_p.
The generating set for the Hecke algebra (on divisors of lattices): all
prime Hecke operators T_p and all prime homotheties R_p.
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Pairwise commutativity of the generating Hecke operators: any two generators
in heckeGenerators commute.
The subring of End ℤ DivL generated by the Hecke operators and homotheties
is commutative.
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Every prime-power Hecke operator T_{p^r} lies in the subring generated by
prime Hecke operators and homotheties.
For every positive n, the Hecke operator T_n lies in the subring
generated by heckeGenerators, by reduction to prime powers and
multiplicativity.
The Hecke operator T_n acting on modular forms of weight k for
SL(2, ℤ).
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The homothety operator R_l acting on modular forms of weight k for
SL(2, ℤ).
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The prime-power Hecke recurrence on modular forms in the form involving the
homothety operator R_p.
The scalar form of the prime-power Hecke recurrence on weight-k modular
forms: T_{p^{r+2}} = T_{p^{r+1}} T_p − p^{k-1} T_{p^r}.
The absolute Galois group of ℚ, Gal(ℚ̄/ℚ).
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Group structure on the absolute Galois group of ℚ.
A choice of Frobenius element in Gal(ℚ̄/ℚ) for the prime p.
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The ℓ-adic Galois representation ρ_{E, ℓ} : Gal(ℚ̄/ℚ) → GL₂(ℤ_ℓ)
associated to an elliptic curve E, arising from the Tate module.
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The mod-ℓ Galois representation attached to E: the composition of the
ℓ-adic representation with reduction modulo ℓ.
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At a prime p of good reduction, the trace of Frobenius on the ℓ-adic
Galois representation of E equals the integer trace of Frobenius a_p(E)
viewed in ℤ_ℓ.
A formal Dirichlet series, wrapping a sequence of complex coefficients.
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The value at s of a Dirichlet series, defined as the L-series of its
coefficient sequence.
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Summability of a Dirichlet series at s: the underlying L-series sum is
summable.
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Unfolding lemma: the eval of a Dirichlet series is definitionally the
L-series of its coefficients.
The Dirichlet series summability predicate unfolds to L-series summability.
If the coefficients are O(n^σ) at infinity, the Dirichlet series is
summable at every s with Re(s) > σ + 1.
Under the same O(n^σ) hypothesis on coefficients, the Dirichlet series is
differentiable on the half-plane Re(s) > σ + 1.
The arithmetic rank of an elliptic curve over ℚ: the ℤ-rank of its
Mordell–Weil group modulo torsion.
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The arithmetic rank is realised by an additive isomorphism
E(ℚ)/tors ≃ ℤ^{rank}.
The analytic continuation of the L-function of E to a function defined on
all of ℂ.
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The extended L-function L_E is differentiable on all of ℂ.
The analytic rank of an elliptic curve over ℚ: the order of vanishing of
its extended L-function at s = 1.
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Specification of the analytic rank: L_E(s) = (s-1)^{rank_an} · g(s) with
g differentiable and g(1) ≠ 0.
The weak form of the Birch–Swinnerton-Dyer conjecture: the arithmetic and
analytic ranks of an elliptic curve over ℚ coincide.
The L-function attached to a cusp form's coefficient data: the Dirichlet L-series of its coefficients.
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The completed L-function of a cusp form:
Λ(f, s) = N^{s/2} (2π)^{-s} Γ(s) L(f, s).
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Hecke's theorem: the L-function attached to a cusp form admits an entire
extension L_ext to ℂ that agrees with the Dirichlet series on the region
of summability and satisfies the functional equation
Λ(s) = ε · Λ(k - s) with sign ε ∈ {±1}.
The principal Dirichlet character of conductor N at the prime p: equals
0 if p | N and 1 otherwise.
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The local Euler factor of a newform at the prime p:
(1 - a_p p^{-s} + χ(p) p^{k-1} p^{-2s})^{-1}.
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The Euler product of a newform: the (unordered) product over primes of the local Euler factors.
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The L-function of a newform, as a Dirichlet series in its Fourier coefficients.
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The L-function of a newform admits an Euler product expansion over primes.
The root number w(E) ∈ {±1} of an elliptic curve over ℚ: the sign in
the functional equation of its L-function.
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The root number of an elliptic curve is ±1.
Parity conjecture: the root number w(E) equals (−1)^{rank(E)},
expressing that the parity of the arithmetic rank matches the sign of the
functional equation.
Cusp forms of weight k for the congruence subgroup Γ₀(N) ≤ SL(2, ℤ).
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The Hecke operator T_n acting on weight-k modular forms for
SL(2, ℤ).
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The Hecke operator T_n acting on weight-k cusp forms for SL(2, ℤ).
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A fundamental domain in the upper half plane for the action of a congruence
subgroup Γ ≤ SL(2, ℤ).
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The Petersson inner product on cusp forms (Definition 24.18): integrates
f(τ) · conj(g(τ)) · y^{k-2} over a fundamental domain for Γ.
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The Petersson inner product is Hermitian: ⟨f, g⟩ = conj(⟨g, f⟩).
The Petersson inner product is additive in its first argument.
The Petersson inner product is ℂ-linear in its first argument.
Positivity: the real part of ⟨f, f⟩ is nonnegative.
⟨f, f⟩ is real: its imaginary part vanishes.
Definiteness: ⟨f, f⟩ = 0 iff f = 0.
The Hecke operator T_n acting on cusp forms for Γ₀(N), defined when
gcd(n, N) = 1.
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Hecke operators on Γ₀(N) at indices coprime to N are self-adjoint
with respect to the Petersson inner product.
Spectral theorem for a family of commuting symmetric operators: the family admits a joint eigenspace decomposition, i.e. the joint eigenspaces give an internal direct sum decomposition of the whole space.
Spectral theorem (supremum version): the supremum of joint eigenspaces of a commuting family of symmetric operators is the whole space.
Spectral theorem for a commuting pair of symmetric operators: the joint
eigenspaces Eig_A(α) ∩ Eig_B(β) form an internal direct sum decomposition.
The normalized (completed) L-function of an elliptic curve over ℚ:
N^{s/2} (2π)^{-s} Γ(s) L(E, s).
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Convert an integral weight-2 newform to abstract cusp-form data (weight 2,
same level, coefficients embedded in ℂ).
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The L-function of an elliptic curve over ℚ extends to an entire function on
ℂ agreeing with L(E, s) on the region of summability and satisfies the
functional equation Λ(s) = w · Λ(2 - s) with sign w ∈ {±1}.
Cusp forms of weight k for an arbitrary subgroup Γ ≤ SL(2, ℤ).
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The n-th q-expansion coefficient a_n(f) of a weight-k cusp form for
SL(2, ℤ).
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For each fixed n, the map f ↦ a_n(f) is ℂ-linear.
A cusp form is determined by its q-expansion: if every coefficient vanishes then the form itself is zero.
A cusp form has zero constant term in its q-expansion: a_0(f) = 0.
Corollary 24.15: for any cusp form f ∈ S_k(Γ₀(1)) and integers m, n with
gcd(m, n) = 1, the Hecke operator satisfies a_m(T_n f) = a_{mn}(f).
Hecke operators on cusp forms commute with each other.
The space S_k(SL(2, ℤ)) of cusp forms is finite-dimensional over ℂ.
A cusp form is a Hecke eigenform if it is nonzero and is an eigenvector of
every Hecke operator T_n.
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A Hecke eigenform is normalized if its first Fourier coefficient is 1.
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For an eigenvector f of T_n with eigenvalue ev, we have
a_n(f) = ev · a_1(f).
The first Fourier coefficient a_1(f) of a Hecke eigenform is nonzero.
Two Hecke eigenforms with the same system of eigenvalues differ by a scalar multiple (eigenspaces in the joint decomposition are one-dimensional).
For a normalized eigenform f, the Hecke eigenvalue at n equals the
Fourier coefficient a_n(f).
The space of cusp forms is spanned by Hecke eigenforms: there exists a finite family of (linearly independent) eigenforms whose span is the whole space.
Every Hecke eigenform admits a normalization to a normalized eigenform: a
nonzero scalar multiple with a_1 = 1.
The space of cusp forms admits a basis of normalized Hecke eigenforms.
The n-th Fourier coefficient of a weight-k cusp form for SL(2, ℤ).
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The 0-th Fourier coefficient of a cusp form vanishes.
Fourier coefficients are additive: a_n(f + g) = a_n(f) + a_n(g).
Fourier coefficients are ℂ-homogeneous: a_n(c · f) = c · a_n(f).
Cusp forms are determined by their full q-expansion: if all Fourier coefficients agree, the cusp forms are equal.
Theorem 24.14: for any cusp form f ∈ S_k(Γ₀(1)) and prime p,
a_n(T_p f) = a_{np}(f) if p ∤ n, otherwise
a_n(T_p f) = a_{np}(f) + p^{k-1} a_{n/p}(f).
The "non-dividing" case of Theorem 24.14: when p ∤ n,
a_n(T_p f) = a_{np}(f).
The "dividing" case of Theorem 24.14: when p ∣ n,
a_n(T_p f) = a_{np}(f) + p^{k-1} a_{n/p}(f).
The Hecke operator at n = 1 is the identity on cusp forms.
Multiplicativity of Hecke operators on cusp forms at coprime indices:
T_{mn} = T_m ∘ T_n when gcd(m, n) = 1.
Prime-power recurrence for Hecke operators on cusp forms:
T_{p^{r+2}} = T_{p^{r+1}} ∘ T_p − p^{k-1} · T_{p^r}.
Fourier coefficients are subtractive: a_n(f - g) = a_n(f) - a_n(g).
For a prime p coprime to m, the Hecke action on Fourier coefficients
specialises to a_m(T_p f) = a_{mp}(f).
Iteration to prime powers: for m coprime to the prime p, the action of
T_{p^r} on Fourier coefficients satisfies a_m(T_{p^r} f) = a_{m p^r}(f).
General statement of Corollary 24.15 (coprime version): for gcd(m, n) = 1,
a_m(T_n f) = a_{mn}(f).
Specialisation of Corollary 24.15 at m = 1: a_1(T_n f) = a_n(f).
The number ν₂(Γ) of elliptic points of order 2 for Γ ≤ SL(2, ℤ).
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The number ν₃(Γ) of elliptic points of order 3 for Γ ≤ SL(2, ℤ).
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The number ν_∞(Γ) of cusps of Γ ≤ SL(2, ℤ).
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The genus g(Γ) of the modular curve Γ\ℍ^*.
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The space of modular forms M_k(Γ) is finite-dimensional over ℂ.
Instance form: the space of modular forms M_k(Γ) is finite-dimensional
over ℂ.
The space of cusp forms S_k(Γ) is finite-dimensional over ℂ.
Instance form: the space of cusp forms S_k(Γ) is finite-dimensional over
ℂ.
M_0(Γ) ≅ ℂ (constants): the space of weight-0 modular forms is
1-dimensional.
There are no nonzero cusp forms of weight 0: S_0(Γ) = 0.
Dimension formula for M_k(Γ) for even positive k, in terms of the genus,
the elliptic point counts, and the number of cusps.
Dimension formula for S_k(Γ) for even k > 2, with cusp contribution
(k/2 − 1).
Dimension formula in weight 2: dim S_2(Γ) = g(Γ).
The new subspace S_k^{new}(Γ₀(N)) of cusp forms of weight k and level
N, complementary to the old subspace coming from divisors of N.
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Additive group structure on the new subspace of cusp forms.
Anonymous instance: additive group structure on the new subspace of cusp forms.
Complex vector space structure on the new subspace of cusp forms.
Anonymous instance: complex vector space structure on the new subspace of cusp forms.
The new subspace S_k^{new}(Γ₀(N)) is finite-dimensional over ℂ.
Anonymous instance: the new subspace S_k^{new}(Γ₀(N)) is
finite-dimensional over ℂ.
The inclusion of the new subspace into the full space of cusp forms for
Γ₀(N).
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The inclusion of the new subspace into the full space of cusp forms is injective.
Hecke operators T_n restrict to endomorphisms of the new subspace.
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The n-th q-expansion coefficient of an element of the new subspace.
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For each n, the map f ↦ a_n(f) on the new subspace is ℂ-linear.
An element of the new subspace is determined by its q-expansion: if every coefficient vanishes, the form itself is zero.
The constant term a_0 of any element of the new subspace vanishes.
A new-subspace eigenform: a nonzero element that is an eigenvector of every Hecke operator restricted to the new subspace.
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A newform of level N and weight k: a new-subspace eigenform with first
Fourier coefficient equal to 1.
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For a newform, every Hecke eigenvalue is equal to the corresponding Fourier coefficient.
Newform basis theorem: the new subspace S_k^{new}(Γ₀(N)) admits a
canonical basis of newforms, each a Hecke eigenvector with eigenvalues equal to
its Fourier coefficients, with corresponding one-dimensional eigenlines, and
uniquely determined by its q-expansion.