Potential good reduction and the inertia image on the Tate module #
Let K be a nonarchimedean local field with residue field k, and let W be a Weierstrass curve
over K. It has potential good reduction when, after a finite extension L/K, a minimal model of
W has good reduction. This file proves the criterion of Néron, Ogg and Shafarevich for potential
good reduction (Silverman AEC VII.7.3): for an elliptic curve W and a prime ℓ ≠ char k, W
has potential good reduction exactly when the inertia subgroup I_K of Gal(K^{alg}/K) acts on
the ℓ-adic Tate module T_ℓ E through a finite quotient.
For the forward implication, embed L into K^{alg}, and let σ ∈ I_K fix L. A point of
W(K^{alg}) lies over a finite normal extension M of K containing L. The restriction of σ
to M lies in the inertia group G_0 of M/K, and since it fixes L it lies in the inertia group
of M/L. Over L the curve has a model with good reduction, so σ fixes the division points of
W prime to char k, as at good reduction
(WeierstrassCurve.Affine.Point.map_eq_self_of_mem_lowerRamificationGroup_zero). Hence the elements
of I_K fixing L act trivially on every T_ℓ E with ℓ ≠ char k. The action of I_K therefore
factors through the restriction to the normal closure of L, whose Galois group over K is finite.
Conversely, suppose that the image of I_K on T_ℓ E is finite, and let M ⊆ K^{alg} be the
field generated by the ℓ²-torsion points. An element of I_K fixing M fixes E[ℓ²], and
ℓ² ≥ 3, so by Serre's lemma it acts trivially on T_ℓ E
(WeierstrassCurve.tateModuleGaloisRepresentation_eq_one_of_finite_image). Identify K^{alg}
with an algebraic closure of M. The inertia subgroup I_M then consists of elements of I_K
fixing M (RingEquiv.mem_inertiaSubgroup_iff_of_semiconj), so I_M acts trivially on the Tate
module of W over M, and W has good reduction over M by the criterion of Néron, Ogg and
Shafarevich (WeierstrassCurve.hasGoodReduction_of_tateModuleGaloisRepresentation_eq_one).
Main results #
WeierstrassCurve.Affine.Point.map_eq_self_of_mem_inertiaSubgroup_of_mem_fixingSubgroup: ifWacquires good reduction over a finite extensionLofKembedded inK^{alg}, an element ofI_KfixingLfixes every pointPofW(K^{alg})for which it fixesn • P, for an integernprime to the residue characteristic.tateModuleGaloisRepresentation_eq_one_of_mem_inertiaSubgroup_of_mem_fixingSubgroup: such an element acts trivially onT_ℓ Efor every primeℓ ≠ char k.WeierstrassCurve.HasPotentialGoodReduction.exists_tateModuleGaloisRepresentation_eq_one: at potential good reduction, the elements ofI_Kfixing some finite extension ofKact trivially on everyT_ℓ Ewithℓ ≠ char k.WeierstrassCurve.HasPotentialGoodReduction.finite_image_tateModuleGaloisRepresentation: at potential good reduction, the image ofI_Kacting onT_ℓ Eis finite.WeierstrassCurve.hasPotentialGoodReduction_of_tateModuleGaloisRepresentation_eq_one: if the elements ofI_Kfixing a finite extension ofKact trivially onT_ℓ Efor one primeℓ ≠ char k, thenWhas potential good reduction.WeierstrassCurve.exists_tateModuleGaloisRepresentation_eq_one_of_finite_image: if the image ofI_KonT_ℓ Eis finite, the elements ofI_Kfixing some finite extension ofKact trivially onT_ℓ E.WeierstrassCurve.hasPotentialGoodReduction_iff_finite_image_tateModuleGaloisRepresentation:Whas potential good reduction exactly when the image ofI_KonT_ℓ Eis finite.
References #
- J. Silverman, The Arithmetic of Elliptic Curves, VII.4.1 and VII.7.3.
- J.-P. Serre and J. Tate, Good reduction of abelian varieties, Ann. of Math. 88 (1968), 492–517.
Inertia over a field of good reduction fixes the division points prime to p. Let L be
a finite extension of K, embedded in K^{alg} by ι, over which W acquires good reduction:
for a change of variables C over L, the equation C • W has coefficients in 𝒪[L] and unit
discriminant. Then every element σ of the inertia subgroup of Gal(K^{alg}/K) fixing ι(L)
fixes every point P of W(K^{alg}) such that σ fixes n • P, for an integer n that is
nonzero in the residue field of K. In particular σ fixes the n-torsion of W(K^{alg}).
The Tate module is unramified over a field of good reduction. Let L be a finite
extension of K, embedded in K^{alg} by ι, over which W acquires good reduction: for a
change of variables C over L, the equation C • W has coefficients in 𝒪[L] and unit
discriminant. Then every element of the inertia subgroup of Gal(K^{alg}/K) fixing ι(L) acts
trivially on the ℓ-adic Tate module of W, for every prime ℓ different from the residue
characteristic of K.
Potential good reduction #
At potential good reduction, the inertia of a finite extension acts trivially on the Tate
modules. If W has potential good reduction, there is a finite extension M of K inside
K^{alg} such that every element of the inertia subgroup of Gal(K^{alg}/K) fixing M acts
trivially on the ℓ-adic Tate module of W, for every prime ℓ different from the residue
characteristic of K.
At potential good reduction, inertia acts on the Tate module through a finite quotient
(Silverman AEC VII.7.3, the implication from potential good reduction). If W has potential good
reduction, then for every prime ℓ different from the residue characteristic of K, the image of
the inertia subgroup of Gal(K^{alg}/K) in the automorphisms of the ℓ-adic Tate module of W is
finite.
The converse: a finite inertia image forces potential good reduction #
Good reduction over a finite extension from an unramified Tate module. Let M be a finite
extension of K inside K^{alg}, and let ℓ be a prime different from the residue characteristic
of K. If every element of the inertia subgroup of Gal(K^{alg}/K) fixing M acts trivially on
the ℓ-adic Tate module of W, then W has potential good reduction.
A finite inertia image is trivial on the inertia of a finite extension. Let ℓ be a prime
invertible in K. If the image of the inertia subgroup of
Gal(K^{alg}/K) in the automorphisms of the ℓ-adic Tate module of W is finite, then there is a
finite extension M of K inside K^{alg} such that every element of the inertia subgroup fixing
M acts trivially on the Tate module.
A finite inertia image forces potential good reduction (Silverman AEC VII.7.3, the
converse implication; Serre–Tate). Let W be an elliptic curve over a nonarchimedean local field
K, and let ℓ be a prime different from the residue characteristic of K. If the image of the
inertia subgroup of Gal(K^{alg}/K) in the automorphisms of the ℓ-adic Tate module of W is
finite, then W has potential good reduction.
The criterion of Néron, Ogg and Shafarevich for potential good reduction (Silverman AEC
VII.7.3). Let W be an elliptic curve over a nonarchimedean local field K, and let ℓ be a
prime different from the residue characteristic of K. Then W has potential good reduction
exactly when the inertia subgroup of Gal(K^{alg}/K) acts on the ℓ-adic Tate module of W
through a finite quotient.