The algorithmic conductor of an elliptic curve over a Dedekind domain #
Let O be a Dedekind domain with fraction field K whose localisations at height-one primes have
perfect residue fields, and let W be an elliptic curve over K. At each height-one prime v,
Tate's algorithm on a minimal equation over Oᵥ = Localization.AtPrime v.asIdeal produces the
algorithmic Ogg exponent fᵥ = v (Δ_min,ᵥ) − mᵥ + 1
(WeierstrassCurve.algorithmicOggExponent). This file assembles those exponents into a single
ideal of O,
𝔑 = ∏ᵥ 𝔭ᵥ ^ fᵥ,
the algorithmic conductor of W (Silverman, Advanced Topics, IV.10). By Ogg's formula it
is the conductor of W; that identification with the ramification-theoretic conductor is not made
here, which is why the ideal is named after the algorithm that computes it.
The exponent fᵥ is 0 at good reduction, 1 at multiplicative reduction and at least 2 at
additive reduction, and it never exceeds v (Δ_min,ᵥ). Accordingly, the primes dividing 𝔑 are
exactly the primes of bad reduction, 𝔑 divides the minimal discriminant ideal, 𝔑 = 1 exactly
when W has good reduction everywhere, and 𝔑 is squarefree exactly when W is semistable,
in which case 𝔑 is the radical of the minimal discriminant ideal.
Like the minimal discriminant ideal, 𝔑 depends only on the K-isomorphism class of W.
Main definitions #
WeierstrassCurve.algorithmicConductor: the ideal∏ᵥ 𝔭ᵥ ^ fᵥofO.
Main results #
WeierstrassCurve.hasFiniteMulSupport_pow_algorithmicOggExponentandWeierstrassCurve.algorithmicConductor_ne_bot: the defining product is finite and nonzero.WeierstrassCurve.count_algorithmicConductor: the exponent of𝔭ᵥin𝔑isfᵥ.WeierstrassCurve.algorithmicConductor_smul:𝔑is invariant under a change of variables.WeierstrassCurve.pow_dvd_algorithmicConductor_iff:𝔭ᵥ ^ n ∣ 𝔑exactly whenn ≤ fᵥ.WeierstrassCurve.dvd_algorithmicConductor_iffandWeierstrassCurve.dvd_algorithmicConductor_iff_dvd_minimalDiscriminantIdeal:𝔭ᵥ ∣ 𝔑exactly whenWhas bad reduction atv, that is, when𝔭ᵥdivides the minimal discriminant ideal.WeierstrassCurve.algorithmicConductor_dvd_minimalDiscriminantIdeal:𝔑divides the minimal discriminant ideal.WeierstrassCurve.algorithmicConductor_eq_top_iff:𝔑 = 1exactly whenWhas good reduction at every height-one prime.WeierstrassCurve.isSemistable_iff_squarefree_algorithmicConductorandWeierstrassCurve.isSemistable_iff_algorithmicConductor_eq_radical:Wis semistable exactly when𝔑is squarefree, equivalently when𝔑is the radical of the minimal discriminant ideal.WeierstrassCurve.IsGlobalMinimal.dvd_algorithmicConductor_iffandWeierstrassCurve.IsGlobalMinimal.sq_dvd_algorithmicConductor_iff: for a globally minimal equation,𝔭ᵥ ∣ 𝔑exactly whenvcontains its discriminant, and𝔭ᵥ² ∣ 𝔑exactly whenvalso contains itsc₄.WeierstrassCurve.one_lt_absNorm_algorithmicConductor_iff: overℚ, the norm of𝔑exceeds1exactly whenWhas bad reduction at some prime.WeierstrassCurve.pow_dvd_absNorm_algorithmicConductor_iff: overℚ, the exponent ofpin the norm of𝔑is the algorithmic Ogg exponent at(p).WeierstrassCurve.IsGlobalMinimal.dvd_absNorm_algorithmicConductor_iffandWeierstrassCurve.IsGlobalMinimal.sq_dvd_absNorm_algorithmicConductor_iff: overℚ, a primepdivides the norm of𝔑exactly when it divides the discriminant of a globally minimal integral equation, andp²does exactly whenpalso divides itsc₄.
References #
- J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, GTM 151, IV.9–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 product defining the algorithmic conductor is finite. Each algorithmic Ogg exponent is bounded by the local minimal discriminant exponent, and those have finite support.
The algorithmic conductor ∏ᵥ 𝔭ᵥ ^ fᵥ of an elliptic curve over the fraction field of a
Dedekind domain O: the product over the height-one primes of O of the prime powers whose
exponents are the algorithmic Ogg exponents fᵥ = v (Δ_min,ᵥ) − mᵥ + 1 read off Tate's algorithm.
The product is finite (hasFiniteMulSupport_pow_algorithmicOggExponent).
Instances For
The defining prime-power factorisation of the algorithmic conductor. Outside this module the
body of algorithmicConductor is not exposed, so this is the interface for unfolding it.
The algorithmic conductor is nonzero.
The exponent of 𝔭ᵥ in the algorithmic conductor is the algorithmic Ogg exponent at v.
This reads the local exponents back off the ideal, so results about the conductor can be proved
without unfolding the defining product.
The algorithmic conductor is invariant under a change of variables, so it is an invariant of the curve and not of the equation presenting it.
𝔭ᵥ ^ n divides the algorithmic conductor exactly when n is at most the algorithmic Ogg
exponent at v.
A prime divides the algorithmic conductor exactly when the curve has bad reduction there.
The algorithmic conductor and the minimal discriminant ideal have the same prime divisors, the primes of bad reduction.
The algorithmic conductor divides the minimal discriminant ideal, as each algorithmic Ogg exponent is at most the valuation of the local minimal discriminant.
The algorithmic conductor is trivial exactly when the curve has good reduction at every height-one prime.
A curve is semistable exactly when its algorithmic conductor is squarefree, since the
algorithmic Ogg exponent is at least 2 exactly at additive reduction.
A curve is semistable exactly when its algorithmic conductor is the radical of its minimal discriminant ideal, that is, the product of its primes of bad reduction.
A prime divides the algorithmic conductor of a globally minimal equation exactly when it contains the discriminant of its integral model, that is, when the curve has bad reduction there.
The square of a prime divides the algorithmic conductor of a globally minimal equation
exactly when the prime contains both the discriminant and c₄ of its integral model, that is,
when the curve has additive reduction there.
The algorithmic conductor of a curve over ℚ exceeds 1 exactly when the curve has bad
reduction at some prime. This is the condition under which the Szpiro ratio is defined.
The local exponents of the algorithmic conductor over ℚ are p-adic valuations of its
norm: for the height-one prime v = (p) of ℤ, p ^ n divides N = |𝔑| exactly when
n ≤ f_p.
A prime divides the conductor of a globally minimal equation over ℚ exactly when it
divides the discriminant of its integral model, that is, when the curve has bad reduction
there.
The square of a prime divides the conductor of a globally minimal equation over ℚ
exactly when the prime divides both the discriminant and c₄ of its integral model, that is,
when the curve has additive reduction there.