Valuation c₄ base Change eq one
WeierstrassCurve.valuation_c₄_baseChange_eq_one
Plain-language statement
The c₄ of the base change has adic valuation 1. Multiplicative reduction of E makes the canonical valuation |E.c₄| = 1 (via adicValuation_eq_one_iff and integralModel_c₄_eq, as in WeierstrassCurve.valuation_c₄_eq_one); this transfers to l by valuation_algebraMap_eq_one, and converts back to the adic valuation over 𝒪[l]. Shared by `isM...
Exact Lean statement
theorem valuation_c₄_baseChange_eq_one [HasMultiplicativeReduction 𝒪[k] E] :
(IsDiscreteValuationRing.maximalIdeal 𝒪[l]).valuation l (E.baseChange l).c₄ = 1Formal artifact
Lean source
theorem valuation_c₄_baseChange_eq_one [HasMultiplicativeReduction 𝒪[k] E] : (IsDiscreteValuationRing.maximalIdeal 𝒪[l]).valuation l (E.baseChange l).c₄ = 1 := by have hk : valuation k E.c₄ = 1 := by have hmul := HasMultiplicativeReduction.multiplicativeReduction (R := 𝒪[k]) (W := E) rw [← integralModel_c₄_eq 𝒪[k] E] at hmul ⊢ exact adicValuation_eq_one_iff.mp hmul have hl : valuation l (E.baseChange l).c₄ = 1 := by rw [show (E.baseChange l).c₄ = algebraMap k l E.c₄ from E.map_c₄ (algebraMap k l)] exact valuation_algebraMap_eq_one hk rw [← integralModel_c₄_eq 𝒪[l] (E.baseChange l)] at hl ⊢ exact adicValuation_eq_one_iff.mpr hl- Project
- Fermat's Last Theorem
- License
- Apache-2.0
- Commit
- 8dd808888295
- Source
- FLT/KnownIn1980s/EllipticCurves/ReductionBaseChange.lean:296-306
Reuse this declaration
Bring the exact result into your workflow
The import identifies the source module. Your project still needs the pinned package dependency shown on this page.
What this badge means
This completion status comes from the project or community source. It has not yet been represented here as an independent rebuild and axiom audit.
Continue in this project
Related declarations
Eq finsum quotient out of bij On
AbstractHeckeOperator.eq_finsum_quotient_out_of_bijOn'
Plain-language statement
If a is fixed by V then ∑ᶠ g ∈ s, g • a is independent of the choice s of coset representatives in G for a subset of G ⧸ V
Source project: Fermat's Last Theorem
Person-level attribution pending.
Comm Group no compact automorphisms
CommGroup.no_compact_automorphisms
Plain-language statement
A connected compact Hausdorff abelian topological group does not admit a nontrivial compact group of automorphisms.
Source project: Fermat's Last Theorem
Person-level attribution pending.
Fermat Last Theorem of p ge 5
FermatLastTheorem.of_p_ge_5
Plain-language statement
If Fermat's Last Theorem is true for primes p ≥ 5, then FLT is true.
Source project: Fermat's Last Theorem
Person-level attribution pending.