Torsion and reduction #
Let A be a Dedekind domain with fraction field F, let u be a height-one prime of A, and let
W be an elliptic curve over F with an integral model at u. A point of W(F) whose order
n lies outside u has integral coordinates at u: carried to the completion F_u, it is a
point of order prime to the residue characteristic, and those avoid the kernel of reduction
E₁(F_u) (WeierstrassCurve.eq_zero_of_mem_kerReduction_of_nsmul_eq_zero).
When the integral model has unit discriminant, so that W has good reduction at u, reduction of
points is a homomorphism W(F) →+ W_k(k) to the points of the reduced curve
(WeierstrassCurve.Affine.Point.reductionHom), whose kernel is the point at infinity together with
the points whose x-coordinate has a pole. Integrality of prime-to-u torsion therefore says that
this homomorphism is injective on the n-torsion W(F)[n] for every n ∉ u: Silverman AEC
VII.3.1(b), read over the global field. When the residue field is finite this bounds the
prime-to-u torsion: #W(F)[n] divides #W_k(k).
Over the completed field F_u, every nonsingular point of the reduced curve is the reduction of a
point of E₀(F_u), and multiplication by n ∉ u is bijective on the formal kernel. Correcting an
arbitrary lift by a point of that kernel therefore gives a unique n-torsion lift in E₀(F_u),
whether the reduction is good or bad. Thus reduction is an additive equivalence from the n-torsion
of E₀(F_u) onto the n-torsion of the nonsingular points of the reduced curve, rather than only
an injection as over a global field. At good reduction E₀(F_u) is all of W(F_u). At bad
reduction this is how points of E₀(F_u) of a given order prime to u are produced, from points
of that order on the nonsingular locus of a singular cubic over the residue field.
Torsion of order divisible by the residue characteristic p is caught as well when the
ramification at u is small relative to p: if p ∉ u ^ (p - 1), the kernel of reduction has no
p-torsion either (Silverman AEC IV.6.1;
WeierstrassCurve.eq_zero_of_mem_kerReduction_of_nsmul_eq_zero_of_forall_prime_dvd). When every
prime p satisfies p ∉ u ^ (p - 1), as at every odd prime of ℤ, reduction is therefore
injective on the whole torsion subgroup of W(F), and its order divides #W_k(k).
At bad reduction, when the integral model has singular reduction, the same integrality makes
reduction injective on the points of W(F)[n] with nonsingular reduction, for n ∉ u. Those land
in the n-torsion of the nonsingular points of a singular cubic over the residue field, whose
order divides n. So the subgroup E₀(F) of points with nonsingular reduction meets W(F)[n] in
at most n points, and the rest of W(F)[n] is seen by the index [E(F) : E₀(F)]:
#W(F)[n] divides n · [E(F) : E₀(F)]. In particular, if W(F)[n] is all of the n² torsion
points, n divides that index. This is the step of the criterion of Néron, Ogg and Shafarevich
(Silverman AEC VII.7.1) that turns unramified torsion of unbounded order into good reduction, the
index being bounded independently of n over unramified extensions.
At additive reduction the singular point of the reduced cubic is a cusp, whose nonsingular points
have no torsion prime to the residue characteristic. Then E₀(F) meets W(F)[n] only in the point
at infinity, for every n ∉ u.
Main results #
Unless indicated otherwise, the results are in WeierstrassCurve.Affine.Point.
valuation_xCoord_le_one_and_valuation_yCoord_le_one_of_nsmul_eq_zero: a nonzero point ofW(F)killed by an integern ∉ uhas coordinates integral atu.eq_zero_of_reductionHom_eq_zero_of_nsmul_eq_zero: at good reduction, a point of order prime touthat reduces to the point at infinity is zero.injOn_reductionHom_torsionBy: at good reduction, reduction is injective onW(F)[n]forn ∉ u.card_torsionBy_dvd_card_reduction: at good reduction,#W(F)[n]divides the number of points of the reduced curve forn ∉ u.WeierstrassCurve.nsmul_ker_nonsingularReductionHom_bijective: multiplication byn ∉ uis bijective on the kernel of nonsingular reduction over the completed field.existsUnique_nonsingularReduction_torsion_lift: over the completed field, with good or bad reduction, every nonsingular reducedn-torsion point has a uniquen-torsion lift with nonsingular reduction, forn ∉ u.WeierstrassCurve.torsionNonsingularReductionAddEquiv: reduction identifies then-torsion ofE₀(F_u)with then-torsion of the nonsingular points of the reduced curve, forn ∉ u.existsUnique_torsion_lift: over the completed field with good reduction, every reducedn-torsion point has a uniquen-torsion lift, forn ∉ u.WeierstrassCurve.torsionReductionAddEquiv: reduction identifies then-torsion over the completed field with then-torsion of its reduction, forn ∉ u.valuation_xCoord_le_one_and_valuation_yCoord_le_one_of_nsmul_eq_zero_of_forall_prime_dvd: a nonzero point ofW(F)killed by an integern ≠ 0whose prime factorspsatisfyp ∉ u ^ (p - 1)has coordinates integral atu.eq_zero_of_reductionHom_eq_zero_of_nsmul_eq_zero_of_forall_prime_dvd: at good reduction, such a point that reduces to the point at infinity is zero.injOn_reductionHom_torsion: at good reduction, if every primepsatisfiesp ∉ u ^ (p - 1), reduction is injective on the torsion subgroup ofW(F).card_torsion_dvd_card_reduction: under the same hypotheses, the order of the torsion subgroup ofW(F)divides the number of points of the reduced curve.eq_zero_of_nonsingularReductionHom_eq_zero_of_nsmul_eq_zero: at bad reduction, a point ofE₀(F)killed by an integer prime toucannot lie in the kernel of reduction.natCard_nonsingularReduction_inf_torsionBy_dvd: at bad reduction, the number of points ofW(F)[n]with nonsingular reduction dividesn, forn ∉ u.nonsingularReduction_inf_torsionBy_eq_bot: at additive reduction, no nonzero point ofW(F)[n]has nonsingular reduction, forn ∉ u.natCard_torsionBy_dvd_mul_index_nonsingularReduction: at bad reduction,#W(F)[n]dividesn · [E(F) : E₀(F)]forn ∉ u.dvd_index_nonsingularReduction: at bad reduction, ifW(F)[n]hasn²points for somen ∉ u, thenndivides[E(F) : E₀(F)].
References #
- J. Silverman, The Arithmetic of Elliptic Curves, IV.6.1, VII.2.1, VII.3.1 and VII.7.1.
Torsion of order prime to u is integral at u: a nonzero point of W(F) killed by an
integer n ∉ u has both coordinates integral at u, for any integral model of W at u.
Torsion is integral at u when ramification is small relative to the prime factors of its
order: a nonzero point of W(F) killed by an integer n ≠ 0 whose prime factors p all satisfy
p ∉ u ^ (p - 1) has both coordinates integral at u, for any integral model of W at u.
At good reduction, reduction has no torsion prime to u in its kernel (Silverman AEC
VII.3.1(b)): a point killed by an integer n ∉ u that reduces to the point at infinity is the
point at infinity.
At good reduction, reduction is injective on torsion prime to u (Silverman AEC
VII.3.1(b)): for n ∉ u, reduction of points is injective on the n-torsion W(F)[n].
At good reduction, the torsion prime to u divides the reduced point count: for n ∉ u,
the order of W(F)[n] divides the number of points of the reduced curve. When the reduced curve
has infinitely many points, Nat.card reads 0 there and the statement is vacuous.
At good reduction, the kernel of reduction has no n-torsion when ramification is small
relative to the prime factors of n (Silverman AEC VII.3.1 and IV.6.1): a point killed by an
integer n ≠ 0 whose prime factors p all satisfy p ∉ u ^ (p - 1) and that reduces to the point
at infinity is the point at infinity.
At good reduction, reduction is injective on the torsion when ramification is small relative
to p (Silverman AEC VII.3.1 and IV.6.1): if every prime p satisfies p ∉ u ^ (p - 1),
that is, if the residue characteristic p of u has ramification index less than p - 1 at u,
then reduction of points is injective on the whole torsion subgroup of W(F). Over ℚ this holds
at every odd prime.
At good reduction, the torsion divides the reduced point count when ramification is small
relative to p: if every prime p satisfies p ∉ u ^ (p - 1), the order of the torsion
subgroup of W(F) divides the number of points of the reduced curve. When the reduced curve has
infinitely many points, Nat.card reads 0 there and the statement is vacuous.
Over the completion, a point with nonsingular reduction lies in the formal kernel of reduction
E₁ exactly when it reduces to the point at infinity.
Multiplication by an integer nonzero in the residue field is bijective on the kernel of nonsingular reduction over the completed field.
Prime-to-residue-characteristic torsion with nonsingular reduction lifts uniquely. Over a
completed discretely valued field, with good or bad reduction, every nonsingular point of the
reduced curve killed by an integer n prime to the residue characteristic is the reduction of
exactly one point with nonsingular reduction killed by n (Silverman VII.2.1 and VII.3.1).
Specialization identifies the prime-to-residue-characteristic torsion with nonsingular
reduction. Over the completed field, with good or bad reduction, reduction is an isomorphism from
the n-torsion of E₀(F_u) onto the n-torsion of the nonsingular points of the reduced curve,
for n prime to the residue characteristic.
Instances For
The torsion specialization equivalence with nonsingular reduction sends a point to its reduction.
The inverse torsion specialization equivalence with nonsingular reduction lifts a reduced point to a point with that reduction.
Over the completion, the kernel of the good-reduction homomorphism is the formal kernel
of reduction E₁.
Prime-to-residue-characteristic torsion lifts uniquely under good reduction. Over a completed discretely valued field, every reduced point killed by an integer prime to the residue characteristic has a unique lift killed by that integer (Silverman VII.3.1).
Specialization identifies the prime-to-residue-characteristic torsion at good
reduction. The reduced curve is the actual integral model selected by reductionHom.
The source is over the completed field; the analogous assertion over a global field need not
hold.
Instances For
The torsion specialization equivalence sends a point to its reduction.
The inverse torsion specialization equivalence lifts a reduced point to a point with that reduction.
At bad reduction, nonsingular reduction has no torsion prime to u in its kernel
(Silverman AEC VII.3.1(b)): a point in E₀(F) killed by an integer n ∉ u that reduces to the
point at infinity is the point at infinity.
At bad reduction, the n-torsion with nonsingular reduction is small (Silverman AEC
VII.7.1): if the integral model of W at u has singular reduction, then for n ∉ u the number of
points of W(F)[n] with nonsingular reduction divides n. In particular E₀(F) ∩ W(F)[n] is
never all of (ℤ / n)² for n > 1.
At additive reduction, E₀ has no torsion prime to u (Silverman AEC III.2.5 and
VII.3.1): if the reduction of the integral model of W at u is a cuspidal cubic, with Δ = 0 and
c₄ = 0, then for n ∉ u the only point of W(F)[n] with nonsingular reduction is the point at
infinity.
At bad reduction, torsion forces a large index of E₀ (Silverman AEC VII.7.1): if the
integral model of W at u has singular reduction, then for n ∉ u the number of points of
W(F)[n] divides n · [E(F) : E₀(F)], where E₀(F) is the subgroup of points with nonsingular
reduction.
At bad reduction, full n-torsion forces n to divide the index of E₀ (Silverman AEC
VII.7.1): if the integral model of W at u has singular reduction and W(F)[n] has n² points
for some n ∉ u, then n divides [E(F) : E₀(F)]. In particular, when that index is finite,
n is at most the index: full torsion of order prime to u beyond the index excludes bad
reduction.