The involution of a positive-definite involution algebra sends 0 to 0.
The structure map ℚ → A of a nontrivial positive-definite involution algebra
is injective. Used to identify ℚ with its image inside A.
The norm map of a nontrivial PD-involution algebra sends 0 to 0.
The norm of any element of a nontrivial PD-involution algebra is nonnegative.
The norm of an element vanishes iff the element itself is zero. Encodes positive-definiteness of the norm form.
A nontrivial PD-involution algebra has no zero divisors: if a * b = 0
then a = 0 or b = 0. Proved using positive-definiteness of the norm.
The norm is invariant under the involution: N(â) = N(a).
Multiplicativity of the norm: N(a * b) = N(a) * N(b).
Every nonzero element of a nontrivial PD-involution algebra is a unit. The
inverse is explicitly N(a)⁻¹ • â.
A nontrivial PD-involution algebra has no zero divisors.
A nontrivial PD-involution algebra is an integral domain.
Explicit two-sided inverse for a nonzero element a: (N(a))⁻¹ • â is both
a left and right inverse of a.
The rational endomorphism algebra End⁰(E) = End(E) ⊗_ℤ ℚ of an elliptic
curve E, an abbreviation for EndomorphismAlgebra E.
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The Rosati dual / canonical involution of an element of End⁰(E),
obtained from the PD-involution algebra structure on the endomorphism algebra.
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The reduced norm N : End⁰(E) → ℚ of an endomorphism, obtained from the
PD-involution algebra structure on End⁰(E). Geometrically this is the
degree of an isogeny (extended ℚ-linearly).
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Defining property of the norm: N(α) = α · α^† after embedding ℚ into
End⁰(E).
The endomorphism algebra End⁰(E) is nontrivial (contains both 0 and 1).
Nonnegativity of the endomorphism norm: N(α) ≥ 0 for all α ∈ End⁰(E).
Bundled statement of the main norm properties on End⁰(E):
nonnegativity, positive-definiteness, invariance under dual, multiplicativity.
Every nonzero element of End⁰(E) is invertible, making End⁰(E) a
division algebra over ℚ.
The reduced trace T : End⁰(E) → ℚ of an endomorphism, obtained from the
PD-involution structure on End⁰(E).
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Defining property of the trace map: T(a) mapped into A equals a + â.
The trace is invariant under the involution: T(â) = T(a).
Additivity of the trace: T(a + b) = T(a) + T(b).
ℚ-linearity of the trace: T(r • a) = r * T(a).
The trace of algebraMap q is 2q, matching the formula T(q · 1) = 2q
for the reduced trace of a scalar.
The trace of â · a is strictly positive for any nonzero a, since
â · a = N(a) · 1 and T(N(a) · 1) = 2 N(a) > 0.
Rationality formula for the trace on End⁰(E):
T(α) = 1 + N(α) - N(1 - α), expressing the trace as a polynomial in norms.
Bundled package: the trace on End⁰(E) is dual-invariant, ℚ-rational via
T(α) = 1 + N(α) - N(1 - α), additive, and ℚ-linear in the scalar action.
The characteristic polynomial of an endomorphism α, as the polynomial
X² - T(α) X + N(α) ∈ ℚ[X].
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α is a root of its own characteristic polynomial under the ℚ-algebra
evaluation Polynomial ℚ → End⁰(E).
The Rosati dual α^† is also a root of α's characteristic polynomial.
Both α and its Rosati dual α^† are roots of the characteristic polynomial
X² - T(α) X + N(α).
The Rosati-fixed elements of End⁰(E) are exactly the rational multiples of
the identity: α^† = α iff α = r • 1 for some r ∈ ℚ.