Points with singular reduction: the index four case Iₙ* #
Let v be a valuation on a field F, with valuation ring O, maximal ideal 𝔪 = (ϖ) and
residue field κ, 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 the index at the reduction symbols Iₙ* with n ≥ 1, which Tate's algorithm reaches
through the subprocedure of its Step 7 (WeierstrassCurve.IsTateOddStage,
WeierstrassCurve.IsTateEvenStage). The subprocedure stops at stage n when a quadratic over
κ has distinct roots:
- at
n = 2k + 1, witha₃ = ϖ^(k+2) A₃anda₆ = ϖ^(2k+4) A₆, the quadraticT² + A₃ T − A₆; - at
n = 2k + 2, witha₂ = ϖ A₂,a₄ = ϖ^(k+3) A₄anda₆ = ϖ^(2k+5) A₆, the quadraticA₂ T² + A₄ T + A₆.
Table 4.1 reports c_v = 4 when this quadratic has a root in κ, and c_v = 2 otherwise. This
file proves that over a Henselian valuation ring, and that the index divides 4 in general.
Every stage of the subprocedure satisfies the normal form ϖ ∣ a₁, ϖ ∥ a₂, ϖ² ∣ a₃,
ϖ³ ∣ a₄, ϖ⁴ ∣ a₆ of stage 1, where the residue cubic is T² (T + A₂) with A₂ = a₂ / ϖ a
unit. A point with singular reduction is (x, y) with x, y ∈ 𝔪. If ϖ² ∤ x, the Weierstrass
equation divided by ϖ² and ϖ³ shows that ϖ² ∣ y and x = ϖ X with X ≡ −A₂: the point
reduces to the simple root of the residue cubic. If ϖ² ∣ x, then ϖ² ∣ y as well. Let S be the
set of points (ϖ² X, ϖ² Y) with X and Y integral. Along the chord through two points the
Weierstrass cubic factors as −(X − x₁)(X − x₂)(X − x₃), so that x₁ + x₂ + x₃ = ℓ² + a₁ ℓ − a₂
and x₁ x₂ x₃ = ν² + a₃ ν − a₆ for the line y = ℓ x + ν. Comparing powers of ϖ in these two
identities shows:
- two points with singular reduction outside
Sadd to a point with nonsingular reduction; - a point with singular reduction outside
Splus a point ofShas singular reduction; - two points of
Sdo not add to a point ofS.
Hence the points outside S form a subgroup H of W(F) containing E₀(F), and both
[H : E₀(F)] and [W(F) : H] divide 2. Over a Henselian valuation ring the simple root of the
residue cubic lifts to a point of H outside E₀(F), so [H : E₀(F)] = 2. Following the
subprocedure on the coordinates of a point of S shows that it is (ϖ^(k+2) X, ϖ^(k+2) Y) with
Y (at n = 2k + 1) or X (at n = 2k + 2) a root of the quadratic above modulo ϖ. So S is
empty when the quadratic has no root in κ. Conversely, over a Henselian valuation ring a root,
which is simple, lifts to a point of S, and then [W(F) : H] = 2.
No discreteness, completeness, minimality or perfectness is assumed. Only the existence of points with singular reduction uses the Henselian hypothesis.
Main results #
All in the namespace WeierstrassCurve.Affine:
index_nonsingularReduction_dvd_four_of_isTateTerminal_IStar: atIₙ*withn ≥ 1, the index[E(F) : E₀(F)]divides4.index_nonsingularReduction_dvd_two_of_isTateTerminal_IStar_two_mul_add_oneandindex_nonsingularReduction_dvd_two_of_isTateTerminal_IStar_two_mul_add_two: if the quadratic of the last stage has no root inκ, the index divides2.index_nonsingularReduction_eq_two_of_isTateTerminal_IStar_two_mul_add_oneandindex_nonsingularReduction_eq_two_of_isTateTerminal_IStar_two_mul_add_two: over a Henselian valuation ring, if the quadratic has no root inκ, the index is2.index_nonsingularReduction_eq_four_of_isTateTerminal_IStar_two_mul_add_oneandindex_nonsingularReduction_eq_four_of_isTateTerminal_IStar_two_mul_add_two: over a Henselian valuation ring, if the quadratic has a root inκ, the index is4.
References #
- 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.
The normal form of the stopping test #
The points with both coordinates divisible by ϖ² #
Points with singular reduction #
The subgroup of points not of the form (ϖ² X, ϖ² Y) #
The index at the symbol Iₙ* #
The stage-one normal form of the Iₙ* subprocedure already forces
[E(F) : E₀(F)] to divide 4, without a terminal stopping condition.
At the symbol Iₙ*, [E(F) : E₀(F)] divides 4. Here ϖ generates the maximal ideal and
Tate's algorithm stops at Iₙ* with n ≥ 1 on the integral model.
At the symbol I*_{2k+1} with no rational root of the residue quadratic,
[E(F) : E₀(F)] divides 2. Here ϖ generates the maximal ideal, Tate's algorithm stops at
I*_{2k+1} on the integral model, and with a₃ = ϖ^(k+2) A₃ and a₆ = ϖ^(2k+4) A₆ the
quadratic T² + A₃ T − A₆ of the last stage of the subprocedure has no root in the residue
field.
At the symbol I*_{2k+1} with no rational root of the residue quadratic,
[E(F) : E₀(F)] = 2, when the valuation ring is Henselian, for instance complete. This is the
local index c_v = 2 that Tate's algorithm reports at I*_{2k+1} in that case. Here ϖ
generates the maximal ideal, Tate's algorithm stops at I*_{2k+1} on the integral model, and with
a₃ = ϖ^(k+2) A₃ and a₆ = ϖ^(2k+4) A₆ the quadratic T² + A₃ T − A₆ has no root in the residue
field.
At the symbol I*_{2k+1} with a rational root of the residue quadratic,
[E(F) : E₀(F)] = 4, when the valuation ring is Henselian, for instance complete. This is the
local index c_v = 4 that Tate's algorithm reports at I*_{2k+1} in that case. Here ϖ
generates the maximal ideal, Tate's algorithm stops at I*_{2k+1} on the integral model, and with
a₃ = ϖ^(k+2) A₃ and a₆ = ϖ^(2k+4) A₆ the quadratic T² + A₃ T − A₆ has a root t in the
residue field.
At the symbol I*_{2k+2} with no rational root of the residue quadratic,
[E(F) : E₀(F)] divides 2. Here ϖ generates the maximal ideal, Tate's algorithm stops at
I*_{2k+2} on the integral model, and with a₂ = ϖ A₂, a₄ = ϖ^(k+3) A₄ and
a₆ = ϖ^(2k+5) A₆ the quadratic A₂ T² + A₄ T + A₆ of the last stage of the subprocedure has no
root in the residue field.
At the symbol I*_{2k+2} with no rational root of the residue quadratic,
[E(F) : E₀(F)] = 2, when the valuation ring is Henselian, for instance complete. This is the
local index c_v = 2 that Tate's algorithm reports at I*_{2k+2} in that case. Here ϖ
generates the maximal ideal, Tate's algorithm stops at I*_{2k+2} on the integral model, and with
a₂ = ϖ A₂, a₄ = ϖ^(k+3) A₄ and a₆ = ϖ^(2k+5) A₆ the quadratic A₂ T² + A₄ T + A₆ has no
root in the residue field.
At the symbol I*_{2k+2} with a rational root of the residue quadratic,
[E(F) : E₀(F)] = 4, when the valuation ring is Henselian, for instance complete. This is the
local index c_v = 4 that Tate's algorithm reports at I*_{2k+2} in that case. Here ϖ
generates the maximal ideal, Tate's algorithm stops at I*_{2k+2} on the integral model, and with
a₂ = ϖ A₂, a₄ = ϖ^(k+3) A₄ and a₆ = ϖ^(2k+5) A₆ the quadratic A₂ T² + A₄ T + A₆ has a root
t in the residue field.