Project-declaredLean 4.32.0 · mathlib@249c48c2

Has Split Multiplicative Reduction quadratic Twist Of of residue

WeierstrassCurve.hasSplitMultiplicativeReduction_quadraticTwistOf_of_residue

number theory arithmetic geometry Fermat's Last Theorem split multiplicative reduction

Plain-language statement

Packaging nodePoly_quadraticTwistOf_map_splits_of_residue: if the base change of the twisted integral model has multiplicative reduction and the residues of (t', n') satisfy the trace and norm relations, then the reduction is split multiplicative.

Exact Lean statement

theorem hasSplitMultiplicativeReduction_quadraticTwistOf_of_residue
    [E.HasMultiplicativeReduction R] (t' n' : R)
    [hW : (((E.integralModel R).quadraticTwistOf t' n')⁄K).HasMultiplicativeReduction 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) :
    (((E.integralModel R).quadraticTwistOf t' n')⁄K).HasSplitMultiplicativeReduction R

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

theorem hasSplitMultiplicativeReduction_quadraticTwistOf_of_residue    [E.HasMultiplicativeReduction R] (t' n' : R)    [hW : (((E.integralModel R).quadraticTwistOf t' n')⁄K).HasMultiplicativeReduction 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) :    (((E.integralModel R).quadraticTwistOf t' n')⁄K).HasSplitMultiplicativeReduction R := by  refine { hW with splitMultiplicativeReduction := ?_ }  rw [show (((E.integralModel R).quadraticTwistOf t' n')⁄K).integralModel R    = (E.integralModel R).quadraticTwistOf t' n' from integralModel_baseChange R _]  exact nodePoly_quadraticTwistOf_map_splits_of_residue E R t' n' hA hB

Project: Fermat's Last Theorem
License: Apache-2.0
Commit: 8dd808888295
Source: FLT/KnownIn1980s/EllipticCurves/QuadraticTwists/SplitMultiplicativeReduction.lean:372-385

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

Project-declaredLean 4.32.0

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

number theoryarithmetic geometryFermat's Last Theorem

Source project: Fermat's Last Theorem

Person-level attribution pending.

View proof record

Project-declaredLean 4.32.0

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.

number theoryarithmetic geometryFermat's Last Theorem

Source project: Fermat's Last Theorem

Person-level attribution pending.

View proof record

Project-declaredLean 4.32.0

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.

number theoryarithmetic geometryFermat's Last Theorem

Source project: Fermat's Last Theorem

Person-level attribution pending.

View proof record