Points with nonsingular reduction #
Let v be a valuation on a field F, with valuation ring O and residue field k, and let W
be a Weierstrass curve over F with an integral model W_O over O. Every point of W(F) reduces
to a k-point of the reduced curve W_k = W_O ⊗ k (WeierstrassCurve.Affine.Point.reduction).
When W_k is singular, the reduction of a point may be its singular point. This file defines the
subgroup
E₀(F) = {P ∈ W(F) | the reduction of P is a nonsingular point of W_k}
and shows that reduction is a group homomorphism E₀(F) →+ W_k,ns(k) to the group of nonsingular
points of W_k, whose kernel is the kernel of reduction E₁(F): the point at infinity together
with the points whose x-coordinate has a pole. This is the left-exact part of Silverman's exact
sequence 0 → E₁(F) → E₀(F) → W_k,ns(k) → 0 (AEC VII.2.1), for an arbitrary valuation and an
arbitrary integral model. The hypotheses are those of the statement: neither ellipticity of W,
minimality of W_O, discreteness of v nor completeness of F is assumed.
The subgroup E₀(F) depends on the integral model, but only up to changes of variables over O:
such a change C carries the integral model of C • W to C applied to that of W, and the
isomorphism (C • W)(F) ≃+ W(F) it induces carries E₀ onto E₀, since it preserves the points
with a pole in x and acts on the reductions of the others by the reduction of C. In particular
the index [W(F) : E₀(F)] is unchanged, which is what lets it be computed on whichever integral
model of W is most convenient, such as the one at which Tate's algorithm stops.
The sequence is also right-exact when O is Henselian, for instance complete: the reduction
homomorphism is then surjective. This is Hensel's lemma applied to the Weierstrass equation in
whichever variable has a nonvanishing partial derivative at the point of W_k
(WeierstrassCurve.Affine.exists_nonsingular_residue_eq).
Main definitions #
WeierstrassCurve.Affine.nonsingularReduction: the subgroupE₀(F)of points whose reduction is a nonsingular point of the reduced curve.WeierstrassCurve.Affine.nonsingularReductionHom: the reduction homomorphismE₀(F) →+ W_k,ns(k).
Main results #
WeierstrassCurve.Affine.Point.reduction_add_of_nonsingularLift: ifPandQreduce to nonsingular points, then the reduction ofP + Qis the sum of their reductions.WeierstrassCurve.Affine.some_mem_nonsingularReduction_iff: an affine point with integral coordinates lies inE₀(F)exactly when the residues of its coordinates form a nonsingular point of the reduced curve.WeierstrassCurve.Affine.mem_nonsingularReduction_of_one_lt: the kernel of reduction is contained inE₀(F).WeierstrassCurve.Affine.nonsingularReduction_congr: equivalent valuations have the sameE₀(F).WeierstrassCurve.Affine.equivVariableChange_mem_nonsingularReduction_iffandWeierstrassCurve.Affine.nonsingularReduction_baseChange_smul: a change of variables overOidentifiesE₀of the two integral models.WeierstrassCurve.Affine.index_nonsingularReduction_baseChange_smul: so the index ofE₀(F)is invariant under changes of variables overO.WeierstrassCurve.Affine.nonsingularReductionHom_eq_zero_iff: the kernel of the reduction homomorphism is the kernel of reductionE₁(F).WeierstrassCurve.Affine.mem_ker_nonsingularReductionHom_iff: the same statement for the kernel subgroupE₁(F)ofE₀(F).WeierstrassCurve.Affine.nonsingularReductionHom_surjective: if the valuation ring is Henselian, the reduction homomorphism is surjective.
References #
- J. Silverman, The Arithmetic of Elliptic Curves, VII.2.1.
- W. Bosma and H. W. Lenstra, Jr., Complete systems of two addition laws for elliptic curves, J. Number Theory 53 (1995), 229–240.
Reduction commutes with addition on points with nonsingular reduction. If P and Q
reduce to nonsingular points of the reduced curve, then the reduction of P + Q is the sum of the
reductions of P and Q on the reduced curve.
The subgroup E₀(F) of points with nonsingular reduction: the points of W(F) whose
reduction is a nonsingular point of the reduced curve.
Instances For
A point lies in E₀(F) exactly when its reduction is a nonsingular point of the reduced curve.
An affine point with integral x-coordinate has nonsingular reduction exactly when the
residues of its coordinates form a nonsingular point of the reduced curve.
The kernel of reduction is contained in E₀(F): a point whose x-coordinate has a pole
reduces to the point at infinity, which is nonsingular.
E₀(F) depends only on the valuation ring: equivalent valuations, which have the same
valuation ring and so the same integral model and reduced curve, have the same subgroup of points
with nonsingular reduction.
Changes of variables over the valuation ring #
A change of variables over the valuation ring preserves nonsingular reduction. For a change
of variables C defined over O, the isomorphism (C • W)(F) ≃ W(F) it induces sends a point to
a point with nonsingular reduction exactly when the point has nonsingular reduction, for the
integral models C • W_O and W_O.
E₀ of C • W is the preimage of E₀ of W under the isomorphism of point groups induced
by a change of variables C over the valuation ring.
The index of E₀ is invariant under changes of variables over the valuation ring:
[(C • W)(F) : E₀] = [W(F) : E₀] for a change of variables C defined over O.
The reduction homomorphism E₀(F) →+ W_k,ns(k), from the points with nonsingular
reduction to the group of nonsingular points of the reduced curve. A point with integral
x-coordinate goes to the residues of its coordinates
(nonsingularReductionHom_some_of_valuation_le_one), and the other points go to the point at
infinity.
Instances For
The reduction homomorphism, read in projective coordinates, is the reduction of points.
A point with nonsingular reduction and integral x-coordinate reduces to the residues of its
coordinates.
The kernel of the reduction homomorphism is the kernel of reduction E₁(F): a point with
nonsingular reduction reduces to the point at infinity exactly when it is the point at infinity or
its x-coordinate has a pole.
The kernel of reduction E₁(F) as a subgroup of E₀(F): a point with nonsingular reduction
lies in the kernel of the reduction homomorphism exactly when it is the point at infinity or its
x-coordinate has a pole.
Reduction onto the nonsingular points. If the valuation ring is Henselian, for instance
complete, every nonsingular point of the reduced curve is the reduction of a point of W(F) with
nonsingular reduction.