Residue c₄ mul residue eq neg c₆
WeierstrassCurve.residue_c₄_mul_residue_eq_neg_c₆
Plain-language statement
The key identity φc₄ · φ(t'² - 4n') = -φc₆ of the twisting datum (t', n'): if its residues satisfy the trace and norm relations cut out by the node polynomial (κ = 54 b₆ - 3 b₂ b₄ + a₂ c₄), then the discriminant identity splitPolynomial_discrim turns them into this identity.
Exact Lean statement
theorem residue_c₄_mul_residue_eq_neg_c₆ [E.HasMultiplicativeReduction R] (t' n' : R)
(hA : residue R (E.integralModel R).c₄ * residue R t'
+ residue R ((E.integralModel R).a₁ * (E.integralModel R).c₄) = 0)
(hB : residue R (E.integralModel R).c₄ * residue R n'
+ residue R (54 * (E.integralModel R).b₆
- 3 * (E.integralModel R).b₂ * (E.integralModel R).b₄
+ (E.integralModel R).a₂ * (E.integralModel R).c₄) = 0) :
residue R (E.integralModel R).c₄ * residue R (t' ^ 2 - 4 * n')
= -residue R (E.integralModel R).c₆Formal artifact
Lean source
theorem residue_c₄_mul_residue_eq_neg_c₆ [E.HasMultiplicativeReduction R] (t' n' : R) (hA : residue R (E.integralModel R).c₄ * residue R t' + residue R ((E.integralModel R).a₁ * (E.integralModel R).c₄) = 0) (hB : residue R (E.integralModel R).c₄ * residue R n' + residue R (54 * (E.integralModel R).b₆ - 3 * (E.integralModel R).b₂ * (E.integralModel R).b₄ + (E.integralModel R).a₂ * (E.integralModel R).c₄) = 0) : residue R (E.integralModel R).c₄ * residue R (t' ^ 2 - 4 * n') = -residue R (E.integralModel R).c₆ := by set c₄' := (E.integralModel R).c₄ with hc₄' set κ' := 54 * (E.integralModel R).b₆ - 3 * (E.integralModel R).b₂ * (E.integralModel R).b₄ + (E.integralModel R).a₂ * c₄' with hκ' simp only [map_mul] at hA have hRid : ((E.integralModel R).a₁ * c₄') ^ 2 + 4 * c₄' * κ' = -(c₄' * (E.integralModel R).c₆) := by rw [hκ', hc₄'] exact splitPolynomial_discrim (E.integralModel R) have hdisc := congrArg (residue R) hRid simp only [map_add, map_mul, map_pow, map_neg, map_ofNat] at hdisc apply mul_left_cancel₀ (residue_integralModel_c₄_ne_zero E R) simp only [map_sub, map_mul, map_pow, map_ofNat] linear_combination hdisc + (residue R c₄' * residue R t' - residue R (E.integralModel R).a₁ * residue R c₄') * hA - 4 * residue R c₄' * hB- Project
- Fermat's Last Theorem
- License
- Apache-2.0
- Commit
- 8dd808888295
- Source
- FLT/KnownIn1980s/EllipticCurves/QuadraticTwists/SplitMultiplicativeReduction.lean:237-260
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