Points with singular reduction at the reduction symbol I₀* #
Let v be a valuation on a field F, with valuation ring O, maximal ideal 𝔪 = (ϖ) and residue
field k, and let W be a Weierstrass curve over F with an integral model W_O over O. The
points with nonsingular reduction form the subgroup E₀(F) of W(F)
(WeierstrassCurve.Affine.nonsingularReduction). When F is complete for a discrete valuation with
perfect residue field and the equation is minimal, the index [E(F) : E₀(F)] is the local index
c_v that Tate's algorithm computes (Silverman, Advanced Topics, IV.9, Table 4.1). This file
computes it at the reduction symbol I₀*, where Tate's algorithm reports
c_v = 1 + #{roots of the residue cubic in k}.
At I₀* the integral model is in the normal form ϖ ∣ a₁, a₂ = ϖ A₂, ϖ² ∣ a₃, a₄ = ϖ² A₄,
a₆ = ϖ³ A₆, and the residue cubic T³ + A₂ T² + A₄ T + A₆ has nonzero discriminant over k
(WeierstrassCurve.IsTateTerminal). A point with singular reduction is an affine point (x, y)
with x, y ∈ 𝔪 (WeierstrassCurve.Affine.some_notMem_nonsingularReduction_iff). Comparing powers
of ϖ in the Weierstrass equation shows x = ϖ X and y = ϖ² Y with X, Y ∈ O, and the residue
of X is a root of the residue cubic.
Two points P = (ϖ X₁, y₁) and Q = (ϖ X₂, y₂) with singular reduction are then congruent modulo
E₀(F) up to sign exactly when X₁ and X₂ have the same residue: P + Q ∈ E₀(F) if and only if
X₁ ≡ X₂ (WeierstrassCurve.Affine.add_mem_nonsingularReduction_iff_of_isTateTerminal_IStar_zero).
If the residues differ, the line through P and Q has slope
ℓ = (y₁ − y₂) / (x₁ − x₂) ∈ ϖ O, since v(x₁ − x₂) = v(ϖ) and ϖ² ∣ y₁ − y₂, so the third
intersection point and with it P + Q has coordinates in 𝔪. If the residues agree and
P + Q = -(ϖ X₃, y₃) has singular reduction as well, then ℓ ∈ 𝔪, and dividing the
factorisation
x³ + a₂ x² + a₄ x + a₆ − (ℓ (x − x₁) + y₁)² − (a₁ x + a₃) (ℓ (x − x₁) + y₁) = (x − x₁) (x − x₂) (x − x₃)
of the Weierstrass cubic along the line by ϖ³ at x = ϖ T shows that the residue cubic is
(T − X₁) (T − X₂) (T − X₃) modulo ϖ. A repeated root would make its discriminant vanish, which
the stopping test for I₀* excludes.
So the cosets of E₀(F) other than E₀(F) itself correspond to the roots of the residue cubic in
k that some point with singular reduction reduces to. Over a Henselian valuation ring every root
t occurs: it lifts to a root X of T³ + A₂ T² + A₄ T + A₆ in O, since the roots of the
residue cubic are simple, and (ϖ X, 0) is a point of W with singular reduction. Hence
[E(F) : E₀(F)] = 1 + #{roots of the residue cubic in k}, which is 1, 2 or 4.
Main results #
WeierstrassCurve.Affine.exists_xCoord_eq_of_isTateTerminal_IStar_zero: a point with singular reduction hasx = ϖ X, with the residue ofXa root of the residue cubic.WeierstrassCurve.Affine.exists_yCoord_eq_of_isTateTerminal_IStar_zero: a point with singular reduction hasy = ϖ² Y.WeierstrassCurve.Affine.add_mem_nonsingularReduction_iff_of_isTateTerminal_IStar_zero: for two points(ϖ X₁, y₁)and(ϖ X₂, y₂)with singular reduction, the sum has nonsingular reduction exactly whenX₁andX₂have the same residue.WeierstrassCurve.Affine.exists_notMem_nonsingularReduction_of_isTateTerminal_IStar_zero: over a Henselian valuation ring, every root of the residue cubic is the residue ofx / ϖfor some point with singular reduction.WeierstrassCurve.Affine.index_nonsingularReduction_eq_of_isTateTerminal_IStar_zero: over a Henselian valuation ring,[E(F) : E₀(F)] = 1 + #{roots of the residue cubic in k}.
References #
- J. Silverman, The Arithmetic of Elliptic Curves, VII.6.
- J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, GTM 151, IV.9.
- 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.
At I₀*, a point with singular reduction has x = ϖ X, with the residue of X a root of the
residue cubic. Here ϖ generates the maximal ideal of the valuation ring, Tate's algorithm stops
at I₀* on the integral model, and a₂ = ϖ A₂, a₄ = ϖ² A₄, a₆ = ϖ³ A₆, so that the residue
cubic is T³ + A₂ T² + A₄ T + A₆ over the residue field.
At I₀*, a point with singular reduction has y = ϖ² Y. Here ϖ generates the maximal
ideal of the valuation ring and Tate's algorithm stops at I₀* on the integral model.
Sums of points with singular reduction #
At I₀*, two points with singular reduction add up to a point with nonsingular reduction
exactly when their values of x / ϖ have the same residue. Here P = (ϖ X₁, y₁) and
Q = (ϖ X₂, y₂) have singular reduction, ϖ generates the maximal ideal of the valuation ring,
and Tate's algorithm stops at I₀* on the integral model. Since -Q = (ϖ X₂, -y₂ - a₁ x₂ - a₃),
this assigns to each coset of E₀(F) other than E₀(F) itself a root of the residue cubic
(exists_xCoord_eq_of_isTateTerminal_IStar_zero), distinct cosets getting distinct roots. That
every root arises needs the Henselian hypothesis of
exists_notMem_nonsingularReduction_of_isTateTerminal_IStar_zero.
Over a Henselian valuation ring #
At I₀* over a Henselian valuation ring, every root of the residue cubic comes from a point
with singular reduction. Here ϖ generates the maximal ideal of the valuation ring, Tate's
algorithm stops at I₀* on the integral model, and a₂ = ϖ A₂, a₄ = ϖ² A₄, a₆ = ϖ³ A₆. A root
t of the residue cubic T³ + A₂ T² + A₄ T + A₆ is simple, since the cubic has nonzero
discriminant, so it lifts to a root X of T³ + A₂ T² + A₄ T + A₆ in the valuation ring, and
(ϖ X, 0) is a point of W with singular reduction.
At I₀* over a Henselian valuation ring, [E(F) : E₀(F)] is one more than the number of
roots of the residue cubic. Here ϖ generates the maximal ideal of the valuation ring, Tate's
algorithm stops at I₀* on the integral model, and a₂ = ϖ A₂, a₄ = ϖ² A₄, a₆ = ϖ³ A₆, so
that the residue cubic is T³ + A₂ T² + A₄ T + A₆ over the residue field. Since the residue cubic
has nonzero discriminant, the index is 1, 2 or 4. When F is complete for a discrete
valuation with perfect residue field and the equation is minimal, this is the local index c_v that
Tate's algorithm reports at I₀*.