Valuation eval Int eq
TateCurve.valuation_evalInt_eq
Plain-language statement
The leading-term principle: if F = X + O(X²) then |F(q)| = |q| on the punctured open unit disc , ultrametrically the leading term dominates the tail, which has valuation at most |q|² by valuation_evalInt_le_pow.
Exact Lean statement
theorem valuation_evalInt_eq (q : k) (hq0 : q ≠ 0) (hq : valuation k q < 1)
{F : ℤ⟦X⟧} (h0 : PowerSeries.constantCoeff F = 0) (h1 : PowerSeries.coeff 1 F = 1) :
valuation k (evalInt q F) = valuation k qFormal artifact
Lean source
theorem valuation_evalInt_eq (q : k) (hq0 : q ≠ 0) (hq : valuation k q < 1) {F : ℤ⟦X⟧} (h0 : PowerSeries.constantCoeff F = 0) (h1 : PowerSeries.coeff 1 F = 1) : valuation k (evalInt q F) = valuation k q := by have hsplit : evalInt q F = q + evalInt q (F - PowerSeries.X) := by conv_lhs => rw [show F = PowerSeries.X + (F - PowerSeries.X) by ring] rw [evalInt_add (summable_evalInt q hq _) (summable_evalInt q hq _), evalInt_X] have hlow : ∀ m < 2, PowerSeries.coeff m (F - PowerSeries.X) = 0 := by intro m hm rcases m with - | - | m · simp [PowerSeries.coeff_zero_eq_constantCoeff, h0] · simp [h1, PowerSeries.coeff_X] · exact absurd hm (by omega) have hr : valuation k (evalInt q (F - PowerSeries.X)) < valuation k q := lt_of_le_of_lt (valuation_evalInt_le_pow q hq hlow) (pow_lt_self_of_lt_one₀ (zero_lt_iff.mpr ((valuation k).ne_zero_iff.mpr hq0)) hq one_lt_two) rw [hsplit, (valuation k).map_add_eq_of_lt_left hr]- Project
- Fermat's Last Theorem
- License
- Apache-2.0
- Commit
- 8dd808888295
- Source
- FLT/KnownIn1980s/EllipticCurves/TateParameter.lean:279-294
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