G₂ q expansion
TateCurve.Blueprint.g₂_q_expansion
Plain-language statement
The q-expansion of g₂ (Silverman, Advanced topics, Theorem I.7.1): g₂(Λ_τ) = (2πi)⁴/12 ⬝ (1 + 240s₃(q)). This is g₂ = 60G₄ and the case k = 4 of G_q_expansion, with row-sum identity sum_int_inv_fourth and 2ζ(4) = π⁴/45 (hasSum_int_inv_fourth).
Exact Lean statement
theorem g₂_q_expansion (τ : ℂ) (hτ : 0 < τ.im) :
(periodPair τ hτ.ne').g₂ =
(2 * (Real.pi : ℂ) * I) ^ 4 / 12 * (1 + 240 * sAn 3 (e τ))Formal artifact
Lean source
theorem g₂_q_expansion (τ : ℂ) (hτ : 0 < τ.im) : (periodPair τ hτ.ne').g₂ = (2 * (Real.pi : ℂ) * I) ^ 4 / 12 * (1 + 240 * sAn 3 (e τ)) := by rw [PeriodPair.g₂, G_q_expansion τ hτ (by norm_num) (by norm_num) ⟨2, by norm_num⟩ (div_ne_zero (pow_ne_zero 4 two_pi_I_ne_zero) (by norm_num : (6 : ℂ) ≠ 0)) sum_int_inv_fourth hasSum_int_inv_fourth, show (2 * (Real.pi : ℂ) * I) ^ 4 = 16 * (Real.pi : ℂ) ^ 4 by rw [show (2 * (Real.pi : ℂ) * I) ^ 4 = ((2 * (Real.pi : ℂ)) ^ 2 * I ^ 2) ^ 2 by ring, Complex.I_sq] ring] ring- Project
- Fermat's Last Theorem
- License
- Apache-2.0
- Commit
- 8dd808888295
- Source
- FLT/KnownIn1980s/EllipticCurves/TateCurveConstruction.lean:868-878
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