G₃ q expansion
TateCurve.Blueprint.g₃_q_expansion
Plain-language statement
The q-expansion of g₃ (Silverman, Advanced topics, Theorem I.7.1): g₃(Λ_τ) = -(2πi)⁶/216 ⬝ (1 - 504s₅(q)). This is g₃ = 140G₆ and the case k = 6 of G_q_expansion, with row-sum identity sum_int_inv_sixth and 2ζ(6) = 2π⁶/945 (hasSum_int_inv_sixth).
Exact Lean statement
theorem g₃_q_expansion (τ : ℂ) (hτ : 0 < τ.im) :
(periodPair τ hτ.ne').g₃ =
-(2 * (Real.pi : ℂ) * I) ^ 6 / 216 * (1 - 504 * sAn 5 (e τ))Formal artifact
Lean source
theorem g₃_q_expansion (τ : ℂ) (hτ : 0 < τ.im) : (periodPair τ hτ.ne').g₃ = -(2 * (Real.pi : ℂ) * I) ^ 6 / 216 * (1 - 504 * sAn 5 (e τ)) := by rw [PeriodPair.g₃, G_q_expansion τ hτ (by norm_num) (by norm_num) ⟨3, by norm_num⟩ (div_ne_zero (pow_ne_zero 6 two_pi_I_ne_zero) (by norm_num : (120 : ℂ) ≠ 0)) sum_int_inv_sixth hasSum_int_inv_sixth, show (2 * (Real.pi : ℂ) * I) ^ 6 = -64 * (Real.pi : ℂ) ^ 6 by rw [show (2 * (Real.pi : ℂ) * I) ^ 6 = ((2 * (Real.pi : ℂ)) ^ 2 * I ^ 2) ^ 3 by ring, Complex.I_sq] ring] ring- Project
- Fermat's Last Theorem
- License
- Apache-2.0
- Commit
- 8dd808888295
- Source
- FLT/KnownIn1980s/EllipticCurves/TateCurveConstruction.lean:920-930
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