Eq zero of forall has Sum zero
TateCurve.Blueprint.eq_zero_of_forall_hasSum_zero
Project documentation
The descent lemma: a formal power series F ∈ ℚ(u)⟦q⟧ vanishes provided that, for infinitely many u₀ : ℂ, the evaluated series ∑ₙ Fₙ(u₀)q₀ⁿ converges with sum 0 for all sufficiently small nonzero q₀. Proof sketch: fix u₀. The function q₀ ↦ ∑ₙ Fₙ(u₀)q₀ⁿ is analytic on ‖q₀‖ < r (a power series converging pointwise on a disc is analytic there)...
Exact Lean statement
theorem eq_zero_of_forall_hasSum_zero (F : (RatFunc ℚ)⟦X⟧) (S : Set ℂ) (hS : S.Infinite)
(h : ∀ u ∈ S, ∃ r > 0, ∀ q : ℂ, 0 < ‖q‖ → ‖q‖ < r →
HasSum (fun n : ℕ ↦ evalAt u ((PowerSeries.coeff n) F) * q ^ n) 0) : F = 0Formal artifact
Lean source
theorem eq_zero_of_forall_hasSum_zero (F : (RatFunc ℚ)⟦X⟧) (S : Set ℂ) (hS : S.Infinite) (h : ∀ u ∈ S, ∃ r > 0, ∀ q : ℂ, 0 < ‖q‖ → ‖q‖ < r → HasSum (fun n : ℕ ↦ evalAt u ((PowerSeries.coeff n) F) * q ^ n) 0) : F = 0 := by ext n refine ratFunc_eq_zero_of_evalAt_eq_zero_on_infinite _ S hS fun u hu ↦ ?_ obtain ⟨r, hr, hsum⟩ := h u hu simpa using congrFun (coeffs_eq_zero_of_hasSum_punctured (fun n : ℕ ↦ evalAt u ((PowerSeries.coeff n) F)) r hr hsum) n- Project
- Fermat's Last Theorem
- License
- Apache-2.0
- Commit
- 8dd808888295
- Source
- FLT/KnownIn1980s/EllipticCurves/TateCurveConstruction.lean:1431-1438
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