The reduced trace tr(a) = N(a + 1) - N(a) - 1, induced by the reduced norm.
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Albert's classification of finite-dimensional positive-definite involution ℚ-algebras A:
either A = ℚ (dimension 1), or A is an imaginary quadratic field (dimension 2 with a
non-rational element squaring to a negative rational), or A is a definite quaternion algebra
(dimension 4 with anticommuting generators squaring to negative rationals).
- rational {A : Type u_1} [Ring A] [Algebra ℚ A] (h : Module.finrank ℚ A = 1) : EndAlgClassification A
- imaginaryQuadratic {A : Type u_1} [Ring A] [Algebra ℚ A] (hdim : Module.finrank ℚ A = 2) (hα : ∃ (α : A), (∀ (q : ℚ), α ≠ (algebraMap ℚ A) q) ∧ ∃ d < 0, α * α = (algebraMap ℚ A) d) : EndAlgClassification A
- quaternionAlgebra {A : Type u_1} [Ring A] [Algebra ℚ A] (hdim : Module.finrank ℚ A = 4) (hαβ : ∃ (α : A) (β : A), (∃ a < 0, α * α = (algebraMap ℚ A) a) ∧ (∃ b < 0, β * β = (algebraMap ℚ A) b) ∧ α * β = -(β * α)) : EndAlgClassification A
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In a positive-definite involution algebra, any element β that commutes with a non-rational
element α lies in the ℚ-subalgebra generated by α: there exist r, s ∈ ℚ with β = r + s α.
This is the crucial step that forces a 4-dimensional algebra to be a quaternion algebra.
Given a pure imaginary element α (anti-fixed by the involution) and a candidate β₁ not in
the subfield ℚ[α], produce a pure imaginary element β anticommuting with α, by Gram-Schmidt
style adjustment. This is a key step in constructing a quaternion basis.
In a positive-definite involution algebra A, the four elements {1, α, β, αβ} are
ℚ-linearly independent whenever α is non-rational, anticommutes with β, β ∉ ℚ[α], and the
norms behave as in a quaternion algebra.
If A is a positive-definite involution algebra containing anticommuting pure imaginary
elements α, β with the right norm properties, then A has ℚ-dimension exactly 4, with basis
{1, α, β, αβ}.
The quaternion case of Albert's classification: given a pure imaginary element α and an
element not in ℚ[α], build a quaternion basis and conclude that A is a definite quaternion
algebra.
Albert's classification of endomorphism algebras: every nontrivial finite-dimensional
positive-definite involution ℚ-algebra is either ℚ itself, an imaginary quadratic field, or a
definite quaternion algebra.
The endomorphism algebra End⁰(E) := End(E) ⊗_ℤ ℚ of an elliptic curve E carries a
positive-definite involution algebra structure: the Rosati involution together with the reduced
degree-norm.
The Rosati involution on the endomorphism algebra of E, packaged as a function.
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The reduced norm End⁰(E) → ℚ (the rational extension of the degree map on End(E)).
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The reduced trace End⁰(E) → ℚ, defined as N(x + 1) - N(x) - 1.
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The Rosati involution is additive.
The Rosati involution fixes the identity.
The defining identity of the reduced norm: a * a^† = N(a) · 1 in the endomorphism algebra.
The integer algebra map into the endomorphism ring is injective: the canonical map
ℤ → End(E) has no kernel.
The endomorphism algebra End⁰(E) is nontrivial: tensoring End(E) (a nontrivial ring with
injective integer algebra map) with ℚ over ℤ remains nontrivial since ℚ is ℤ-flat.
The endomorphism algebra End⁰(E) is a finite-dimensional ℚ-vector space.
Specialization of Albert's classification to the endomorphism algebra of an elliptic curve:
End⁰(E) is ℚ, an imaginary quadratic field, or a definite quaternion algebra.
The endomorphism ring of an elliptic curve is torsion-free as a ℤ-module.
The endomorphism ring of an elliptic curve is finitely generated as a ℤ-module.
A finitely generated torsion-free ℤ-module is free; in particular, End(E) is ℤ-free.
The ℤ-rank of End(E) is 1, 2, or 4, matching the three Albert cases.
Combined freeness and rank statement for End(E): it is a free ℤ-module of rank 1, 2,
or 4.