Inertia acts trivially on the Tate module at good reduction #
Let K be a nonarchimedean local field with residue field k, and let W be a Weierstrass
equation over K with good reduction: its coefficients lie in πͺ[K] and its discriminant is a
unit there. This file proves that the inertia subgroup I_K of Gal(K^{alg}/K) fixes every point
of W(K^{alg}) whose order is prime to the characteristic of k, and hence acts trivially on the
Tate module T_β E for every prime β β char k. This is the implication from good reduction to
an unramified β-adic representation in the criterion of NΓ©ron, Ogg and Shafarevich (Silverman
AEC VII.7.1).
The finite-level statement is WeierstrassCurve.Affine.Point.map_eq_self_of_mem_inertiaSubgroup,
for the inertia group of a height-one prime of a Dedekind domain. A finite extension L of K
is the fraction field of the discrete valuation ring πͺ[L], whose maximal ideal gives the
height-one prime, and the inertia group G_0 of L/K is the inertia group of that prime. So
G_0 fixes the torsion of W(L) of order prime to char k, and more generally every point P
for which it fixes n β’ P with n prime to char k
(WeierstrassCurve.Affine.Point.map_eq_self_of_mem_lowerRamificationGroup_zero). In particular a
finite extension of K generated by the n-division points of K-rational points of W has
trivial G_0, and when it is Galois over K, as K([n]β»ΒΉ W(K)) is, it is unramified, because
#G_0 = e(L/K) (Silverman AEC VIII.1.5(b)). A point over K^{alg} is defined over
the normal closure of the field generated by its coordinates, a finite normal extension of K, on
which every element of I_K restricts into G_0
(TauCeti.restrictNormal_mem_lowerRamificationGroup_zero).
Main results #
WeierstrassCurve.Affine.Point.map_eq_self_of_mem_lowerRamificationGroup_zero: for a finite extensionL/Kof local fields, an element of the inertia groupG_0ofL/Kfixes every pointPofW(L)for which it fixesn β’ P, for an integernprime to the residue characteristic; for instance every point killed byn.WeierstrassCurve.Affine.Point.map_eq_self_of_mem_lowerRamificationGroup_zero_of_variableChange: the same whenWoverLonly acquires good reduction after a change of variables overL, for an element of the inertia groupG_0of a finite extensionM/Lacting onW(M).WeierstrassCurve.Affine.Point.map_restrictScalars_eq_self_of_mem_lowerRamificationGroup_zero: the same for a curve over a subfieldKofL, with the element acting as aK-automorphism.WeierstrassCurve.lowerRamificationGroup_zero_eq_bot_of_adjoin_coords_eq_top: a finite extension ofKgenerated by pointsQofWsuch thatG_0fixesn β’ Qhas trivial inertia groupG_0.WeierstrassCurve.isUnramified_of_adjoin_coords_eq_top: such an extension is unramified when it is Galois overK.WeierstrassCurve.isUnramified_of_adjoin_coords_eq_top_of_nsmul_eq_baseChange: in particular, a finite Galois extension ofKgenerated by then-division points ofK-rational points ofWis unramified.WeierstrassCurve.Affine.Point.map_eq_self_of_mem_inertiaSubgroup_algebraicClosure: an element of the inertia subgroupI_Kfixes every pointPofW(K^{alg})for which it fixesn β’ P, for such ann; for instance every point killed byn.WeierstrassCurve.tateModuleGaloisRepresentation_eq_one_of_mem_inertiaSubgroup:I_Kacts trivially onT_β Efor every primeβdifferent from the residue characteristic.
References #
- J. Silverman, The Arithmetic of Elliptic Curves, VII.4.1, VII.7.1 and VIII.1.5.
Division points prime to p are unramified in a finite extension, at good reduction. Let
L/K be a finite extension of nonarchimedean local fields and let W have good reduction over
πͺ[K]. An element Ο of the inertia group G_0 of L/K fixes every point P of W(L) such
that Ο fixes n β’ P, for an integer n that is nonzero in the residue field of K; in
particular it fixes the n-torsion of W(L).
A field generated by points with inertia-fixed multiples has trivial inertia at good
reduction. Let L/K be a finite extension of nonarchimedean local fields generated by the
coordinates of a set S of points Q of W(L) such that every element of the inertia group G_0
of L/K fixes n β’ Q, for an integer n that is nonzero in the residue field of K. If W has
good reduction over πͺ[K], then G_0 is trivial. This group consists of automorphisms of L/K,
so it measures ramification only when L/K is Galois; see
WeierstrassCurve.isUnramified_of_adjoin_coords_eq_top.
A Galois field generated by points with inertia-fixed multiples is unramified at good
reduction. Let L/K be a finite Galois extension of nonarchimedean local fields generated by the
coordinates of a set S of points Q of W(L) such that every element of the inertia group G_0
of L/K fixes n β’ Q, for an integer n that is nonzero in the residue field of K. If W has
good reduction over πͺ[K], then L/K is unramified.
A Galois field of division points is unramified at good reduction (Silverman AEC
VIII.1.5(b), over a local field). Let L/K be a finite Galois extension of nonarchimedean local
fields generated by the coordinates of a set S of points Q of W(L) with n β’ Q in the image
of W(K), for an integer n that is nonzero in the residue field of K, as L = K([n]β»ΒΉ W(K))
is. If W has good reduction over πͺ[K], then L/K is unramified.
An element Ο of the inertia group G_0 of a finite extension M/L fixes every point Q of
W(M) such that Ο fixes n β’ Q, for an integer n that is nonzero in the residue field of L,
provided W acquires good reduction over L after a change of variables C over L: the
equation C β’ W has coefficients in πͺ[L] and unit discriminant.
An element Ο of the inertia group G_0 of a finite extension M/L fixes every point Q of
W(M) such that Ο fixes n β’ Q, for an integer n that is nonzero in the residue field of K,
provided W acquires good reduction over L: for a change of variables C over L, the equation
C β’ W has coefficients in πͺ[L] and unit discriminant. Here Ο acts on W(M) as an automorphism
of M over K.
Division points prime to p are unramified at good reduction (Silverman AEC VII.4.1 and
VIII.1.5(b)). If W has good reduction over πͺ[K], every element Ο of the inertia subgroup of
Gal(K^{alg}/K) fixes every point P of W(K^{alg}) such that Ο fixes n β’ P, for an integer
n that is nonzero in the residue field of K. In particular Ο fixes the n-torsion of
W(K^{alg}) and the n-division points of the K-rational points of W.
The Tate module is unramified at good reduction (Silverman AEC VII.4.1, and the forward
implication of VII.7.1). If W has good reduction over πͺ[K] and β is a prime different from
the residue characteristic of K, the inertia subgroup of Gal(K^{alg}/K) acts trivially on the
β-adic Tate module of W.