The reduction symbol and the algorithmic Ogg exponent of an elliptic curve #
Let R be a discrete valuation ring with fraction field K and perfect residue field, and let W
be an elliptic curve over K. Tate's algorithm, run on a minimal equation of W over R, stops
with a symbol (WeierstrassCurve.exists_isTateTerminal), and this symbol does not depend on the
minimal equation, on the changes of variables made along the way, or on the uniformiser
(WeierstrassCurve.IsTateTerminal.eq_of_isMinimal_smul). This file names it as an invariant of
the curve, W.reductionSymbol R, and with it the algorithmic Ogg exponent
f = v(Δ_min) − m + 1,
where m is the component count of the symbol (TauCeti.ReductionSymbol.componentCount) and
v(Δ_min) the valuation of the minimal discriminant
(WeierstrassCurve.localMinimalDiscriminantValuation). It is the exponent read off Tate's
algorithm (Silverman, Advanced Topics, IV.9); by Ogg's formula (Advanced Topics, IV.11) it is
the exponent of the conductor, an identification not made here.
Both are invariant under a change of variables over K, and both detect the reduction type: the
symbol is I₀ at good reduction, Iₙ with n = v(Δ_min) ≥ 1 at multiplicative reduction, and
none of the Iₙ at additive reduction; correspondingly the exponent is 0, 1, or at least 2.
The exponent never exceeds v(Δ_min), so it vanishes at every prime of good reduction.
Main definitions #
WeierstrassCurve.reductionSymbol: the symbol at which Tate's algorithm stops on a minimal equation ofW.WeierstrassCurve.algorithmicOggExponent: the algorithmic Ogg exponentv(Δ_min) − m + 1.
Main results #
WeierstrassCurve.reductionSymbol_eq_of_isTateTerminalandWeierstrassCurve.reductionSymbol_eq_iff: the symbol is the output of Tate's algorithm on any minimal equation ofW, with any uniformiser.WeierstrassCurve.reductionSymbol_smul: the symbol is invariant under a change of variables.WeierstrassCurve.reductionSymbol_eq_I_iff,WeierstrassCurve.reductionSymbol_eq_I_zero_iff,WeierstrassCurve.exists_reductionSymbol_eq_I_add_one_iffandWeierstrassCurve.forall_reductionSymbol_ne_I_iff: the symbol detects the reduction type.WeierstrassCurve.reductionSymbol_eq_I_zero_of_hasGoodReductionandWeierstrassCurve.reductionSymbol_eq_I_of_hasMultiplicativeReduction: direct simp evaluations of the symbol under good or multiplicative reduction hypotheses on the curve.WeierstrassCurve.natCast_algorithmicOggExponent: the exponent isv(Δ_min) − m + 1.WeierstrassCurve.algorithmicOggExponent_eq_zero_iff,WeierstrassCurve.algorithmicOggExponent_eq_one_iffandWeierstrassCurve.two_le_algorithmicOggExponent_iff: the exponent is0at good,1at multiplicative and at least2at additive reduction.WeierstrassCurve.algorithmicOggExponent_le_localMinimalDiscriminantValuation:f ≤ v(Δ_min).
References #
- J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, GTM 151, IV.9 and IV.11.
- J. Tate, Algorithm for determining the type of a singular fibre in an elliptic pencil, in Modular Functions of One Variable IV, LNM 476 (1975), 33–52.
The reduction symbol of an elliptic curve: the symbol at which Tate's algorithm stops on a
minimal equation of W over R. It does not depend on the minimal equation, on the changes of
variables made by the algorithm, or on the uniformiser (reductionSymbol_eq_of_isTateTerminal).
Instances For
Tate's algorithm computes the reduction symbol: it stops with output W.reductionSymbol R
on some minimal equation of W over R.
The reduction symbol is the output of Tate's algorithm on any minimal equation. If W' is
an equation over R, minimal over K and related to W by a change of variables over K, and
Tate's algorithm, run with any uniformiser, stops on W' with output s, then s is the
reduction symbol of W.
Characterisation of the reduction symbol: W.reductionSymbol R = s exactly when Tate's
algorithm stops with output s on some minimal equation of W.
The reduction symbol is an invariant of the curve: it does not change under a change of variables.
The reduction symbol detects additive reduction: it is none of the Iₙ exactly when the
minimal equation of W has additive reduction.
The reduction symbol is Iₙ exactly at good or multiplicative reduction, with n the
valuation of the minimal discriminant.
The reduction symbol is I₀ exactly at good reduction.
Tate's canonical output is I₀ for an equation with good reduction.
Tate's canonical output at multiplicative reduction has index equal to the minimal-discriminant valuation.
The reduction symbol is Iₙ with n ≥ 1 exactly at multiplicative reduction.
The algorithmic Ogg exponent of an elliptic curve, v(Δ_min) − m + 1, where m is the
component count of the reduction symbol and v(Δ_min) the valuation of the minimal discriminant
(natCast_algorithmicOggExponent). It is 0 at good reduction, 1 at multiplicative reduction
and at least 2 at additive reduction.
Instances For
The algorithmic Ogg exponent is v(Δ_min) − m + 1, with m the component count of the
reduction symbol: the integer v(Δ_min) − m + 1 is nonnegative, so no truncation occurs.
The algorithmic Ogg exponent is an invariant of the curve: it does not change under a change of variables.
The algorithmic Ogg exponent vanishes exactly at good reduction.
The algorithmic Ogg exponent is positive exactly at bad reduction.
The algorithmic Ogg exponent is 1 exactly at multiplicative reduction.
The algorithmic Ogg exponent is at least 2 exactly at additive reduction.
The algorithmic Ogg exponent is at most the valuation of the minimal discriminant, since every reduction symbol has at least one component.