ReductionBaseChange.lean`

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12 declarations, 0 contain sorry, 0 declared as axioms

Reduction theory under base change

Let k → l be a valuative extension of nonarchimedean local fields and let E/k be an elliptic curve with split multiplicative reduction. This file proves that the two reduction instances currently sorried in FLT.KnownIn1980s.EllipticCurves.TateCurve transfer to the base change (stated for an arbitrary such E, not just Tate curves):

(E.baseChange l).IsMinimal 𝒪[l] (isMinimal_baseChange, and the HasMultiplicativeReduction instance which yields it by parent projection)
(E.baseChange l).HasSplitMultiplicativeReduction 𝒪[l]

Neither is provable by assembling existing declarations: mathlib's reduction API (Mathlib.AlgebraicGeometry.EllipticCurve.Reduction) has no minimality criterion and no base-change results at all, and mathlib's ValuativeExtension provides only the map of value groups mapValueGroupWithZero, not the induced ring map 𝒪[k] →+* 𝒪[l] of rings of integers. FLT PR #1081 (branch logos2) also leaves both instances sorried.

Mathematical route

The whole point is that minimality by itself is not preserved by base change — y² = x³ + p is minimal over ℚ_p but not over ℚ_p(p^{1/6}) — so one cannot argue about minimality abstractly. The multiplicative reduction hypothesis rescues everything through a single mechanism: it forces |c₄| = 1, and "integral with |c₄| = 1" is a property that (i) forces minimality and (ii) is visibly preserved by any valuative extension.

Mathlib phrases HasMultiplicativeReduction at the field level — its two non-parent fields are statements about the adic valuation of W.Δ and W.c₄ as elements of the field — and this branch already has both halves of the translation apparatus:

the bridges ValuativeRel.adicValuation_lt_one_iff and adicValuation_eq_one_iff (FLT.Mathlib.RingTheory.Valuation.ValuativeRel.Basic) convert between the adic valuation of the DVR 𝒪[k], in which mathlib's reduction theory is phrased, and the canonical valuation of k, on the two distinctions that matter (< 1 and = 1);
strict monotonicity of ValuativeExtension.mapValueGroupWithZero transfers the canonical valuation along k → l.

So everything except the minimality criterion and the splitness clause is a field-level valuation chase along the chain adic over k → canonical over k → canonical over l → adic over l. In detail:

Transfer of the canonical valuation (ValuativeExtension.valuation_algebraMap_le_one, _lt_one, _eq_one — proved below, not sorried). A valuative extension preserves |x| ≤ 1, |x| < 1 and |x| = 1, because mapValueGroupWithZero is a strictly monotone monoid-with-zero homomorphism sending valuation k x to valuation l (algebraMap k l x). (The < 1 case duplicates TateCurve.valuation_algebraMap_lt_one from TateCurveBaseChange; when de-blueprinting, consolidate all three next to the bridges in FLT.Mathlib.RingTheory.Valuation.ValuativeRel.Basic.)
Functoriality of the integers (ValuativeExtension.integerMap), needed only for the splitness clause (step 5). By valuation_algebraMap_le_one, algebraMap k l restricts to a map 𝒪[k] →+* 𝒪[l]; it is a local homomorphism (by valuation_algebraMap_lt_one) and hence induces a map of residue fields (residueMap). The IsMinimal and HasMultiplicativeReduction transfers below do not use any of this.
The minimality criterion (WeierstrassCurve.isMinimal_of_valuation_c₄_eq_one). An integral Weierstrass equation whose c₄ has adic valuation 1 is minimal — the unit-c₄ case of the Kraus–Laska criterion "v(c₄) < 4 or v(Δ) < 12 implies minimal" (Silverman AEC, Remark VII.1.1). This is the one genuinely new piece of mathematics, and it is mathlib-worthy. It is stated at the field level to match mathlib's HasMultiplicativeReduction.multiplicativeReduction, which is exactly the fact consumers hold.
Base change of the integral model. Integrality transfers coefficientwise: each coefficient of E.baseChange l is algebraMap k l of the corresponding coefficient of E, and valuation_algebraMap_le_one keeps it integral (isIntegral_of_exists_lift). For the splitness clause we also record integralModel_baseChange_map: the integral model of the base change is the integerMap-image of the integral model, by uniqueness of lifts along the injective map 𝒪[l] → l.
Multiplicative reduction. badReduction and multiplicativeReduction transfer along the chain above: adic |Δ| < 1 over k converts to the canonical valuation (adicValuation_lt_one_iff + integralModel_Δ_eq), moves along the extension (valuation_algebraMap_lt_one + map_Δ), and converts back over l (adicValuation_lt_one_iff again); same for |c₄| = 1 with the eq_one lemmas. Minimality is step 2 applied to the transferred |c₄| = 1. No integerMap, no IsLocalHom. Note multiplicative reduction is preserved by arbitrary extensions of local fields (Silverman AEC, VII.5.4(b)) — no unramifiedness is needed, in contrast to good reduction.
Splitness. Mathlib's HasSplitMultiplicativeReduction demands that the explicit quadratic c₄T² + a₁c₄T - (54b₆ - 3b₂b₄ + a₂c₄) (tangent directions at the node) split over the residue field. By 3 the quadratic of the l-model is the image of the quadratic of the k-model under the induced map of residue fields (IsLocalRing.ResidueField.map (integerMap k l)), and a split polynomial stays split under any ring map (Polynomial.Splits.map). Splitness can only be gained, never lost, when the residue field grows. This is the only step that uses integerMap, residueMap and integralModel_baseChange_map.

The base-change transfer is packaged as theorems here rather than instances, so that the consuming instances live in FLT.KnownIn1980s.EllipticCurves.TateCurve, where this file is imported: the two sorried instances there are filled by isMinimal_baseChange and hasSplitMultiplicativeReduction_baseChange (which also yields IsMinimal and HasMultiplicativeReduction by class-parent projection).

The k-side adic → canonical conversions of valuation_c₄_baseChange_eq_one and valuation_Δ_baseChange_lt_one duplicate the downstream WeierstrassCurve.valuation_c₄_eq_one and valuation_Δ_lt_one of TateCurve; they are reduction-theoretic rather than Tate-theoretic, so consider extracting the k-side halves as shared lemmas (or moving those TateCurve lemmas here) when de-blueprinting.

References

Silverman, The arithmetic of elliptic curves, VII §1 (minimal Weierstrass equations, Remark VII.1.1 for the minimality criterion) and VII.5.4 (behaviour of reduction type under field extension).

Imports (4)

Imported by 2 files

def ValuativeExtension.integerMap Documented declaration: integerMap line 170 source
Source documentation
The restriction of algebraMap k l to the rings of integers: 𝒪[k] is the subring (valuation k).integer of k, and valuation_algebraMap_le_one says algebraMap k l maps it into (valuation l).integer.
Implementation note: it may be worth defining the Algebra 𝒪[k] 𝒪[l] instance directly instead, so that WeierstrassCurve.baseChange (rather than map) can be used on integral models in integralModel_baseChange_map below; whichever is chosen, this is the underlying ring homomorphism.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, ValuativeRel, Algebra, ValuativeExtension, RingHom, Subtype, Membership.mem, Subring, Valuation.integer, ValuativeRel.ValueGroupWithZero, ValuativeRel.valuation
Proof uses: RingHom.restrict, Algebra.algebraMap
29 instances, coercions and auxiliary constants left out.
```
noncomputable def integerMap : 𝒪[k] →+* 𝒪[l] :=
…
```
theorem ValuativeExtension.integerMap_coe integerMap is the algebra map on underlying elements line 177 source
Source documentation
integerMap is the algebra map on underlying elements.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, ValuativeRel, Algebra, ValuativeExtension, Subtype, Membership.mem, Subring, Valuation.integer, ValuativeRel.ValueGroupWithZero, ValuativeRel.valuation, Eq, RingHom, ValuativeExtension.integerMap, Algebra.algebraMap
Proof uses: rfl
40 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
```
theorem integerMap_coe (x : 𝒪[k]) :
    (integerMap k l x : l) = algebraMap k l (x : k) :=
…
```
Mentions: ValuativeExtension.integerMap
instance ValuativeExtension.(instance at line 186) line 186 source
Source documentation
integerMap is a local homomorphism: if integerMap x is a unit then x is a unit. Non-units of the integer subring are exactly the elements of valuation < 1 (Valuation.Integer.not_isUnit_iff_valuation_lt_one), and valuation k x < 1 gives valuation l (integerMap k l x) < 1 by valuation_algebraMap_lt_one. Needed only to make IsLocalRing.ResidueField.map applicable in residueMap.
```
instance : IsLocalHom (integerMap k l) where
…
```
Mentions: ValuativeExtension.integerMap
theorem ValuativeExtension.algebraMap_integerMap Documented declaration: algebraMap_integerMap line 198 source
Source documentation
The commuting square algebraMap 𝒪[l] l ∘ integerMap = algebraMap k l ∘ algebraMap 𝒪[k] k underlying integerMap: on underlying elements both send x to algebraMap k l x (integerMap_coe, and algebraMap 𝒪[·] · is the subring inclusion). This is the functoriality used to transfer integral models along the base change.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, ValuativeRel, Algebra, ValuativeExtension, Subtype, Membership.mem, Subring, Valuation.integer, ValuativeRel.ValueGroupWithZero, ValuativeRel.valuation, Eq, RingHom, Algebra.algebraMap, ValuativeExtension.integerMap
Proof uses: ValuativeExtension.integerMap_coe
32 instances, coercions and auxiliary constants left out.
```
theorem algebraMap_integerMap (x : 𝒪[k]) :
    algebraMap 𝒪[l] l (integerMap k l x) = algebraMap k l (algebraMap 𝒪[k] k x) :=
…
```
Mentions: ValuativeExtension.integerMap
def ValuativeExtension.residueMap The induced map of residue fields 𝒪[k]/𝓂[k] →+* 𝒪[l]/𝓂[l], from the local homomorphism integerMap line 219 source
Source documentation
The induced map of residue fields 𝒪[k]/𝓂[k] →+* 𝒪[l]/𝓂[l], from the local homomorphism integerMap. (An honest definition, not a sorry: it is IsLocalRing.ResidueField.map of integerMap k l.)

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, ValuativeRel, TopologicalSpace, IsNonarchimedeanLocalField, Algebra, ValuativeExtension, RingHom, IsLocalRing.ResidueField, Subtype, Membership.mem, Subring, Valuation.integer, ValuativeRel.ValueGroupWithZero, ValuativeRel.valuation
Proof uses: IsLocalRing.ResidueField.map, ValuativeExtension.integerMap
32 instances, coercions and auxiliary constants left out.

Tactic trace: 10 steps
Open to load the checked before-and-after goal states.
```
noncomputable def residueMap :
    IsLocalRing.ResidueField 𝒪[k] →+* IsLocalRing.ResidueField 𝒪[l] :=
…
```
instance WeierstrassCurve.(instance at line 257) line 257 source
Source documentation
Base change along a valuative extension preserves integrality, coefficientwise: each coefficient of E.baseChange l is algebraMap k l of the corresponding coefficient of E (map_a₁, …), which is integral over k (integralModel_a₁_eq, …, and elements of 𝒪[k] have valuation k · ≤ 1), so stays integral over l by valuation_algebraMap_le_one; conclude with isIntegral_of_exists_lift. (No functoriality of map is needed.)
```
instance [IsIntegral 𝒪[k] E] : IsIntegral 𝒪[l] (E.baseChange l) :=
…
```
theorem WeierstrassCurve.integralModel_baseChange_map The integral model of the base change is the base change of the integral model line 277 source
Source documentation
The integral model of the base change is the base change of the integral model. Both sides are lifts of E.baseChange l along the injective map 𝒪[l] → l (injectivity from IsFractionRing), and lifts along an injective map are unique: compare coefficientwise via integralModel_a₁_eq on both sides and the commuting square algebraMap_integerMap. (Only the splitness clause of step 5 consumes this.)

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, ValuativeRel, Algebra, ValuativeExtension, WeierstrassCurve, WeierstrassCurve.IsIntegral, Subtype, Membership.mem, Subring, Valuation.integer, ValuativeRel.ValueGroupWithZero, ValuativeRel.valuation, Eq, WeierstrassCurve.integralModel, WeierstrassCurve.baseChange, WeierstrassCurve.map, ValuativeExtension.integerMap
Proof uses: WeierstrassCurve.map_injective, Algebra.algebraMap, IsFractionRing.injective, id, RingHom, RingHom.comp, RingHom.ext, ValuativeExtension.algebraMap_integerMap, Eq.mpr, congrArg, WeierstrassCurve.map_map, Eq.symm, WeierstrassCurve.baseChange_integralModel_eq
39 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
```
theorem integralModel_baseChange_map [IsIntegral 𝒪[k] E] :
    integralModel 𝒪[l] (E.baseChange l) =
      (integralModel 𝒪[k] E).map (integerMap k l) := by
…
```
theorem WeierstrassCurve.valuation_c₄_baseChange_eq_one The c₄ of the base change has adic valuation 1 line 296 source
Source documentation
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 isMinimal_baseChange and multiplicative reduction of the base change.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, ValuativeRel, TopologicalSpace, IsNonarchimedeanLocalField, Algebra, ValuativeExtension, WeierstrassCurve, WeierstrassCurve.HasMultiplicativeReduction, Subtype, Membership.mem, Subring, Valuation.integer, ValuativeRel.ValueGroupWithZero, ValuativeRel.valuation, Eq, WithZero, Multiplicative, Int, Valuation, IsDedekindDomain.HeightOneSpectrum.valuation and 4 more
Proof uses: WeierstrassCurve.HasMultiplicativeReduction.multiplicativeReduction, Eq.mpr, RingHom, Algebra.algebraMap, WeierstrassCurve.integralModel, id, congrArg, Eq.symm, WeierstrassCurve.integralModel_c₄_eq, Iff.mp, ValuativeRel.adicValuation_eq_one_iff, Eq.mp, WeierstrassCurve.map_c₄, ValuativeExtension.valuation_algebraMap_eq_one, Iff.mpr
96 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
```
theorem valuation_c₄_baseChange_eq_one [HasMultiplicativeReduction 𝒪[k] E] :
    (IsDiscreteValuationRing.maximalIdeal 𝒪[l]).valuation l (E.baseChange l).c₄ = 1 := by
…
```
theorem WeierstrassCurve.valuation_Δ_baseChange_lt_one The discriminant of the base change has adic valuation < 1 line 312 source
Source documentation
The discriminant of the base change has adic valuation < 1. Multiplicative reduction of E makes |E.Δ| < 1 (via adicValuation_lt_one_iff and integralModel_Δ_eq, as in WeierstrassCurve.valuation_Δ_lt_one); this transfers to l by valuation_algebraMap_lt_one, and converts back to the adic valuation over 𝒪[l].

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, ValuativeRel, TopologicalSpace, IsNonarchimedeanLocalField, Algebra, ValuativeExtension, WeierstrassCurve, WeierstrassCurve.HasMultiplicativeReduction, Subtype, Membership.mem, Subring, Valuation.integer, ValuativeRel.ValueGroupWithZero, ValuativeRel.valuation, LT.lt, WithZero, Multiplicative, Int, Valuation, IsDedekindDomain.HeightOneSpectrum.valuation and 4 more
Proof uses: WeierstrassCurve.HasMultiplicativeReduction.badReduction, Eq.mpr, RingHom, Algebra.algebraMap, WeierstrassCurve.integralModel, id, Eq, congrArg, Eq.symm, WeierstrassCurve.integralModel_Δ_eq, Iff.mp, ValuativeRel.adicValuation_lt_one_iff, Eq.mp, WeierstrassCurve.map_Δ, ValuativeExtension.valuation_algebraMap_lt_one, Iff.mpr
105 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
```
theorem valuation_Δ_baseChange_lt_one [HasMultiplicativeReduction 𝒪[k] E] :
    (IsDiscreteValuationRing.maximalIdeal 𝒪[l]).valuation l (E.baseChange l).Δ < 1 := by
…
```
theorem WeierstrassCurve.isMinimal_baseChange Documented declaration: isMinimal_baseChange line 334 source
Source documentation
Base change along a valuative extension preserves minimality in the presence of multiplicative reduction (minimality alone is not preserved: y² = x³ + p is minimal over ℚ_p but not over ℚ_p(p^{1/6})): multiplicative reduction makes |c₄| = 1 (valuation_c₄_baseChange_eq_one), and unit c₄ forces minimality (isMinimal_of_valuation_c₄_eq_one).
A theorem rather than an instance: the HasMultiplicativeReduction instance below yields IsMinimal 𝒪[l] (E.baseChange l) again by class-parent projection, and a second instance path to it would overlap (cf. the overlappingInstances linter and mathlib#41391).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, ValuativeRel, TopologicalSpace, IsNonarchimedeanLocalField, Algebra, ValuativeExtension, WeierstrassCurve, WeierstrassCurve.HasMultiplicativeReduction, Subtype, Membership.mem, Subring, Valuation.integer, ValuativeRel.ValueGroupWithZero, ValuativeRel.valuation, WeierstrassCurve.IsMinimal, WeierstrassCurve.baseChange
Proof uses: WeierstrassCurve.isMinimal_of_valuation_c₄_eq_one, WeierstrassCurve.valuation_c₄_baseChange_eq_one
37 instances, coercions and auxiliary constants left out.
```
theorem isMinimal_baseChange [HasMultiplicativeReduction 𝒪[k] E] :
    IsMinimal 𝒪[l] (E.baseChange l) :=
…
```
theorem WeierstrassCurve.hasMultiplicativeReduction_baseChange Documented declaration: hasMultiplicativeReduction_baseChange line 347 source
Source documentation
Base change along a valuative extension preserves multiplicative reduction (Silverman AEC, VII.5.4(b); true for arbitrary extensions of local fields — contrast good reduction, which needs unramifiedness): minimality is isMinimal_baseChange, and the badReduction/multiplicativeReduction fields are valuation_Δ_baseChange_lt_one and valuation_c₄_baseChange_eq_one.
Stated as a theorem rather than an instance so that the consuming instance lives in FLT.KnownIn1980s.EllipticCurves.TateCurve; use it there via WeierstrassCurve.hasMultiplicativeReduction_baseChange.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, ValuativeRel, TopologicalSpace, IsNonarchimedeanLocalField, Algebra, ValuativeExtension, WeierstrassCurve, WeierstrassCurve.HasMultiplicativeReduction, Subtype, Membership.mem, Subring, Valuation.integer, ValuativeRel.ValueGroupWithZero, ValuativeRel.valuation, WeierstrassCurve.baseChange
Proof uses: WeierstrassCurve.IsMinimal, WeierstrassCurve.isMinimal_baseChange, WeierstrassCurve.HasMultiplicativeReduction.mk, WeierstrassCurve.valuation_Δ_baseChange_lt_one, WeierstrassCurve.valuation_c₄_baseChange_eq_one
34 instances, coercions and auxiliary constants left out.
```
theorem hasMultiplicativeReduction_baseChange [HasMultiplicativeReduction 𝒪[k] E] :
    HasMultiplicativeReduction 𝒪[l] (E.baseChange l) where
…
```
theorem WeierstrassCurve.hasSplitMultiplicativeReduction_baseChange Base change along a valuative extension preserves split multiplicative reduction: the residue field only grows, and a split quadratic stays split under any ring map line 368 source
Source documentation
Base change along a valuative extension preserves split multiplicative reduction: the residue field only grows, and a split quadratic stays split under any ring map.
Proof: multiplicative reduction is hasMultiplicativeReduction_baseChange. For splitness, by integralModel_baseChange_map the integral model over 𝒪[l] is integerMap of the one over 𝒪[k], so the node polynomial transfers as map_nodePoly. The residue square residue 𝒪[l] ∘ integerMap = residueMap ∘ residue 𝒪[k] (IsLocalRing.ResidueField.map_residue) and Polynomial.map_map identify the reduced quadratic over 𝓀[l] with the residueMap-image of the reduced quadratic over 𝓀[k], which splits by hypothesis; conclude with Polynomial.Splits.map. This is the only place integerMap, residueMap and integralModel_baseChange_map are used.
Stated as a theorem rather than an instance so that the consuming instance lives in FLT.KnownIn1980s.EllipticCurves.TateCurve; use it there via WeierstrassCurve.hasSplitMultiplicativeReduction_baseChange.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, ValuativeRel, TopologicalSpace, IsNonarchimedeanLocalField, Algebra, ValuativeExtension, WeierstrassCurve, WeierstrassCurve.HasSplitMultiplicativeReduction, Subtype, Membership.mem, Subring, Valuation.integer, ValuativeRel.ValueGroupWithZero, ValuativeRel.valuation, WeierstrassCurve.baseChange
Proof uses: WeierstrassCurve.HasMultiplicativeReduction, WeierstrassCurve.hasMultiplicativeReduction_baseChange, WeierstrassCurve.HasSplitMultiplicativeReduction.mk, Polynomial.Splits, IsLocalRing.ResidueField, Polynomial.map, Algebra.algebraMap, WeierstrassCurve.nodePoly, WeierstrassCurve.integralModel, WeierstrassCurve.HasSplitMultiplicativeReduction.splitMultiplicativeReduction, Eq, RingHom, RingHom.comp, IsLocalRing.residue, ValuativeExtension.integerMap, ValuativeExtension.residueMap, RingHom.ext, IsLocalRing.ResidueField.map_residue, id, HSub.hSub and 23 more
77 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
```
theorem hasSplitMultiplicativeReduction_baseChange [HasSplitMultiplicativeReduction 𝒪[k] E] :
    HasSplitMultiplicativeReduction 𝒪[l] (E.baseChange l) where
…
```