Points reducing to the singular point #

Let v be a valuation on a field F, with valuation ring O, maximal ideal 𝔪 and residue field k, and let W be a Weierstrass curve over F with an integral model W_O over O. When the reduced curve W_k is singular, the points of W(F) with nonsingular reduction form the subgroup E₀(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. This file gives conditions on the equation under which no point of W(F) reduces to the singular point, so that E₀(F) = W(F) and the index is one.

The input is a computation at an integral solution (x₀, y₀) of W_O. Translating it to the origin by x ↦ x + x₀, y ↦ y + y₀ gives an equation with a₆ = 0, whose a₃ and -a₄ are the values at (x₀, y₀) of the two partial derivatives of the Weierstrass polynomial, and whose discriminant is that of W. For such an equation

Δ = b₂² (a₄² + a₁ a₃ a₄ − a₂ a₃²) − 8 (2 a₄ + a₁ a₃)³ − 27 a₃⁴ + 9 b₂ (2 a₄ + a₁ a₃) a₃²,

so Δ lies in the square of any ideal containing both partial values (WeierstrassCurve.Affine.Δ_mem_sq_of_equation). If (x₀, y₀) reduces to the singular point of W_k, both partial values lie in 𝔪, hence Δ ∈ 𝔪². If moreover the singular point is a nonsplit node, the tangent form a₄² + a₁ a₃ a₄ − a₂ a₃² is anisotropic modulo 𝔪, so its value is a unit times the square of a₃ or of a₄; the remaining terms are of higher order, so Δ is a unit times a square. This gives three criteria for E₀(F) = W(F):

Δ ∉ 𝔪²; for a discrete valuation with normalized additive valuation ord this is ord(Δ) ≤ 1, the reduction symbols I₀ and I₁;
the reduction is a nonsplit node and Δ is not a unit times a square; for a discrete valuation this is nonsplit multiplicative reduction of type Iₙ with n = ord(Δ) odd;
the reduction is singular at the origin and a₆ ∉ 𝔪²; on such an equation with b₂ ∈ 𝔪 this is Tate's test for the reduction symbol II.

In each case Tate's algorithm reports the local index 1 (Silverman, Advanced Topics, IV.9, Table 4.1), and the theorems here show that E₀(F) = W(F) there, for an arbitrary valuation and without completeness.

When the reduction is singular at the origin, that is a₃, a₄, a₆ ∈ 𝔪, the points outside E₀(F) are the affine points (x, y) with x, y ∈ 𝔪, since the singular point of W_k is unique.

Main results #

WeierstrassCurve.Δ_mem_maximalIdeal_sq_of_isSingular: over a local ring, a solution reducing to a singular point forces Δ ∈ 𝔪².
WeierstrassCurve.exists_associated_sq_Δ_of_isSingular: over a valuation ring, a solution reducing to a nonsplit node forces Δ to be a unit times a square.
WeierstrassCurve.mem_maximalIdeal_of_isSingular: over a local ring, if the reduction is singular at the origin, a point reducing to a singular point has coordinates in 𝔪.
WeierstrassCurve.a₆_mem_maximalIdeal_sq_of_isSingular: over a local ring, if the reduction is singular at the origin, a solution reducing to it forces a₆ ∈ 𝔪².
WeierstrassCurve.Affine.nonsingularReduction_eq_top_of_forall_not_isSingular: if no integral solution reduces to a singular point, then E₀(F) = W(F).
WeierstrassCurve.Affine.nonsingularReduction_eq_top_of_forall_mem_maximalIdeal_not_equation: if the reduction is singular at the origin and no integral solution has both coordinates in 𝔪, then E₀(F) = W(F).
WeierstrassCurve.Affine.nonsingularReduction_eq_top_of_Δ_notMem, WeierstrassCurve.Affine.nonsingularReduction_eq_top_of_not_splits and WeierstrassCurve.Affine.nonsingularReduction_eq_top_of_a₆_notMem: the three criteria above.
WeierstrassCurve.Affine.some_notMem_nonsingularReduction_iff: when the reduction is singular at the origin, an affine point has singular reduction exactly when both of its coordinates lie in the maximal ideal, and WeierstrassCurve.Affine.exists_eq_some_of_notMem_nonsingularReduction writes such a point with integral coordinates.
WeierstrassCurve.Affine.some_coe_notMem_nonsingularReduction_iff: the same criterion for points with coordinates in the valuation ring.
WeierstrassCurve.Affine.exists_chord_of_add_notMem_nonsingularReduction: the identities satisfied by the chord through two points whose sum has singular reduction.
WeierstrassCurve.Affine.two_nsmul_notMem_of_slope_mem: twice a point with singular reduction whose tangent slope lies in the maximal ideal has singular reduction.

References #

J. Silverman, The Arithmetic of Elliptic Curves, VII.2 and 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.

Points reducing to the singular point #

Main results #

References #

Solutions reducing to a nonsplit node #

Criteria for every point to have nonsingular reduction #

Points with singular reduction #