Research proof index

Rearrangement for Y: for 0 < ‖q‖ < ‖u‖ < 1 with u transcendental, the coefficients of the formal series TateCurve.Y evaluated at u sum to Yₐ(u, q). Proof: as for hasSum_X_eval, using v²/(1-v)³ = ∑_{m ≥ 1} (m choose 2) vᵐ for the rows n ≥ 1, the rational-function identity v²/(1-v)³ = -v⁻¹/(1-v⁻¹)³ together with `v/(1-v)³ = ∑_{m ≥ 1} ((m...

The q-expansion of the Weierstrass ℘-function (Silverman, Advanced topics, Theorem I.6.2): for τ in the upper half plane and 0 < im z < im τ (which forces z ∉ Λ_τ), ℘(z; Λ_τ) = (2πi)² (1/12 + Xₐ(e z, e τ)). Proof: group the absolutely convergent sum defining ℘ into rows ω = nτ + m, n : ℤ (Fubini). The condition 0 < im z < im τ guaran...

The point (X(u,q), Y(u,q)) satisfies the Weierstrass equation y² + xy = x³ + a₄x + a₆ of the Tate curve, as an identity in ℚ(u)⟦q⟧.