statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
C_simp : tactic unit | `[simp only [C_0, C_1, C_neg, C_add, C_sub, C_mul, C_pow]] | def | C_simp | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
weierstrass_curve (R : Type u) | (a₁ a₂ a₃ a₄ a₆ : R) | structure | weierstrass_curve | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [] | A Weierstrass curve $Y^2 + a_1XY + a_3Y = X^3 + a_2X^2 + a_4X + a_6$ with parameters $a_i$. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
b₂ : R | W.a₁ ^ 2 + 4 * W.a₂ | def | weierstrass_curve.b₂ | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [] | The `b₂` coefficient of a Weierstrass curve. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
b₄ : R | 2 * W.a₄ + W.a₁ * W.a₃ | def | weierstrass_curve.b₄ | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [] | The `b₄` coefficient of a Weierstrass curve. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
b₆ : R | W.a₃ ^ 2 + 4 * W.a₆ | def | weierstrass_curve.b₆ | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [] | The `b₆` coefficient of a Weierstrass curve. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
b₈ : R | W.a₁ ^ 2 * W.a₆ + 4 * W.a₂ * W.a₆ - W.a₁ * W.a₃ * W.a₄ + W.a₂ * W.a₃ ^ 2 - W.a₄ ^ 2 | def | weierstrass_curve.b₈ | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [] | The `b₈` coefficient of a Weierstrass curve. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
b_relation : 4 * W.b₈ = W.b₂ * W.b₆ - W.b₄ ^ 2 | by { simp only [b₂, b₄, b₆, b₈], ring1 } | lemma | weierstrass_curve.b_relation | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
c₄ : R | W.b₂ ^ 2 - 24 * W.b₄ | def | weierstrass_curve.c₄ | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [] | The `c₄` coefficient of a Weierstrass curve. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
c₆ : R | -W.b₂ ^ 3 + 36 * W.b₂ * W.b₄ - 216 * W.b₆ | def | weierstrass_curve.c₆ | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [] | The `c₆` coefficient of a Weierstrass curve. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Δ : R | -W.b₂ ^ 2 * W.b₈ - 8 * W.b₄ ^ 3 - 27 * W.b₆ ^ 2 + 9 * W.b₂ * W.b₄ * W.b₆ | def | weierstrass_curve.Δ | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [] | The discriminant `Δ` of a Weierstrass curve. If `R` is a field, then this polynomial vanishes
if and only if the cubic curve cut out by this equation is singular. Sometimes only defined up to
sign in the literature; we choose the sign used by the LMFDB. For more discussion, see
[the LMFDB page on discriminants](https:/... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
c_relation : 1728 * W.Δ = W.c₄ ^ 3 - W.c₆ ^ 2 | by { simp only [b₂, b₄, b₆, b₈, c₄, c₆, Δ], ring1 } | lemma | weierstrass_curve.c_relation | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
variable_change : weierstrass_curve R | { a₁ := ↑u⁻¹ * (W.a₁ + 2 * s),
a₂ := ↑u⁻¹ ^ 2 * (W.a₂ - s * W.a₁ + 3 * r - s ^ 2),
a₃ := ↑u⁻¹ ^ 3 * (W.a₃ + r * W.a₁ + 2 * t),
a₄ := ↑u⁻¹ ^ 4 * (W.a₄ - s * W.a₃ + 2 * r * W.a₂ - (t + r * s) * W.a₁ + 3 * r ^ 2 - 2 * s * t),
a₆ := ↑u⁻¹ ^ 6 * (W.a₆ + r * W.a₄ + r ^ 2 * W.a₂ + r ^ 3 - t * W.a₃ - t ^ 2 - r * t * W.a... | def | weierstrass_curve.variable_change | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"weierstrass_curve"
] | The Weierstrass curve over `R` induced by an admissible linear change of variables
$(X, Y) \mapsto (u^2X + r, u^3Y + u^2sX + t)$ for some $u \in R^\times$ and some $r, s, t \in R$. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
variable_change_b₂ : (W.variable_change u r s t).b₂ = ↑u⁻¹ ^ 2 * (W.b₂ + 12 * r) | by { simp only [b₂, variable_change_a₁, variable_change_a₂], ring1 } | lemma | weierstrass_curve.variable_change_b₂ | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
variable_change_b₄ :
(W.variable_change u r s t).b₄ = ↑u⁻¹ ^ 4 * (W.b₄ + r * W.b₂ + 6 * r ^ 2) | by { simp only [b₂, b₄, variable_change_a₁, variable_change_a₃, variable_change_a₄], ring1 } | lemma | weierstrass_curve.variable_change_b₄ | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
variable_change_b₆ :
(W.variable_change u r s t).b₆ = ↑u⁻¹ ^ 6 * (W.b₆ + 2 * r * W.b₄ + r ^ 2 * W.b₂ + 4 * r ^ 3) | by { simp only [b₂, b₄, b₆, variable_change_a₃, variable_change_a₆], ring1 } | lemma | weierstrass_curve.variable_change_b₆ | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
variable_change_b₈ :
(W.variable_change u r s t).b₈
= ↑u⁻¹ ^ 8 * (W.b₈ + 3 * r * W.b₆ + 3 * r ^ 2 * W.b₄ + r ^ 3 * W.b₂ + 3 * r ^ 4) | by { simp only [b₂, b₄, b₆, b₈, variable_change_a₁, variable_change_a₂, variable_change_a₃,
variable_change_a₄, variable_change_a₆], ring1 } | lemma | weierstrass_curve.variable_change_b₈ | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
variable_change_c₄ : (W.variable_change u r s t).c₄ = ↑u⁻¹ ^ 4 * W.c₄ | by { simp only [c₄, variable_change_b₂, variable_change_b₄], ring1 } | lemma | weierstrass_curve.variable_change_c₄ | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
variable_change_c₆ : (W.variable_change u r s t).c₆ = ↑u⁻¹ ^ 6 * W.c₆ | by { simp only [c₆, variable_change_b₂, variable_change_b₄, variable_change_b₆], ring1 } | lemma | weierstrass_curve.variable_change_c₆ | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
variable_change_Δ : (W.variable_change u r s t).Δ = ↑u⁻¹ ^ 12 * W.Δ | by { dsimp, ring1 } | lemma | weierstrass_curve.variable_change_Δ | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
base_change : weierstrass_curve A | ⟨algebra_map R A W.a₁, algebra_map R A W.a₂, algebra_map R A W.a₃, algebra_map R A W.a₄,
algebra_map R A W.a₆⟩ | def | weierstrass_curve.base_change | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"algebra_map",
"weierstrass_curve"
] | The Weierstrass curve over `R` base changed to `A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
base_change_b₂ : (W.base_change A).b₂ = algebra_map R A W.b₂ | by { simp only [b₂, base_change_a₁, base_change_a₂], map_simp } | lemma | weierstrass_curve.base_change_b₂ | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"algebra_map",
"map_simp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
base_change_b₄ : (W.base_change A).b₄ = algebra_map R A W.b₄ | by { simp only [b₄, base_change_a₁, base_change_a₃, base_change_a₄], map_simp } | lemma | weierstrass_curve.base_change_b₄ | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"algebra_map",
"map_simp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
base_change_b₆ : (W.base_change A).b₆ = algebra_map R A W.b₆ | by { simp only [b₆, base_change_a₃, base_change_a₆], map_simp } | lemma | weierstrass_curve.base_change_b₆ | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"algebra_map",
"map_simp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
base_change_b₈ : (W.base_change A).b₈ = algebra_map R A W.b₈ | by { simp only [b₈, base_change_a₁, base_change_a₂, base_change_a₃, base_change_a₄, base_change_a₆],
map_simp } | lemma | weierstrass_curve.base_change_b₈ | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"algebra_map",
"map_simp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
base_change_c₄ : (W.base_change A).c₄ = algebra_map R A W.c₄ | by { simp only [c₄, base_change_b₂, base_change_b₄], map_simp } | lemma | weierstrass_curve.base_change_c₄ | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"algebra_map",
"map_simp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
base_change_c₆ : (W.base_change A).c₆ = algebra_map R A W.c₆ | by { simp only [c₆, base_change_b₂, base_change_b₄, base_change_b₆], map_simp } | lemma | weierstrass_curve.base_change_c₆ | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"algebra_map",
"map_simp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
base_change_Δ : (W.base_change A).Δ = algebra_map R A W.Δ | by { simp only [Δ, base_change_b₂, base_change_b₄, base_change_b₆, base_change_b₈], map_simp } | lemma | weierstrass_curve.base_change_Δ | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"algebra_map",
"map_simp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
base_change_self : W.base_change R = W | by ext; refl | lemma | weierstrass_curve.base_change_self | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
base_change_base_change : (W.base_change A).base_change B = W.base_change B | by ext; exact (is_scalar_tower.algebra_map_apply R A B _).symm | lemma | weierstrass_curve.base_change_base_change | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"is_scalar_tower.algebra_map_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
two_torsion_polynomial : cubic R | ⟨4, W.b₂, 2 * W.b₄, W.b₆⟩ | def | weierstrass_curve.two_torsion_polynomial | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"cubic"
] | A cubic polynomial whose discriminant is a multiple of the Weierstrass curve discriminant. If
`W` is an elliptic curve over a field `R` of characteristic different from 2, then its roots over a
splitting field of `R` are precisely the $X$-coordinates of the non-zero 2-torsion points of `W`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
two_torsion_polynomial_disc : W.two_torsion_polynomial.disc = 16 * W.Δ | by { dsimp [two_torsion_polynomial, cubic.disc], ring1 } | lemma | weierstrass_curve.two_torsion_polynomial_disc | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"cubic.disc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
two_torsion_polynomial_disc_is_unit [invertible (2 : R)] :
is_unit W.two_torsion_polynomial.disc ↔ is_unit W.Δ | begin
rw [two_torsion_polynomial_disc, is_unit.mul_iff, show (16 : R) = 2 ^ 4, by norm_num1],
exact and_iff_right (is_unit_of_invertible $ 2 ^ 4)
end | lemma | weierstrass_curve.two_torsion_polynomial_disc_is_unit | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"invertible",
"is_unit",
"is_unit.mul_iff",
"is_unit_of_invertible"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
two_torsion_polynomial_disc_ne_zero [nontrivial R] [invertible (2 : R)] (hΔ : is_unit W.Δ) :
W.two_torsion_polynomial.disc ≠ 0 | (W.two_torsion_polynomial_disc_is_unit.mpr hΔ).ne_zero | lemma | weierstrass_curve.two_torsion_polynomial_disc_ne_zero | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"invertible",
"is_unit",
"ne_zero",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
polynomial : R[X][Y] | Y ^ 2 + C (C W.a₁ * X + C W.a₃) * Y - C (X ^ 3 + C W.a₂ * X ^ 2 + C W.a₄ * X + C W.a₆) | def | weierstrass_curve.polynomial | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"polynomial"
] | The polynomial $W(X, Y) := Y^2 + a_1XY + a_3Y - (X^3 + a_2X^2 + a_4X + a_6)$ associated to a
Weierstrass curve `W` over `R`. For ease of polynomial manipulation, this is represented as a term
of type `R[X][X]`, where the inner variable represents $X$ and the outer variable represents $Y$.
For clarity, the alternative n... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
polynomial_eq : W.polynomial = cubic.to_poly
⟨0, 1, cubic.to_poly ⟨0, 0, W.a₁, W.a₃⟩, cubic.to_poly ⟨-1, -W.a₂, -W.a₄, -W.a₆⟩⟩ | by { simp only [weierstrass_curve.polynomial, cubic.to_poly], C_simp, ring1 } | lemma | weierstrass_curve.polynomial_eq | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"C_simp",
"cubic.to_poly",
"weierstrass_curve.polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
polynomial_ne_zero [nontrivial R] : W.polynomial ≠ 0 | by { rw [polynomial_eq], exact cubic.ne_zero_of_b_ne_zero one_ne_zero } | lemma | weierstrass_curve.polynomial_ne_zero | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"cubic.ne_zero_of_b_ne_zero",
"nontrivial",
"one_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
degree_polynomial [nontrivial R] : W.polynomial.degree = 2 | by { rw [polynomial_eq], exact cubic.degree_of_b_ne_zero' one_ne_zero } | lemma | weierstrass_curve.degree_polynomial | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"cubic.degree_of_b_ne_zero'",
"nontrivial",
"one_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_degree_polynomial [nontrivial R] : W.polynomial.nat_degree = 2 | by { rw [polynomial_eq], exact cubic.nat_degree_of_b_ne_zero' one_ne_zero } | lemma | weierstrass_curve.nat_degree_polynomial | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"cubic.nat_degree_of_b_ne_zero'",
"nontrivial",
"one_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monic_polynomial : W.polynomial.monic | by { nontriviality R, simpa only [polynomial_eq] using cubic.monic_of_b_eq_one' } | lemma | weierstrass_curve.monic_polynomial | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"cubic.monic_of_b_eq_one'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
irreducible_polynomial [is_domain R] : irreducible W.polynomial | begin
by_contra h,
rcases (W.monic_polynomial.not_irreducible_iff_exists_add_mul_eq_coeff W.nat_degree_polynomial).mp
h with ⟨f, g, h0, h1⟩,
simp only [polynomial_eq, cubic.coeff_eq_c, cubic.coeff_eq_d] at h0 h1,
apply_fun degree at h0 h1,
rw [cubic.degree_of_a_ne_zero' $ neg_ne_zero.mpr $ one_ne_ze... | lemma | weierstrass_curve.irreducible_polynomial | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"by_contra",
"cubic.coeff_eq_c",
"cubic.coeff_eq_d",
"cubic.degree_of_a_ne_zero'",
"cubic.degree_of_b_eq_zero'",
"irreducible",
"is_domain",
"one_ne_zero'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eval_polynomial (x y : R) :
(W.polynomial.eval $ C y).eval x
= y ^ 2 + W.a₁ * x * y + W.a₃ * y - (x ^ 3 + W.a₂ * x ^ 2 + W.a₄ * x + W.a₆) | by { simp only [weierstrass_curve.polynomial], eval_simp, rw [add_mul, ← add_assoc] } | lemma | weierstrass_curve.eval_polynomial | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"eval_simp",
"weierstrass_curve.polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eval_polynomial_zero : (W.polynomial.eval 0).eval 0 = -W.a₆ | by simp only [← C_0, eval_polynomial, zero_add, zero_sub, mul_zero, zero_pow (nat.zero_lt_succ _)] | lemma | weierstrass_curve.eval_polynomial_zero | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"mul_zero",
"zero_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equation (x y : R) : Prop | (W.polynomial.eval $ C y).eval x = 0 | def | weierstrass_curve.equation | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [] | The proposition that an affine point $(x, y)$ lies in `W`. In other words, $W(x, y) = 0$. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equation_iff' (x y : R) :
W.equation x y ↔ y ^ 2 + W.a₁ * x * y + W.a₃ * y - (x ^ 3 + W.a₂ * x ^ 2 + W.a₄ * x + W.a₆) = 0 | by rw [equation, eval_polynomial] | lemma | weierstrass_curve.equation_iff' | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equation_iff (x y : R) :
W.equation x y ↔ y ^ 2 + W.a₁ * x * y + W.a₃ * y = x ^ 3 + W.a₂ * x ^ 2 + W.a₄ * x + W.a₆ | by rw [equation_iff', sub_eq_zero] | lemma | weierstrass_curve.equation_iff | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equation_zero : W.equation 0 0 ↔ W.a₆ = 0 | by rw [equation, C_0, eval_polynomial_zero, neg_eq_zero] | lemma | weierstrass_curve.equation_zero | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equation_iff_variable_change (x y : R) :
W.equation x y ↔ (W.variable_change 1 x 0 y).equation 0 0 | begin
rw [equation_iff', ← neg_eq_zero, equation_zero, variable_change_a₆, inv_one, units.coe_one],
congr' 2,
ring1
end | lemma | weierstrass_curve.equation_iff_variable_change | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"inv_one",
"units.coe_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equation_iff_base_change [nontrivial A] [no_zero_smul_divisors R A] (x y : R) :
W.equation x y ↔ (W.base_change A).equation (algebra_map R A x) (algebra_map R A y) | begin
simp only [equation_iff],
refine ⟨λ h, _, λ h, _⟩,
{ convert congr_arg (algebra_map R A) h; { map_simp, refl } },
{ apply no_zero_smul_divisors.algebra_map_injective R A, map_simp, exact h }
end | lemma | weierstrass_curve.equation_iff_base_change | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"algebra_map",
"map_simp",
"no_zero_smul_divisors",
"no_zero_smul_divisors.algebra_map_injective",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equation_iff_base_change_of_base_change [nontrivial B] [no_zero_smul_divisors A B] (x y : A) :
(W.base_change A).equation x y
↔ (W.base_change B).equation (algebra_map A B x) (algebra_map A B y) | by rw [equation_iff_base_change (W.base_change A) B, base_change_base_change] | lemma | weierstrass_curve.equation_iff_base_change_of_base_change | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"algebra_map",
"no_zero_smul_divisors",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
polynomial_X : R[X][Y] | C (C W.a₁) * Y - C (C 3 * X ^ 2 + C (2 * W.a₂) * X + C W.a₄) | def | weierstrass_curve.polynomial_X | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [] | The partial derivative $W_X(X, Y)$ of $W(X, Y)$ with respect to $X$.
TODO: define this in terms of `polynomial.derivative`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eval_polynomial_X (x y : R) :
(W.polynomial_X.eval $ C y).eval x = W.a₁ * y - (3 * x ^ 2 + 2 * W.a₂ * x + W.a₄) | by { simp only [polynomial_X], eval_simp } | lemma | weierstrass_curve.eval_polynomial_X | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"eval_simp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eval_polynomial_X_zero : (W.polynomial_X.eval 0).eval 0 = -W.a₄ | by simp only [← C_0, eval_polynomial_X, zero_add, zero_sub, mul_zero, zero_pow zero_lt_two] | lemma | weierstrass_curve.eval_polynomial_X_zero | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"mul_zero",
"zero_lt_two",
"zero_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
polynomial_Y : R[X][Y] | C (C 2) * Y + C (C W.a₁ * X + C W.a₃) | def | weierstrass_curve.polynomial_Y | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [] | The partial derivative $W_Y(X, Y)$ of $W(X, Y)$ with respect to $Y$.
TODO: define this in terms of `polynomial.derivative`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eval_polynomial_Y (x y : R) :
(W.polynomial_Y.eval $ C y).eval x = 2 * y + W.a₁ * x + W.a₃ | by { simp only [polynomial_Y], eval_simp, rw [← add_assoc] } | lemma | weierstrass_curve.eval_polynomial_Y | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"eval_simp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eval_polynomial_Y_zero : (W.polynomial_Y.eval 0).eval 0 = W.a₃ | by simp only [← C_0, eval_polynomial_Y, zero_add, mul_zero] | lemma | weierstrass_curve.eval_polynomial_Y_zero | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"mul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonsingular (x y : R) : Prop | W.equation x y ∧ ((W.polynomial_X.eval $ C y).eval x ≠ 0 ∨ (W.polynomial_Y.eval $ C y).eval x ≠ 0) | def | weierstrass_curve.nonsingular | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [] | The proposition that an affine point $(x, y)$ on `W` is nonsingular.
In other words, either $W_X(x, y) \ne 0$ or $W_Y(x, y) \ne 0$. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nonsingular_iff' (x y : R) :
W.nonsingular x y
↔ W.equation x y
∧ (W.a₁ * y - (3 * x ^ 2 + 2 * W.a₂ * x + W.a₄) ≠ 0 ∨ 2 * y + W.a₁ * x + W.a₃ ≠ 0) | by rw [nonsingular, equation_iff', eval_polynomial_X, eval_polynomial_Y] | lemma | weierstrass_curve.nonsingular_iff' | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonsingular_iff (x y : R) :
W.nonsingular x y
↔ W.equation x y ∧ (W.a₁ * y ≠ 3 * x ^ 2 + 2 * W.a₂ * x + W.a₄ ∨ y ≠ -y - W.a₁ * x - W.a₃) | by { rw [nonsingular_iff', sub_ne_zero, ← @sub_ne_zero _ _ y], congr' 4; ring1 } | lemma | weierstrass_curve.nonsingular_iff | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonsingular_zero : W.nonsingular 0 0 ↔ W.a₆ = 0 ∧ (W.a₃ ≠ 0 ∨ W.a₄ ≠ 0) | by rw [nonsingular, equation_zero, C_0, eval_polynomial_X_zero, neg_ne_zero, eval_polynomial_Y_zero,
or_comm] | lemma | weierstrass_curve.nonsingular_zero | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonsingular_iff_variable_change (x y : R) :
W.nonsingular x y ↔ (W.variable_change 1 x 0 y).nonsingular 0 0 | begin
rw [nonsingular_iff', equation_iff_variable_change, equation_zero, ← neg_ne_zero, or_comm,
nonsingular_zero, variable_change_a₃, variable_change_a₄, inv_one, units.coe_one],
congr' 4; ring1
end | lemma | weierstrass_curve.nonsingular_iff_variable_change | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"inv_one",
"units.coe_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonsingular_iff_base_change [nontrivial A] [no_zero_smul_divisors R A] (x y : R) :
W.nonsingular x y ↔ (W.base_change A).nonsingular (algebra_map R A x) (algebra_map R A y) | begin
rw [nonsingular_iff, nonsingular_iff, and_congr $ W.equation_iff_base_change A x y],
refine ⟨or.imp (not_imp_not.mpr $ λ h, _) (not_imp_not.mpr $ λ h, _),
or.imp (not_imp_not.mpr $ λ h, _) (not_imp_not.mpr $ λ h, _)⟩,
any_goals { apply no_zero_smul_divisors.algebra_map_injective R A, map_simp, exact h }... | lemma | weierstrass_curve.nonsingular_iff_base_change | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"algebra_map",
"map_simp",
"no_zero_smul_divisors",
"no_zero_smul_divisors.algebra_map_injective",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonsingular_iff_base_change_of_base_change [nontrivial B] [no_zero_smul_divisors A B]
(x y : A) : (W.base_change A).nonsingular x y
↔ (W.base_change B).nonsingular (algebra_map A B x) (algebra_map A B y) | by rw [nonsingular_iff_base_change (W.base_change A) B, base_change_base_change] | lemma | weierstrass_curve.nonsingular_iff_base_change_of_base_change | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"algebra_map",
"no_zero_smul_divisors",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonsingular_zero_of_Δ_ne_zero (h : W.equation 0 0) (hΔ : W.Δ ≠ 0) : W.nonsingular 0 0 | by { simp only [equation_zero, nonsingular_zero] at *, contrapose! hΔ, simp [h, hΔ] } | lemma | weierstrass_curve.nonsingular_zero_of_Δ_ne_zero | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonsingular_of_Δ_ne_zero {x y : R} (h : W.equation x y) (hΔ : W.Δ ≠ 0) : W.nonsingular x y | (W.nonsingular_iff_variable_change x y).mpr $
nonsingular_zero_of_Δ_ne_zero _ ((W.equation_iff_variable_change x y).mp h) $
by rwa [variable_change_Δ, inv_one, units.coe_one, one_pow, one_mul] | lemma | weierstrass_curve.nonsingular_of_Δ_ne_zero | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"inv_one",
"one_mul",
"one_pow",
"units.coe_one"
] | A Weierstrass curve is nonsingular at every point if its discriminant is non-zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coordinate_ring : Type u | adjoin_root W.polynomial | def | weierstrass_curve.coordinate_ring | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"adjoin_root"
] | The coordinate ring $R[W] := R[X, Y] / \langle W(X, Y) \rangle$ of `W`.
Note that `derive comm_ring` generates a reducible instance of `comm_ring` for `coordinate_ring`.
In certain circumstances this might be extremely slow, because all instances in its definition are
unified exponentially many times. In this case, on... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
function_field : Type u | fraction_ring W.coordinate_ring | abbreviation | weierstrass_curve.function_field | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"fraction_ring",
"function_field"
] | The function field $R(W) := \mathrm{Frac}(R[W])$ of `W`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_domain_of_field {F : Type u} [field F] (W : weierstrass_curve F) :
is_domain W.coordinate_ring | by { classical, apply_instance } | instance | weierstrass_curve.coordinate_ring.is_domain_of_field | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"field",
"is_domain",
"weierstrass_curve"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
X_class : W.coordinate_ring | adjoin_root.mk W.polynomial $ C $ X - C x | def | weierstrass_curve.coordinate_ring.X_class | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"adjoin_root.mk"
] | The class of the element $X - x$ in $R[W]$ for some $x \in R$. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
X_class_ne_zero [nontrivial R] : X_class W x ≠ 0 | adjoin_root.mk_ne_zero_of_nat_degree_lt W.monic_polynomial (C_ne_zero.mpr $ X_sub_C_ne_zero x) $
by { rw [nat_degree_polynomial, nat_degree_C], norm_num1 } | lemma | weierstrass_curve.coordinate_ring.X_class_ne_zero | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"adjoin_root.mk_ne_zero_of_nat_degree_lt",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Y_class : W.coordinate_ring | adjoin_root.mk W.polynomial $ Y - C y | def | weierstrass_curve.coordinate_ring.Y_class | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"adjoin_root.mk"
] | The class of the element $Y - y(X)$ in $R[W]$ for some $y(X) \in R[X]$. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Y_class_ne_zero [nontrivial R] : Y_class W y ≠ 0 | adjoin_root.mk_ne_zero_of_nat_degree_lt W.monic_polynomial (X_sub_C_ne_zero y) $
by { rw [nat_degree_polynomial, nat_degree_X_sub_C], norm_num1 } | lemma | weierstrass_curve.coordinate_ring.Y_class_ne_zero | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"adjoin_root.mk_ne_zero_of_nat_degree_lt",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
X_ideal : ideal W.coordinate_ring | span {X_class W x} | def | weierstrass_curve.coordinate_ring.X_ideal | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"ideal"
] | The ideal $\langle X - x \rangle$ of $R[W]$ for some $x \in R$. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Y_ideal : ideal W.coordinate_ring | span {Y_class W y} | def | weierstrass_curve.coordinate_ring.Y_ideal | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"ideal"
] | The ideal $\langle Y - y(X) \rangle$ of $R[W]$ for some $y(X) \in R[X]$. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
XY_ideal (x : R) (y : R[X]) : ideal W.coordinate_ring | span {X_class W x, Y_class W y} | def | weierstrass_curve.coordinate_ring.XY_ideal | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"ideal"
] | The ideal $\langle X - x, Y - y(X) \rangle$ of $R[W]$ for some $x \in R$ and $y(X) \in R[X]$. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
algebra' : algebra R W.coordinate_ring | quotient.algebra R | instance | weierstrass_curve.coordinate_ring.algebra' | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"algebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_XY_ideal_equiv {x : R} {y : R[X]}
(h : (W.polynomial.eval y).eval x = 0) : (W.coordinate_ring ⧸ XY_ideal W x y) ≃ₐ[R] R | (quotient_equiv_alg_of_eq R $
by simpa only [XY_ideal, X_class, Y_class, ← set.image_pair, ← map_span]).trans $
(double_quot.quot_quot_equiv_quot_of_leₐ R $ (span_singleton_le_iff_mem _).mpr $
mem_span_C_X_sub_C_X_sub_C_iff_eval_eval_eq_zero.mpr h).trans $
((quotient_span_C_X_sub_C_alg_equiv (X - C x) y).re... | def | weierstrass_curve.coordinate_ring.quotient_XY_ideal_equiv | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"double_quot.quot_quot_equiv_quot_of_leₐ",
"restrict_scalars",
"set.image_pair"
] | The $R$-algebra isomorphism from $R[W] / \langle X - x, Y - y(X) \rangle$ to $R$ obtained by
evaluation at $y(X)$ and at $x$ provided that $W(x, y(x)) = 0$. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
basis : basis (fin 2) R[X] W.coordinate_ring | (subsingleton_or_nontrivial R).by_cases (λ _, by exactI default) $ λ _, by exactI
((adjoin_root.power_basis' W.monic_polynomial).basis.reindex $
fin_congr W.nat_degree_polynomial) | def | weierstrass_curve.coordinate_ring.basis | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"adjoin_root.power_basis'",
"basis",
"basis.reindex",
"fin_congr",
"subsingleton_or_nontrivial"
] | The basis $\{1, Y\}$ for the coordinate ring $R[W]$ over the polynomial ring $R[X]$.
Given a Weierstrass curve `W`, write `W^.coordinate_ring.basis` for this basis. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
basis_apply (n : fin 2) :
W^.coordinate_ring.basis n = (adjoin_root.power_basis' W.monic_polynomial).gen ^ (n : ℕ) | begin
classical,
nontriviality R,
simpa only [coordinate_ring.basis, or.by_cases, dif_neg (not_subsingleton R),
basis.reindex_apply, power_basis.basis_eq_pow]
end | lemma | weierstrass_curve.coordinate_ring.basis_apply | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"adjoin_root.power_basis'",
"basis.reindex_apply",
"not_subsingleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basis_zero : W^.coordinate_ring.basis 0 = 1 | by simpa only [basis_apply] using pow_zero _ | lemma | weierstrass_curve.coordinate_ring.basis_zero | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basis_one : W^.coordinate_ring.basis 1 = adjoin_root.mk W.polynomial Y | by simpa only [basis_apply] using pow_one _ | lemma | weierstrass_curve.coordinate_ring.basis_one | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"adjoin_root.mk",
"pow_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_basis :
(W^.coordinate_ring.basis : fin 2 → W.coordinate_ring) = ![1, adjoin_root.mk W.polynomial Y] | by { ext n, fin_cases n, exacts [basis_zero W, basis_one W] } | lemma | weierstrass_curve.coordinate_ring.coe_basis | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"adjoin_root.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul (x : R[X]) (y : W.coordinate_ring) : x • y = adjoin_root.mk W.polynomial (C x) * y | (algebra_map_smul W.coordinate_ring x y).symm | lemma | weierstrass_curve.coordinate_ring.smul | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"adjoin_root.mk",
"algebra_map_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_basis_eq_zero {p q : R[X]}
(hpq : p • 1 + q • adjoin_root.mk W.polynomial Y = 0) : p = 0 ∧ q = 0 | begin
have h := fintype.linear_independent_iff.mp (coordinate_ring.basis W).linear_independent ![p, q],
erw [fin.sum_univ_succ, basis_zero, fin.sum_univ_one, basis_one] at h,
exact ⟨h hpq 0, h hpq 1⟩
end | lemma | weierstrass_curve.coordinate_ring.smul_basis_eq_zero | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"adjoin_root.mk",
"linear_independent"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_smul_basis_eq (x : W.coordinate_ring) :
∃ p q : R[X], p • 1 + q • adjoin_root.mk W.polynomial Y = x | begin
have h := (coordinate_ring.basis W).sum_equiv_fun x,
erw [fin.sum_univ_succ, fin.sum_univ_one, basis_zero, basis_one] at h,
exact ⟨_, _, h⟩
end | lemma | weierstrass_curve.coordinate_ring.exists_smul_basis_eq | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"adjoin_root.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_basis_mul_C (p q : R[X]) :
(p • 1 + q • adjoin_root.mk W.polynomial Y) * adjoin_root.mk W.polynomial (C y)
= ((p * y) • 1 + (q * y) • adjoin_root.mk W.polynomial Y) | by { simp only [smul, _root_.map_mul], ring1 } | lemma | weierstrass_curve.coordinate_ring.smul_basis_mul_C | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"adjoin_root.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_basis_mul_Y (p q : R[X]) :
(p • 1 + q • adjoin_root.mk W.polynomial Y) * adjoin_root.mk W.polynomial Y
= (q * (X ^ 3 + C W.a₂ * X ^ 2 + C W.a₄ * X + C W.a₆)) • 1
+ (p - q * (C W.a₁ * X + C W.a₃)) • adjoin_root.mk W.polynomial Y | begin
have Y_sq : adjoin_root.mk W.polynomial Y ^ 2 = adjoin_root.mk W.polynomial
(C (X ^ 3 + C W.a₂ * X ^ 2 + C W.a₄ * X + C W.a₆) - C (C W.a₁ * X + C W.a₃) * Y) :=
adjoin_root.mk_eq_mk.mpr ⟨1, by { simp only [weierstrass_curve.polynomial], ring1 }⟩,
simp only [smul, add_mul, mul_assoc, ← sq, Y_sq, map_sub, ... | lemma | weierstrass_curve.coordinate_ring.smul_basis_mul_Y | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"adjoin_root.mk",
"mul_assoc",
"weierstrass_curve.polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_smul_basis (p q : R[X]) :
algebra.norm R[X] (p • 1 + q • adjoin_root.mk W.polynomial Y)
= p ^ 2 - p * q * (C W.a₁ * X + C W.a₃)
- q ^ 2 * (X ^ 3 + C W.a₂ * X ^ 2 + C W.a₄ * X + C W.a₆) | begin
simp_rw [algebra.norm_eq_matrix_det W^.coordinate_ring.basis, matrix.det_fin_two,
algebra.left_mul_matrix_eq_repr_mul, basis_zero, mul_one, basis_one, smul_basis_mul_Y,
map_add, finsupp.add_apply, map_smul, finsupp.smul_apply, ← basis_zero, ← basis_one,
basis.repr_self_apply, if... | lemma | weierstrass_curve.coordinate_ring.norm_smul_basis | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"adjoin_root.mk",
"algebra.left_mul_matrix_eq_repr_mul",
"algebra.norm",
"algebra.norm_eq_matrix_det",
"basis.repr_self_apply",
"finsupp.add_apply",
"finsupp.smul_apply",
"matrix.det_fin_two",
"mul_one",
"one_ne_zero",
"smul_eq_mul",
"zero_ne_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_norm_smul_basis (p q : R[X]) :
↑(algebra.norm R[X] $ p • 1 + q • adjoin_root.mk W.polynomial Y)
= adjoin_root.mk W.polynomial
((C p + C q * X) * (C p + C q * (-Y - C (C W.a₁ * X + C W.a₃)))) | adjoin_root.mk_eq_mk.mpr
⟨C q ^ 2, by { rw [norm_smul_basis, weierstrass_curve.polynomial], C_simp, ring1 }⟩ | lemma | weierstrass_curve.coordinate_ring.coe_norm_smul_basis | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"C_simp",
"adjoin_root.mk",
"algebra.norm",
"weierstrass_curve.polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
degree_norm_smul_basis [is_domain R] (p q : R[X]) :
(algebra.norm R[X] $ p • 1 + q • adjoin_root.mk W.polynomial Y).degree
= max (2 • p.degree) (2 • q.degree + 3) | begin
have hdp : (p ^ 2).degree = 2 • p.degree := degree_pow p 2,
have hdpq : (p * q * (C W.a₁ * X + C W.a₃)).degree ≤ p.degree + q.degree + 1,
{ simpa only [degree_mul] using add_le_add_left degree_linear_le (p.degree + q.degree) },
have hdq : (q ^ 2 * (X ^ 3 + C W.a₂ * X ^ 2 + C W.a₄ * X + C W.a₆)).degree = 2... | lemma | weierstrass_curve.coordinate_ring.degree_norm_smul_basis | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"adjoin_root.mk",
"algebra.norm",
"is_domain",
"max_bot_left",
"max_bot_right",
"mul_zero",
"one_mul",
"one_ne_zero'",
"zero_lt_two",
"zero_mul",
"zero_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
degree_norm_ne_one [is_domain R] (x : W.coordinate_ring) : (algebra.norm R[X] x).degree ≠ 1 | begin
rcases exists_smul_basis_eq x with ⟨p, q, rfl⟩,
rw [degree_norm_smul_basis],
rcases p.degree with (_ | _ | _ | _); cases q.degree,
any_goals { rintro (_ | _) },
exact (lt_max_of_lt_right dec_trivial).ne'
end | lemma | weierstrass_curve.coordinate_ring.degree_norm_ne_one | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"algebra.norm",
"is_domain",
"lt_max_of_lt_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_degree_norm_ne_one [is_domain R] (x : W.coordinate_ring) :
(algebra.norm R[X] x).nat_degree ≠ 1 | mt (degree_eq_iff_nat_degree_eq_of_pos zero_lt_one).mpr $ degree_norm_ne_one x | lemma | weierstrass_curve.coordinate_ring.nat_degree_norm_ne_one | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"algebra.norm",
"is_domain",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
elliptic_curve (R : Type u) [comm_ring R] extends weierstrass_curve R | (Δ' : Rˣ) (coe_Δ' : ↑Δ' = to_weierstrass_curve.Δ) | structure | elliptic_curve | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"comm_ring",
"weierstrass_curve"
] | An elliptic curve over a commutative ring. Note that this definition is only mathematically
accurate for certain rings whose Picard group has trivial 12-torsion, such as a field or a PID. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
j : R | ↑E.Δ'⁻¹ * E.c₄ ^ 3 | def | elliptic_curve.j | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [] | The j-invariant `j` of an elliptic curve, which is invariant under isomorphisms over `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
two_torsion_polynomial_disc_ne_zero [nontrivial R] [invertible (2 : R)] :
E.two_torsion_polynomial.disc ≠ 0 | E.two_torsion_polynomial_disc_ne_zero $ E.coe_Δ' ▸ E.Δ'.is_unit | lemma | elliptic_curve.two_torsion_polynomial_disc_ne_zero | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"invertible",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonsingular [nontrivial R] {x y : R} (h : E.equation x y) : E.nonsingular x y | E.nonsingular_of_Δ_ne_zero h $ E.coe_Δ' ▸ E.Δ'.ne_zero | lemma | elliptic_curve.nonsingular | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
variable_change : elliptic_curve R | ⟨E.variable_change u r s t, u⁻¹ ^ 12 * E.Δ',
by rw [units.coe_mul, units.coe_pow, coe_Δ', E.variable_change_Δ]⟩ | def | elliptic_curve.variable_change | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"elliptic_curve",
"units.coe_mul",
"units.coe_pow"
] | The elliptic curve over `R` induced by an admissible linear change of variables
$(X, Y) \mapsto (u^2X + r, u^3Y + u^2sX + t)$ for some $u \in R^\times$ and some $r, s, t \in R$.
When `R` is a field, any two Weierstrass equations isomorphic to `E` are related by this. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_variable_change_Δ' : (↑(E.variable_change u r s t).Δ' : R) = ↑u⁻¹ ^ 12 * E.Δ' | by rw [variable_change_Δ', units.coe_mul, units.coe_pow] | lemma | elliptic_curve.coe_variable_change_Δ' | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"units.coe_mul",
"units.coe_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inv_variable_change_Δ' : (↑(E.variable_change u r s t).Δ'⁻¹ : R) = u ^ 12 * ↑E.Δ'⁻¹ | by rw [variable_change_Δ', mul_inv, inv_pow, inv_inv, units.coe_mul, units.coe_pow] | lemma | elliptic_curve.coe_inv_variable_change_Δ' | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"inv_inv",
"inv_pow",
"mul_inv",
"units.coe_mul",
"units.coe_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
variable_change_j : (E.variable_change u r s t).j = E.j | begin
rw [j, coe_inv_variable_change_Δ'],
have hu : (u * ↑u⁻¹ : R) ^ 12 = 1 := by rw [u.mul_inv, one_pow],
linear_combination E.j * hu with { normalization_tactic := `[dsimp, ring1] }
end | lemma | elliptic_curve.variable_change_j | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"one_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
base_change : elliptic_curve A | ⟨E.base_change A, units.map ↑(algebra_map R A) E.Δ',
by rw [units.coe_map, ring_hom.coe_monoid_hom, coe_Δ', E.base_change_Δ]⟩ | def | elliptic_curve.base_change | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"algebra_map",
"elliptic_curve",
"ring_hom.coe_monoid_hom",
"units.coe_map",
"units.map"
] | The elliptic curve over `R` base changed to `A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.