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import FormalConjecturesUtilModularity conjecture
The Modularity conjecture (also know as the Shimura-Taniyama-Weil conjecture) states that
every rational elliptic curve is modular, meaning that it can be
associated with a modular form. We state the a_p version of the conjecture, which relates the
coefficients of the modular form to the number of points on the elliptic curve over finite fields.
Since we don't have the conductor of the elliptic curve, our definition of a_p(E) differs from
that in the literature at primes of bad reduction. For this reason, we state the conjecture with the
assumption that p ∤ N, in order to give an equivalent statement.
References:
namespace ModularityConjectureopen Complex CongruenceSubgroup ModularFormClass ModularityConjecture UpperHalfPlaneopen scoped Real ModularForm CongruenceSubgroup
The n-th Fourier coefficient of a modular forms (around the cusp at infinity).
noncomputable def modularFormAn (n : ℕ) {N : ℕ} {k : ℤ} (f : CuspForm (Gamma0 N) k) : ℂ :=
(qExpansion N f).coeff nlocal notation:73 "a_[" n:0 "]" f:72 => modularFormAn n f
We need to reduce a rational modulo p, in practice we won't be dividing by zero since the
conductor of the elliptic curve saves us.
def ratRed (q : ℚ) (p : ℕ) : ZMod p :=
(q.num : ZMod p) * (q.den : ZMod p)⁻¹
The set of points on an elliptic curve over ZMod n.
def setOfPointsModN (E : WeierstrassCurve ℚ) [E.IsElliptic] (n : ℕ) :=
{P : ZMod n × ZMod n |
let ⟨x, y⟩ := P
y ^ 2 + ratRed E.a₁ n * x * y + ratRed E.a₃ n * y =
x ^ 3 + ratRed E.a₂ n * x ^ 2 + ratRed E.a₄ n * x + ratRed E.a₆ n}
The set of point mod n is finite.
E:WeierstrassCurve ℚinst✝:E.IsEllipticp:ℕ+⊢ Fintype
↑{(x, y) |
y ^ 2 + ratRed E.a₁ ↑p * x * y + ratRed E.a₃ ↑p * y =
x ^ 3 + ratRed E.a₂ ↑p * x ^ 2 + ratRed E.a₄ ↑p * x + ratRed E.a₆ ↑p}
All goals completed! 🐙
Note that normally this is written as p + 1 - #E(𝔽ₚ), but since we don't have a point at
infinity on this affine curve we only have p. The trace is integer-valued and can be negative.
noncomputable def WeierstrassCurve.ap (E : WeierstrassCurve ℚ) [E.IsElliptic] (p : ℕ) : ℤ :=
(p : ℤ) - Cardinal.toNat (Cardinal.mk (setOfPointsModN E p))Since we don't have Hecke operators yet, we define this via the q-expansion coefficients. See Proposition 5.8.5 of [diamondshurman2005].
def IsNormalisedEigenform {N : ℕ} {k : ℤ} (f : CuspForm (Gamma0 N) k) : Prop :=
a_[1]f = 1 ∧
(∀ (m n : ℕ), m.Coprime n → a_[n * m]f = a_[n]f * a_[m]f) ∧
(∀ (p r : ℕ), p.Prime → 2 ≤ r → (N : ZMod p) ≠ 0 →
a_[p ^ r]f = a_[p]f * a_[p ^ (r - 1)]f - p ^ (k - 1) * a_[p ^ (r - 2)]f) ∧
∀ (p r : ℕ), p.Prime → 2 ≤ r → (N : ZMod p) = 0 → a_[p ^ r]f = (a_[p]f) ^ rSee theorem 8.8.1 of [diamondshurman2005].
def modularityConjecture (E : WeierstrassCurve ℚ) [E.IsElliptic] : Prop :=
∃ (N : ℕ+) (f : CuspForm (Gamma0 N) 2), IsNormalisedEigenform f ∧
∀ (p : ℕ), p.Prime → (N : ZMod p) ≠ 0 → a_[p]f = E.ap pThe Modularity Theorem (formerly Shimura-Taniyama-Weil conjecture): every elliptic curve over $\mathbb{Q}$ is modular.
@[category research solved, AMS 11]
theorem modularity_conjecture (E : WeierstrassCurve ℚ) [E.IsElliptic] : modularityConjecture E := E:WeierstrassCurve ℚinst✝:E.IsElliptic⊢ modularityConjecture E
All goals completed! 🐙end ModularityConjecture