Analytic weierstrass
TateCurve.Blueprint.analytic_weierstrass
Project documentation
The analytic form of the main theorem (Silverman, Advanced topics, Theorem V.1.1(a)): for 0 < ‖q‖ < ‖u‖ < 1, Yₐ² + XₐYₐ = Xₐ³ - 5s₃(q)Xₐ - (5s₃(q) + 7s₅(q))/12. Proof sketch: the hypotheses ensure u ∉ qᶻ, and we may choose z, τ with e z = u, e τ = q, 0 < im z < im τ (so z ∉ Λ_τ). Substitute the four q-expansions into the differential...
Exact Lean statement
theorem analytic_weierstrass {u q : ℂ} (h0 : 0 < ‖q‖) (h1 : ‖q‖ < ‖u‖) (h2 : ‖u‖ < 1) :
YAn u q ^ 2 + XAn u q * YAn u q =
XAn u q ^ 3 - 5 * sAn 3 q * XAn u q - (5 * sAn 3 q + 7 * sAn 5 q) / 12Formal artifact
Lean source
theorem analytic_weierstrass {u q : ℂ} (h0 : 0 < ‖q‖) (h1 : ‖q‖ < ‖u‖) (h2 : ‖u‖ < 1) : YAn u q ^ 2 + XAn u q * YAn u q = XAn u q ^ 3 - 5 * sAn 3 q * XAn u q - (5 * sAn 3 q + 7 * sAn 5 q) / 12 := by have him : ∀ {v : ℂ}, 0 < ‖v‖ → ‖v‖ < 1 → 0 < (Complex.log v / (2 * (Real.pi : ℂ) * I)).im := fun hv0 hv1 ↦ by rw [log_div_two_pi_I_im] exact div_pos (neg_pos.2 ((Real.log_neg_iff hv0).2 hv1)) (by positivity) refine analytic_weierstrass_of_exp (τ := Complex.log q / (2 * (Real.pi : ℂ) * I)) (z := Complex.log u / (2 * (Real.pi : ℂ) * I)) (him h0 (h1.trans h2)) (him (h0.trans h1) h2) ?_ (e_log_div_two_pi_I (norm_pos_iff.mp (h0.trans h1))) (e_log_div_two_pi_I (norm_pos_iff.mp h0)) rw [log_div_two_pi_I_im, log_div_two_pi_I_im] exact div_lt_div_of_pos_right (neg_lt_neg (Real.log_lt_log h0 h1)) (by positivity)- Project
- Fermat's Last Theorem
- License
- Apache-2.0
- Commit
- 8dd808888295
- Source
- FLT/KnownIn1980s/EllipticCurves/TateCurveConstruction.lean:1002-1014
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