NEW VALUES FOR THE LEVEL AND

SUBLEVEL OF COMPOSITION ALGEBRAS

James O’Shea

Abstract Constructions of quaternion and octonion algebras, suggested to have new level and sublevel values, are proposed and justified. In particular, octonion algebras of level and sublevel 6 and 7 are constructed. In addition, Hoffmann’s proof of the existence of infinitely many new values for the level of a quaternion algebra is generalised and adapted.

1 Introduction

In [Lew1], Lewis constructed quaternion algebras of level 2^{k} and 2^{k} + 1, both of sublevel 2^{k}, for all integers k ≥ 0. Laghribi and Mammone recovered these values as the level of a quaternion algebra using function field techniques (see [LM]). Pumplün employed their methodology in [Pu] to construct octonion algebras of level 2^{k} and 2^{k} + 1. By constructing a quaternion algebra of sublevel 3, Krüskemper and Wadsworth produced the first example of a quaternion algebra whose sublevel was not a power of 2 (see [KW]). This construction was subsequently employed to prove the existence of an octonion algebra of sublevel 3, and indeed one of sublevel 5 was also produced (see [O’S]). Heretofore, these remained the only known values for the level and sublevel of quaternion and octonion algebras.

In Section 3.1, we introduce certain quaternion and octonion algebras, denoted Q(n) and O(n) respectively, which we conjecture to be of level and sublevel n, for all n.

In Section 3.1.1, we proceed to precisely determine the level and sublevel of these algebras for certain values of n. In particular, we produce the first example of a composition algebra whose known level is not equal to 2^{k} or 2^{k} + 1 for some k.

Section 3.1.2 sees us outline the argument employed by Hoffmann [H3] to bound the level of Q(n), proving the existence of infinitely many values for the level of Q(n) which are not of the form 2^{k} or 2^{k} + 1 for some k. We extend this argument to O(n), and refine these bounds for particular n-values.

We conclude with Section 3.2, wherein we introduce alternative constructions of quaternion and octonion algebras, which we suggest to be of level and sublevel l · 2^{k}. Some merits of these constructions are outlined.

2 Preliminaries

Let F denote a field of characteristic ≠ 2, F_{0} a formally real field and C a unital, not necessarily associative, F -algebra.

A map ∗ is called an involution on C if it is an anti-automorphism of period 2. We have C = Sym(C, ∗) ⊕ Skew(C, ∗), with Sym(C, ∗) = {x ∈ C|x^{∗} = x} and Skew(C, ∗) = {x ∈ C|x^{∗} = −x}. An involution ∗ is called scalar if x^{∗}x ∈ F and x^{∗} + x ∈ F for all x ∈ C. For an algebra C with scalar involution ∗, we call t_{C}(x) = x + x^{∗} the trace of C and the quadratic form T_{C} : C → F, T_{C}(x) = t_{C}(x^{2}) the trace form.

The Cayley-Dickson doubling process is an algorithm for constructing new algebras with scalar involution from old ones. Applying the process to an algebra C with scalar involution ∗, together with a chosen scalar μ ∈ F^{×}, produces a new algebra, Cay(C, μ), the Cayley-Dickson double of C, whose scalar involution we will also denote by ∗. We note that Cay(C, μ) is the F -module C × C, with multiplication defined by (u, v)(u′, v′) = (uu′ + μv^{′∗}v, v′u + vu^{′∗}) and involution given by (u, v)^{∗} = (u^{∗}, −v), for u, u′, v, v′ ∈ C.

An algebra C is a composition algebra if there exists a nondegenerate quadratic form q on C which allows composition, that is q(x y) = q(x) q(y) for all x, y ∈ C. Composition algebras are of rank 1, 2, 4 or 8. The composition algebras of rank 2 are the quadratic étale F -algebras; the composition algebras of rank 4 are the (non-commutative) quaternion algebras and those of rank 8 are the (noncommutative and non-associative) octonion algebras. For a, b ∈ F^{×}, the quaternion algebra Q = (\frac{a,b}{F}) over F is a 4-dimensional F -vector space with basis \{1, i, j, k\}, satisfying i^{2} = a, j^{2} = b and ij = −ji = k. For a, b, c ∈ F^{×}, the octonion algebra (\frac{a,b,c}{F}) over F is defined as (\frac{a,b,c}{F}) := Cay ((\frac{a,b}{F}) , c). We note that Cay ((\frac{a,b}{F}) , c) = (\frac{a,b}{F}) ⊕ (\frac{a,b}{F}) e is an 8-dimensional F -vector space with basis {1, i, j, k, e, ie, je, ke}, satisfying i^{2} = a, j^{2} = b and e^{2} = c. Applying the Cayley-Dickson doubling process to a composition algebra C over F, yields another composition algebra (if the dimension of the new algebra is at most 8), or alternatively what is known as a generalised Cayley-Dickson algebra.

For C = (\frac{a,b}{F}), we have T_{C} ≃ 2⟨1, a, b, −ab⟩, whereas for C = (\frac{a,b,c}{F}), T_{C} ≃ 2⟨1, a, b, −ab, c, −ac, −bc, abc⟩. Since we are concerned with the isotropy of trace forms, we may disregard this scalar factor of 2. We define the pure trace form of a composition algebra C, denoted T_{P}, via the following relation: T_{C} = ⟨1⟩ ⊥ T_{P}.

For C a division algebra, we define the level of C, denoted s(C), as the least integer n such that −1 is a sum of n squares in C. If no such integer exists, we say that s(C) = ∞. The sublevel of C, denoted \underline{s}(C), is the least integer n for which 0 is a sum of n + 1 squares of elements in C. If 0 is not expressible in this manner, we say that \underline{s}(C) = ∞. Note that \underline{s}(C) ≤ s(C).

In [Pf], Pfister showed that the level of a field, if finite, is a power of 2, and moreover that any prescribed power of 2 may be realised as the level of a field. This classification extends to the case where C is a quadratic étale F -algebra. A composition algebra is split if it contains a composition subalgebra which is isomorphic to F ⊕ F, which is the case if and only if C contains zero divisors. Split quaternion and octonion algebras have level 1 (a consequence of a split quaternion algebra being isomorphic to M_{2}F). Hence, in order to complete the classification of the level of composition algebras, we consider quaternion and octonion algebras which are division.

For φ a regular n-dimensional quadratic form over F, we can consider φ as a homogeneous polynomial of degree 2, that is φ(X) = φ(X_{1}, …, X_{n}) = ∑_{i,j} α_{ij}X_{i}X_{j}, where α_{ij} ∈ F^{×}. For n > 1 and φ ≄ ⟨1, −1⟩, the function field of φ, denoted F (φ), is defined to be the quotient field of the integral domain

F [X]/(φ(X)) = F [X_{1}, …, X_{n}]/(φ(X_{1}, …, X_{n})).

φ is said to be isotropic if there exists a non-zero vector x such that φ(x) = 0. By construction, every quadratic form is isotropic over its function field. The Witt index of φ, denoted i_{W} (φ), is the dimension of a maximal totally isotropic subform of φ. The first Witt index of φ, denoted i_{1}(φ), is the Witt index of φ over its function field. The essential dimension of φ is given by the following relation: dim_{es}(φ) = dim(φ) − i_{1}(φ) + 1. The value set of φ, denoted D_{F} (φ), is the set of elements of F^{×} which are represented by φ. By D_{F} (φ)D_{F} (ψ) we will denote the usual set product of D_{F} (φ) and D_{F} (ψ).

We highlight the following recent progress towards classifying when one quadratic form becomes isotropic over the function field of another:

Theorem 2.1. [I, Lemma 5.3] Let ψ_{0}, ψ_{1}, φ_{0}, φ_{1} be anisotropic quadratic forms over F. For ψ := ψ_{0} ⊥ T ψ_{1} and φ := φ_{0} ⊥ T φ_{1}, the following are equivalent:

• ψ isotropic over F (T)(φ).

• ψ isotropic over F ((T))(φ).

• D_{F′}(φ_{0})D_{F′}(φ_{1}) ⊆ D_{F′}(ψ_{0})D_{F′}(ψ_{1}) for every extension F′ of F.

Theorem 2.2. [R, Theorem 5.1.5] ψ is isotropic over F (φ) ⟺ D_{F′}(φ)D_{F′}(φ) ⊆ D_{F′}(ψ)D_{F′}(ψ) for every extension F′ of F.

In addition, Karpenko and Merkurjev employed advanced algebro-geometric techniques involving Chow groups to prove the following powerful result:

Theorem 2.3. [KM, Theorem 4.1] Let φ and ψ be anisotropic over F and suppose that ψ is isotropic over F (φ). Then

(i) dim_{es}(φ) ≤ dim_{es}(ψ);

(ii) moreover, the equality dim_{es}(φ) = dim_{es}(ψ) holds if and only if φ is isotropic over F (ψ).

For an overview of function fields of quadrics or for further definitions and notation regarding quadratic forms, we refer the reader to [Lam].

3 Results

3.1 Q(n) and O(n)

In this section, we will introduce certain families of quaternion and octonion algebras, before proceeding to investigate what values their level and sublevel may take.

3.1.1 Introducing Q(n) and O(n)

As motivated above, we will alternate between considering quaternion and octonion algebras which are division. We will let C denote a quaternion or octonion algebra which is division. For those occasions in which we will need to discriminate between these alternatives, we will use Q to denote a quaternion division algebra and O an octonion division algebra. By T_{P} we will denote the pure trace form of the algebra in question, which should be clear from the context.

Since D_{F} (n × T_{P}) = {∑_{i=1}^{n} p_{i}^{2}|p_{i} ∈ Skew(C, ∗)} for ∗ given by conjugation, if ⟨1⟩ ⊥ n × T_{P} is isotropic then −1 is expressible as a sum of n squares of pure elements in C, yielding that s(C) ≤ n.

For n = 2^{k} −1, Lewis stated that the converse holds for quaternion division algebras (see [Lew2]). This result extends to composition algebras (see [O’S, Lemma 3.9]), allowing us to conclude that

s(C) ≤ n ⟺ ⟨1⟩ ⊥ n × T_{P} is isotropic for n = 2^{k} − 1.

Furthermore, for n = 2^{k} Lewis showed that s(Q) ≤ n implies that ⟨1⟩ ⊥ n × T_{P} or (n + 1) × ⟨1⟩ ⊥ (n − 1) × T_{P} is isotropic (see [Lew2, Lemma 3]), with Leep proving the converse in [Le, Theorem 2.2]. This equivalence also admits generalisation to the class of composition algebras:

s(C) ≤ 2^{k} ⟺ ⟨1⟩ ⊥ 2^{k} × T_{P} or (2^{k} + 1) × ⟨1⟩ ⊥ (2^{k} − 1) × T_{P} is isotropic(1)

In light of the above results, a natural question which then arose was whether s(C) ≤ n implies that ⟨1⟩ ⊥ n×T_{P} is isotropic for all n. By producing a quaternion algebra of level 2^{k} such that ⟨1⟩ ⊥ 2^{k} × T_{P} is anisotropic for all k, Koprowski [Kop] answered this question in the negative. Pumplün later showed that there also exist octonion algebras of level 2^{k} such that ⟨1⟩ ⊥ 2^{k} × T_{P} is anisotropic for all k (see [Pu]).

More specifically, Koprowski’s example and its octonion algebra analogue are directly relevant to (1), proving that if s(C) ≤ 2^{k} it remains necessary, for all values of k, to consider whether (2^{k} + 1) × ⟨1⟩ ⊥ (2^{k} − 1) × T_{P} becomes isotropic in consequence. As an interesting aside, we note that if s(C) ≤ 2^{k}, it is actually the consideration of the isotropy of ⟨1⟩ ⊥ 2^{k} × T_{P} which becomes redundant for k sufficiently large, as the following attests:

Theorem 3.1. For Q a quaternion division algebra, s(Q) ≤ 2^{k} ⟺ (2^{k} + 1) × ⟨1⟩ ⊥ (2^{k} − 1) × T_{P} is isotropic, where k ≥ 2. For O an octonion division algebra, s(O) ≤ 2^{k} ⟺ (2^{k} + 1) × ⟨1⟩ ⊥ (2^{k} − 1) × T_{P} is isotropic, where k ≥ 3.

Proof. Since s(C) ≤ 2^{k} ⟺ (2^{k} + 1) × ⟨1⟩ ⊥ (2^{k} − 1) × T_{P} or ⟨1⟩ ⊥ 2^{k} × T_{P} is isotropic, it suffices to prove that the isotropy of ⟨1⟩ ⊥ 2^{k} × T_{P} implies that of (2^{k} + 1) × ⟨1⟩ ⊥ (2^{k} − 1) × T_{P}, for k as above. Suppose ⟨1⟩ ⊥ 2^{k} × T_{P} is isotropic, implying that 2^{k} ×T_{C} ≃ (2^{k} ×⟨1⟩)⊗T_{C} is isotropic. We know that i_{W} (2^{k} ×T_{C}) ≥ 2^{k} by [EL, Proposition 1.4]. For k as above, 2^{k} × ⟨1⟩ ⊥ (2^{k} − 1) × T_{P} is a subform of 2^{k} × T_{C} of codimension < 2^{k}, and thus is also isotropic.

Since s(C) = 2^{k} ⇒ \underline{s}(C) = 2^{k} ([Le, Theorem 2.5] for C = Q and [O’S, Theorem 3.11] for C = O), Koprowski’s example and its octonion algebra analogue prove that neither s(C) = n nor \underline{s}(C) = n implies that ⟨1⟩ ⊥ n × T_{P} is isotropic for all n. Of course, this does not preclude the existence of composition algebras of level and sublevel n such that ⟨1⟩ ⊥ n × T_{P} is isotropic, for all values of n. As we have seen, the isotropy of ⟨1⟩ ⊥ n × T_{P} encodes n as an upper bound for the level and sublevel of a composition algebra. Indeed, since D_{F} (n × T_{P}) = {∑_{i=1}^{n} p_{i}^{2}|p_{i} ∈ Skew(C, ∗)}, ⟨1⟩ ⊥ n × T_{P} is the most natural quadratic form whose isotropy bounds the level. Thus it seems reasonable to suggest that if one were to construct composition algebras, subject to ⟨1⟩ ⊥ n × T_{P} being isotropic, in a suitably general fashion, then their level and sublevel might actually equal n. In defining Q(n) and O(n), we are aiming for just such a construction. In an attempt to make Q(n) and O(n) suitably general, the better to eradicate factors which could serve to reduce the level or sublevel below n, we choose the generators of our algebra to be transcendental over a formally real ground field, and extend to the function field of ⟨1⟩ ⊥ n × T_{P}, the natural method of generically encoding its isotropy. Hence, we let

Q(n) := (\frac{x, y}{F} ) ⊗_{F} F (⟨1⟩ ⊥ n × T_{P})

and

O(n) := (\frac{x, y, z}{F} ) ⊗_{F} F (⟨1⟩ ⊥ n × T_{P}) ,

where T_{P} is the pure trace form of (\frac{x,y}{F} ) and (\frac{x,y,z}{F} ) respectively, and F denotes F_{0}(x, y) and F_{0}(x, y, z) respectively. We posit the following conjecture:

Conjecture 3.2. s (Q(n)) = \underline{s} (Q(n)) = s (O(n)) = \underline{s} (O(n)) = n for all n.

3.1.2 Determined Values

We begin our investigation of Q(n) and O(n) by noting that if the sublevel of these algebras equals n, then their level and sublevel must coincide.

We will now proceed to establish the first seven cases of the octonionic component of Conjecture 3.2. In doing so, we prove that 6 and 7 are realisable as both the level and sublevel of octonion algebras. We note that these are the first examples of composition algebras whose level is not of the form 2^{k} or 2^{k} + 1 for some k.

Theorem 3.3. s (O(n)) = \underline{s} (O(n)) = n, for n ≤ 7.

Proof. For ease of notation we let φ := ⟨1⟩ ⊥ n × T_{P}.

By construction \underline{s} (O(n)) ≤ n.

Suppose \underline{s} (O(n)) ≤ n − 1. Hence

∑_{i=1}^{n} c_{i}^{2} + ∑_{i=1}^{n} p_{i}^{2} = 0 (2)

and

∑_{i=1}^{n} c_{i}p_{i} = 0, (3)

where c_{i} ∈ F (φ) and p_{i} ∈ P, the F (φ)-vector space spanned by the pure octonions.

If c_{i} = 0 ∀ i, (2) implies that n × T_{P} is isotropic over F (φ).

Alternatively, if there exists an i such that c_{i} ≠ 0, (3) implies the existence of V, an (n − 1)-dimensional F (φ)-subspace of P, containing p_{1}, …, p_{n}. Let β : V → F (φ), p ↦ p^{2}. Hence, β is an (n − 1)-dimensional subform of T_{P} and (2) implies that ϑ := n × (⟨1⟩ ⊥ β) is isotropic over F (φ).

Note that n×T_{P} and 8×(⟨−1⟩ ⊥ T_{P}) are anisotropic over F by Springer’s Theorem.

If n × T_{P} becomes isotropic over F (φ), then the Pfister form 8 × (⟨1⟩ ⊥ −T_{P}) becomes hyperbolic over F (φ). Invoking the Cassels-Pfister Subform Theorem (see [S, p.155]), α(φ) should be a subform of 8 × (⟨1⟩ ⊥ −T_{P}) for any α ∈ D_{F} (8 × (⟨1⟩ ⊥ −T_{P})) . In particular, since −x ∈ D_{F} (8 × (⟨1⟩ ⊥ −T_{P})) , −x(φ) ≃ ⟨−x⟩ ⊥ n × ⟨−1, −xy, y, −xz, z, xyz, −yz⟩ should be a subform of 8 × (⟨1⟩ ⊥ −T_{P}), implying that n × ⟨−1⟩ is also a subform, which is clearly false. Hence, n × T_{P} also remains anisotropic over F (φ).

Thus, ϑ must be isotropic over F (φ) . By Theorem 2.2 we must have

D_{F′} (φ) D_{F′} (φ) ⊆ D_{F′} (ϑ) D_{F′} (ϑ) for every extension F′ of F.

Let F′ = F.

Since T_{P} ≃ ⟨x, y, −xy, z, −xz, −yz, xyz⟩, we have x ∈ D_{F} (φ) D_{F} (φ). Suppose ⟨x⟩ is not a subform of β. Hence, all multiples of x in D_{F} (ϑ) D_{F} (ϑ) are of the form −x (∑_{i=1}^{m} a_{i}^{2}), for a_{i} ∈ F (note that ⟨(y)(−xy)⟩ ≃ ⟨(z)(−xz)⟩ ≃ ⟨(−yz)(xyz)⟩ ≃ ⟨−x⟩). If x is positive with respect to some ordering on F, then −x (∑_{i=1}^{m} a_{i}^{2}) is necessarily negative, and vice-versa. Thus x ∉ D_{F} (ϑ) D_{F} (ϑ), which is a contradiction. Hence ⟨x⟩ is a subform of β.

Note that y ∈ D_{F} (φ) D_{F} (φ). Now suppose that ⟨y⟩ is not a subform of β. Hence, all multiples of y in D_{F} (ϑ) D_{F} (ϑ) are of the form −y (∑_{i=1}^{m} a_{i}^{2}), for a_{i} ∈ F (note that ⟨(x)(−xy)⟩ ≃ ⟨(z)(−yz)⟩ ≃ ⟨(−xz)(xyz)⟩ ≃ ⟨−y⟩). Again, if y is positive with respect to some ordering on F, then −y (∑_{i=1}^{m} a_{i}^{2}) is negative, and vice-versa. Thus y ∉ D_{F} (ϑ) D_{F} (ϑ), which is a contradiction. Hence ⟨y⟩ is a subform of β.

Similarly, since −xy, z, −xz, −yz and xyz ∈ D_{F} (φ) D_{F} (φ), ⟨−xy⟩, ⟨z⟩, ⟨−xz⟩, ⟨−yz⟩ and ⟨xyz⟩ must also be subforms of β.

Thus, T_{P} ≃ ⟨x⟩ ⊥ ⟨y⟩ ⊥ ⟨−xy⟩ ⊥ ⟨z⟩ ⊥ ⟨−xz⟩ ⊥ ⟨−yz⟩ ⊥ ⟨xyz⟩ is a subform of β, which contradicts the fact that β is (n − 1)-dimensional.

Thus \underline{s} (O(n)) ≰ n − 1, implying that \underline{s} (O(n)) = n.

Showing that ϑ remains anisotropic over F (φ) is a key step in the above proof. The machinery with which we successfully endeavoured to achieve this end is Roussey’s Theorem 2.2. Hence, all the ingredients in the above proof lie within the “classical” theory of quadratic forms.

If we instead choose to invoke Theorem 2.3 of Karpenko and Merkurjev, whose proof relies upon deep algebro-geometric techniques, in order to prove anisotropy of forms over F (⟨1⟩ ⊥ n × T_{P}), we obtain a more succinct, if less elementary, argument:

Proof. As detailed above, to prove that \underline{s} (O(n)) ≰ n−1 it suffices to show that n× T_{P} and n×(⟨1⟩ ⊥ β) are anisotropic over F (⟨1⟩ ⊥ n × T_{P}). Since i_{1} (⟨1⟩ ⊥ n × T_{P}) = 1 (see Lemma 3.8), the fact that dim (⟨1⟩ ⊥ n × T_{P}) > dim (n × T_{P}) implies that n× T_{P} is anisotropic over F (⟨1⟩ ⊥ n × T_{P}) by Theorem 2.3. Similarly since dim (⟨1⟩ ⊥ n × T_{P}) > dim (n × (⟨1⟩ ⊥ β)), we can conclude that n × (⟨1⟩ ⊥ β) is also anisotropic over F (⟨1⟩ ⊥ n × T_{P}) by Theorem 2.3. Hence \underline{s} (O(n)) ≰ n − 1.

Both of the above methods of proof apply to quaternion algebras as well, yielding:

Theorem 3.4. s (Q(n)) = \underline{s} (Q(n)) = n for n ≤ 3.

We note that the above results may be extended to their respective next cases. Indeed, the following result shows that Conjecture 3.2 is true for all n = 2^{k}:

Theorem 3.5. s (Q(n)) = \underline{s} (Q(n)) = s (O(n)) = \underline{s} (O(n)) = n, for n = 2^{k}.

Proof. We may assume that k ≥ 2 by Theorem 3.3 and Theorem 3.4.

Suppose \underline{s} (Q(2^{k})) ≤ 2^{k} − 1. Thus, ⟨1⟩ ⊥ (2^{k} − 1) × T_{P} is isotropic over F (φ) by [O’S, Theorem 3.5], where φ := ⟨1⟩ ⊥ 2^{k} × T_{P}. Since ψ_{⟨x⟩} := ⟨1⟩ ⊥ 2^{k} × ⟨x⟩ is a subform of φ, there exists an F -place from F (φ) to F (ψ_{⟨x⟩}), by [K, Theorem 3.3]. Thus ⟨1⟩ ⊥ (2^{k} − 1) × T_{P} is isotropic over F (ψ_{⟨x⟩}), by [K, Theorem 3.3]. Springer’s Theorem implies that either ⟨1⟩ ⊥ (2^{k} − 1) × ⟨x⟩ or (2^{k} − 1) × ⟨1, −x⟩ is isotropic over F_{0}(x) (ψ_{⟨x⟩}). However, ⟨1⟩ ⊥ (2^{k} − 1) × ⟨x⟩ is anisotropic over F_{0}(x) (ψ_{⟨x⟩}) by [H1, Theorem 1]. Hence, (2^{k} − 1) × ⟨1, −x⟩ must be isotropic over F_{0}(x) (ψ_{⟨x⟩}), implying that 2^{k} × ⟨1, −x⟩ is hyperbolic over F_{0}(x) (ψ_{⟨x⟩}). The Cassels-Pfister Subform Theorem implies that ψ_{⟨x⟩} is a subform of 2^{k} × ⟨1, −x⟩, giving a desired contradiction. Hence \underline{s} (Q(2^{k})) = 2^{k}.

Now suppose \underline{s} (O(2^{k})) ≤ 2^{k} − 1. Thus, ⟨1⟩ ⊥ (2^{k} − 1) × T_{P} is isotropic over F (⟨1⟩ ⊥ 2^{k} × T_{P} ) by [O’S, Theorem 3.5]. Since ψ_{⟨x,y,−xy⟩} := ⟨1⟩ ⊥ 2^{k} ×⟨x, y, −xy⟩ is a subform of ⟨1⟩ ⊥ 2^{k} × T_{P}, there exists an F -place from F (⟨1⟩ ⊥ 2^{k} × T_{P} ) to F (ψ_{⟨x,y,−xy⟩}), by [K, Theorem 3.3]. Thus ⟨1⟩ ⊥ (2^{k} − 1) × T_{P} is isotropic over F (ψ_{⟨x,y,−xy⟩}), by [K, Theorem 3.3]. Springer’s Theorem implies that either ⟨1⟩ ⊥ (2^{k}−1)×⟨x, y, −xy⟩ or (2^{k}−1)×⟨1, −x, −y, xy⟩ is isotropic over F_{0}(x, y) (ψ_{⟨x,y,−xy⟩}). In proving the quaternion case above, we showed that ⟨1⟩ ⊥ (2^{k} − 1) × ⟨x, y, −xy⟩ is anisotropic over F_{0}(x, y) (ψ_{⟨x,y,−xy⟩}). Hence, (2^{k} − 1) × ⟨1, −x, −y, xy⟩ must be isotropic over F_{0}(x, y) (ψ_{⟨x,y,−xy⟩}), implying that 2^{k} × ⟨1, −x, −y, xy⟩ is hyperbolic over F_{0}(x, y) (ψ_{⟨x,y,−xy⟩}). The Cassels-Pfister Subform Theorem implies that ψ_{⟨x,y,−xy⟩} is a subform of 2^{k} × ⟨1, −x, −y, xy⟩, furnishing a contradiction. Hence \underline{s} (O(2^{k})) = 2^{k}.

Karpenko and Merkurjev’s result, Theorem 2.3, provides another method of proof:

Proof. Since i_{1} (⟨1⟩ ⊥ 2^{k} × T_{P} ) = 1 (see Lemma 3.7 and Lemma 3.8), the fact that dim (⟨1⟩ ⊥ 2^{k} × T_{P} ) > dim (⟨1⟩ ⊥ (2^{k} − 1) × T_{P} ) implies that ⟨1⟩ ⊥ (2^{k} − 1) × T_{P} remains anisotropic over F (⟨1⟩ ⊥ 2^{k} × T_{P} ) by Theorem 2.3. Hence \underline{s} (Q(2^{k})) ≰ 2^{k} − 1 and \underline{s} (O(2^{k})) ≰ 2^{k} − 1.

In their construction of a quaternion algebra of level 2^{k} + 1 for k ≥ 1, Laghribi and Mammone considered Q(2^{k} +1), proving that s (Q(2^{k} + 1)) = 2^{k} +1 [LM, Theorem 2.1]. Furthermore, Pumplün adapted their argument to show that s (O(2^{k} + 1)) = 2^{k} + 1 for k ≥ 1 [Pu, Theorem 3.1]. We note that these results extend to the case where k = 0, as this is merely the level component of Theorem 3.5 when k = 1. We now present a simplified method of proving these results:

Theorem 3.6. s (Q(n)) = s (O(n)) = n, for n = 2^{k} + 1.

Proof. Suppose s (Q(2^{k} + 1)) ≤ 2^{k}. Hence (2^{k} + 1) × ⟨1⟩ ⊥ (2^{k} − 1) × T_{P} is isotropic over F (⟨1⟩ ⊥ (2^{k} + 1) × T_{P} ) by Theorem 3.1 (we can assume that k ≥ 2 by Theorem 3.4). Since y(2^{k}+1)×⟨1, −x⟩ is a subform of ⟨1⟩ ⊥ (2^{k}+1)×T_{P}, there exists an F -place from F (⟨1⟩ ⊥ (2^{k} + 1) × T_{P} ) to F ((2^{k} + 1) × ⟨1, −x⟩), by [K, Theorem 3.3]. Thus (2^{k}+1)×⟨1⟩ ⊥ (2^{k}−1)×T_{P} is isotropic over F ((2^{k} + 1) × ⟨1, −x⟩), by [K, Theorem 3.3]. Springer’s Theorem implies that either (2^{k} +1)×⟨1⟩ ⊥ (2^{k} −1)×⟨x⟩ or (2^{k} − 1) × ⟨1, −x⟩ is isotropic over F_{0}(x) ((2^{k} + 1) × ⟨1, −x⟩). However, both forms are anisotropic by [H1, Theorem 1]. Hence s (Q(2^{k} + 1)) ≰ 2^{k}.

The same method applies in the octonion case.

The values of n specified in the above results remain the only ones for which we can precisely determine s (Q(n)) , \underline{s} (Q(n)) , s (O(n)) or \underline{s} (O(n)). For other values of n, bounds may be placed on these quantities via Theorem 2.3, as we will now proceed to outline.

3.1.3 Bounding Undetermined Values

Given the similarities between sums of squares in quaternion and octonion algebras, that there exist octonion algebras whose level is not of the form 2^{k} or 2^{k}+1, as proved in Theorem 3.3, suggests that the same is true for quaternion algebras. Indeed, at a recent seminar in University College Dublin [H2], Detlev Hoffmann kindly communicated his method of showing the existence of infinitely many quaternion algebras whose level is neither 2^{k} nor 2^{k} + 1 for some k, answering a question posed by David Leep in [Le]. The family of quaternion algebras that Hoffmann considers is {Q(n)}.

A key step in Hoffmann’s argument is the following:

Lemma 3.7. [H3, Lemma 4.1] i_{1} (⟨1⟩ ⊥ n × T_{P}) = 1 for all n, where T_{P} represents the pure trace form of Q(n).

Employing the same method of proof, one can obtain the corresponding result for O(n), as we will now proceed to outline:

Lemma 3.8. i_{1} (⟨1⟩ ⊥ n × T_{P}) = 1 for all n, where T_{P} represents the pure trace form of O(n).

Proof. For P an ordering of F_{0}, let P′ represent an extension to F such that x, y and z are negative. Hence, ⟨1⟩ ⊥ n × T_{P} is indefinite with respect to P′, implying that P′ extends to F (⟨1⟩ ⊥ n × T_{P}) by [ELW, Theorem 3.5 and Remark 3.6]. Now, since ⟨1⟩ ⊥ n × T_{P} has only one positive coefficient with respect to P′, we can conclude that i_{1} (⟨1⟩ ⊥ n × T_{P}) = 1.

Hoffmann invokes Lemma 3.7 to prove the following result:

Theorem 3.9. [H3, Corollary 4.3] For n = m + 1 + ⌊\frac{m}{3}⌋, s (Q(n)) ∈ [m + 1, n].

The choice of m + 1 + ⌊\frac{m}{3}⌋ for n is justifiable since it is the least value which ensures that the essential dimension of ⟨1⟩ ⊥ n × T_{P} is greater than or equal to the dimension of ⟨1⟩ ⊥ m × T_{Q}, thereby ensuring that s(Q) ≰ m via Theorem 2.3. For the sake of consistency, we reformulate the above result in the following manner:

s (Q(n)) ∈ [n − ⌊\frac{n}{4}⌋ , n] for all n.

Of course, we may similarly invoke Lemma 3.8 to the same effect, allowing us to obtain the analogous bounds for s (O(n)):

Theorem 3.10. s (O(n)) ∈ [n − ⌊\frac{n}{8}⌋ , n] for all n.

Proof. Since i_{1} (⟨1⟩ ⊥ n × T_{P}) = 1 for all n, if dim (⟨1⟩ ⊥ n × T_{P}) > dim (⟨1⟩ ⊥ m × T_{O(n)}) then Theorem 2.3 implies that ⟨1⟩ ⊥ m×T_{O(n)} remains anisotropic over F (⟨1⟩ ⊥ n × T_{P}), bounding s (O(n)) in [m + 1, n]. Comparing dimensions, we see that n − ⌊\frac{n}{8}⌋ − 1 is the greatest value of m such that dim (⟨1⟩ ⊥ n × T_{P}) > dim (⟨1⟩ ⊥ m × T_{O(n)}). Hence, s (O(n)) ∈ [n − ⌊\frac{n}{8}⌋ , n].

We note that the above result recovers the level component of Theorem 3.3. In a similar fashion, we may further apply Hoffmann’s methodology to considerations of the sublevel of Q(n):

Theorem 3.11. \underline{s} (Q(n)) ∈ [n − ⌊\frac{n+3}{4}⌋ , n] for all n.

Proof. Since i_{1} (⟨1⟩ ⊥ n × T_{P}) = 1 for all n, if dim (⟨1⟩ ⊥ n × T_{P}) > dim ((m + 1) × T_{Q(n)}) then Theorem 2.3 implies that (m+1)×T_{Q(n)} remains anisotropic over F (⟨1⟩ ⊥ n × T_{P}), bounding \underline{s} (Q(n)) in [m + 1, n]. Comparing dimensions, we see that n − ⌊\frac{n+3}{4}⌋ − 1 is the greatest value of m such that dim (⟨1⟩ ⊥ n × T_{P}) > dim ((m + 1) × T_{Q(n)}). Hence, \underline{s} (Q(n)) ∈ [n − ⌊\frac{n+3}{4}⌋ , n].

The corresponding argument yields the following bounds for the sublevel of O(n):

Theorem 3.12. \underline{s} (O(n)) ∈ [n − ⌊\frac{n+7}{8}⌋ , n] for all n.

Without placing any restrictions on the value that n may take, the above results represent the sharpest bounds on the level and sublevel of Q(n) and O(n) currently available. For a large class of values however, we may further reduce the intervals in which these quantities are known to lie. We achieve this reduction by exploiting Theorem 2.3 more fully: For m = 2^{k}d, m×T_{C} ≃ 2^{k}×⟨1⟩⊗(d×T_{C}). In consequence, i_{1} (m × T_{C}) ≥ 2^{k} by [EL, Theorem 1.4]. Thus, invoking Theorem 2.3 in conjunction with this observation, we may increase the lower bounds on \underline{s} (Q(n)) and \underline{s} (O(n)) for certain n, and consequently those on s (Q(n)) and s (O(n)).

Theorem 3.13. s (Q(n)) and \underline{s} (Q(n)) ∈ [m, n] for n ≥ m + ⌊\frac{m−2^{k}}{3} ⌋ + 1, where k is the 2-adic order of m.

Proof. [EL, Theorem 1.4] implies that i_{1} (m × T_{Q(n)}) ≥ 2^{k}. Comparing the essential dimensions of ⟨1⟩ ⊥ n × T_{P} and m × T_{Q(n)}, we see that m + ⌊\frac{m−2^{k}}{3} ⌋ + 1 is the least value of n such that dim (⟨1⟩ ⊥ n × T_{P}) − 1 > dim (m × T_{Q(n)}) − 2^{k}. Hence, for n ≥ m + ⌊\frac{m−2^{k}}{3} ⌋ + 1, m × T_{Q(n)} is anisotropic over F (⟨1⟩ ⊥ n × T_{P}) by Theorem 2.3, implying that \underline{s} (Q(n)) ∈ [m, n]. Since \underline{s} (Q(n)) ≤ s (Q(n)), s (Q(n)) ∈ [m, n] also.

Arguing in a likewise manner, we obtain the following bounds for the level and sublevel of O(n):

Theorem 3.14. s (O(n)) and \underline{s} (O(n)) ∈ [m, n] for n ≥ m + ⌊\frac{m−2^{k}}{7} ⌋ + 1, where k is the 2-adic order of m.

Of course, if n = 2^{k} + h for h ≥ 0 sufficiently small, we may yet again increase the lower bounds on the levels and sublevels of Q(n) and O(n) relative to those derived from the above results:

Theorem 3.15. s (Q(n)) , \underline{s} (Q(n)) , s (O(n)) and \underline{s} (O(n)) ∈ [2^{k}, n], where 2^{k} is the largest 2-power ≤ n.

Proof. In the proof of Theorem 3.5, we showed that ⟨1⟩ ⊥ (2^{k}−1)×T_{P} is anisotropic over F (⟨1⟩ ⊥ 2^{k} × T_{P} ). For n ≥ 2^{k}, ⟨1⟩ ⊥ 2^{k} × T_{P} is a subform of ⟨1⟩ ⊥ n × T_{P}, implying the existence of an F -place from F (⟨1⟩ ⊥ n × T_{P}) to F (⟨1⟩ ⊥ 2^{k} × T_{P} ) by [K, Theorem 3.3]. Hence, ⟨1⟩ ⊥ (2^{k} −1)×T_{P} is anisotropic over F (⟨1⟩ ⊥ n × T_{P}) by [K, Proposition 3.1], implying the result.

Similarly, if n = 2^{k} + h for h ≥ 1 sufficiently small, we may interpret Theorem 3.6 as a further increase to earlier lower bounds on the levels of Q(n) and O(n):

Theorem 3.16. s (Q(n)) and s (O(n)) ∈ [2^{k} + 1, n], where 2^{k} is the largest 2power < n.

Proof. As in the proof of Theorem 3.6, (2^{k} + 1) × ⟨1⟩ ⊥ (2^{k} − 1) × T_{P} is anisotropic over F (⟨1⟩ ⊥ (2^{k} + 1) × T_{P} ). Hence, as in the previous proof, it is also anisotropic over F (⟨1⟩ ⊥ n × T_{P}) for n ≥ 2^{k} + 1.

For certain values of n, the lower bounds obtained for s (Q(n)) , \underline{s} (Q(n)) , s (O(n)) and \underline{s} (O(n)) in the above results are actually optimal with respect to the standard isotropy tests.

For example, if we consider the case where n = 15, we note that s (Q(15)) and \underline{s} (Q(15)) ∈ [12, 15] as a consequence of Theorem 3.13. We cannot hope to show that s (Q(15)) or \underline{s} (Q(15)) ≠ 12 via the standard methodology, since both ⟨1⟩ ⊥ 12 × T_{Q(15)} and 13 × T_{Q(15)} are isotropic over F (⟨1⟩ ⊥ 15 × T_{P}), as the following argument demonstrates:

16 × T_{Q(15)} is clearly isotropic over F (⟨1⟩ ⊥ 15 × T_{P}). [EL, Theorem 1.4] implies that i_{W} (16 × T_{Q(15)}) ≥ 16. Since ⟨1⟩ ⊥ 12 × T_{Q(15)} and 13 × T_{Q(15)} are both subforms of 16 × T_{Q(15)} of codimension < 16, they too are isotropic over F (⟨1⟩ ⊥ 15 × T_{P}).

Hence, more refined quadratic form theoretic consequences of the sublevels of Q(n) and O(n) being bound above by n − 1 are required, if we hope to prove Conjecture 3.2.

3.2 Constructions for the case where n = l · 2^{k}

As outlined in Section 3.1.1, if we seek to construct quaternion and octonion algebras of level and sublevel n for all values of n, the most natural candidates to consider are Q(n) and O(n). However, if we restrict our attention to certain values of n, other constructions appear to be equally, if not more, appropriate.

As a first step in arriving at our constructions of Q(n) and O(n), we observed that the isotropy of the quadratic form ⟨1⟩ ⊥ n × T_{P} placed an upper bound on the level and sublevel of these algebras. Of course, it is possible to find other, though admittedly less natural, quadratic forms whose isotropy encodes such an upper bound on the level and sublevel of associated quaternion or octonion algebras.

As highlighted earlier, one such quadratic form is (2^{k} + 1) × ⟨1⟩ ⊥ (2^{k} − 1) × T_{P}, whose isotropy implies that s(C) ≤ 2^{k} ([Le, Theorem 2.2] for the Q case and [Pu, Proposition 2.12] for the O case). This result, and its method of proof, admit the following generalisation:

Theorem 3.17. If (2^{h} + 1) × ⟨1⟩ ⊥ (l · 2^{k} − 1) × T_{P} is isotropic, where h denotes the 2-adic order of l · 2^{k}, then s(C) ≤ l · 2^{k}.

Proof. Consider the case where C is a quaternion algebra. If (2^{h} + 1) × ⟨1⟩ ⊥ (l · 2^{k} − 1) × T_{P} is isotropic, then there exists −A ∈ D_{F} (⟨1⟩ ⊥ (l · 2^{k} − 1) × T_{P} ) for some nonzero A ∈ D_{F} (2^{h} × ⟨1⟩) . Hence for some α ∈ F and B, C, D ∈ D_{F} ((l · 2^{k} − 1) × ⟨1⟩) ∪ {0} we have −A = α^{2} + aB + bC − abD, that is −1 = \frac{1}{A^{2}} (α^{2}A + aAB + bAC − abAD). Let A = ∑_{λ=1}^{2^{h}} x_{λ}^{2}. We show that there exist y_{λ} ∈ F such that ∑_{λ=1}^{l·2^{k}} y_{λ}^{2} = AB and x⃗ · y⃗ = 0, where x⃗ · y⃗ represents the scalar product of the two vectors, and we take x_{2^{h}+1}, …, x_{l·2^{k}} = 0 if 2^{h} < l · 2^{k}. If B = 0, let each y_{λ} = 0. If B ≠ 0, then ⟨A, AB⟩ ≅ A · ⟨1, B⟩ is a subform of A · (l · 2^{k}) × ⟨1⟩ ≅ (l · 2^{k}) × ⟨1⟩, since B ∈ D_{F} ((l · 2^{k} − 1) × ⟨1⟩) and A · 2^{h} × ⟨1⟩ ≅ 2^{h} × ⟨1⟩. Therefore such a y⃗ exists. Similarly z⃗, w⃗ exist such that ∑_{λ=1}^{l·2^{k}} z_{λ}^{2} = AC, ∑_{λ=1}^{l·2^{k}} w_{λ}^{2} = AD and x⃗ · z⃗ = x⃗ · w⃗ = 0. It follows that ∑_{λ=1}^{l·2^{k}} (\frac{αx_{λ}}{A} + \frac{y_{λ}}{A} i + \frac{z_{λ}}{A} j + \frac{w_{λ}}{A} k)^{2} = \frac{1}{A^{2}} (α^{2}A + aAB + bAC − abAD) = −1. Therefore s(C) ≤ l · 2^{k}.

The same method of proof applies in the octonionic case.

As the above form bounds the level of composition algebras, we may adapt our constructions of Q(n) and O(n) by hardwiring in the isotropy of this form as opposed to that of ⟨1⟩ ⊥ n × T_{P}. We introduce the following notation to signify this alteration: let

Q′ := (\frac{x, y}{F} ) ⊗_{F} F ((2^{h} + 1) × ⟨1⟩ ⊥ (l · 2^{k} − 1) × T_{P} )

and

O′ := (\frac{x, y, z}{F} ) ⊗_{F} F ((2^{h} + 1) × ⟨1⟩ ⊥ (l · 2^{k} − 1) × T_{P} ) ,

where h is the 2-adic order of l · 2^{k}, T_{P} is respectively the pure trace form of (\frac{x,y}{F} ) and (\frac{x,y,z}{F} ) and F denotes F_{0}(x, y) and F_{0}(x, y, z) respectively.

By design, the level of these constructions is bounded above by l · 2^{k}. There is no a priori reason why the level or sublevel of either Q′ or O′ might be strictly less than l · 2^{k} for any values of l or k. Moreover, given the general fashion of these constructions, we propose that Q′ and O′ are actually of level and sublevel l · 2^{k}:

Conjecture 3.18. s (Q′) = \underline{s} (Q′) = s (O′) = \underline{s} (O′) = l · 2^{k} for all l and k.

In the case where l is a 2-power, Q′ represents a construction of Laghribi and Mammone (see [LM]), with O′ coinciding with one of Pumplün (see [Pu]), allowing us to conclude that s(Q′) = s(O′) = 2^{k}. Moreover, we yield that \underline{s}(Q′) = \underline{s}(O′) = 2^{k} immediately for k ≥ 2, by [Le, Theorem 2.5] for the Q case and [O’S, Theorem 3.11] for the O case, with an ad hoc argument proving the case where k = 1.

Since the conjecture is true in the above case, we will restrict our attention to the situation where l is not a 2-power. Moreover, we will henceforth assume without loss of generality that l is odd, whereby k is the 2-adic order of l · 2^{k}.

Having introduced Q′ and O′, it is reasonable to wonder why we might consider these algebras as opposed to Q(l·2^{k}) and O(l·2^{k}) in seeking quaternion and octonion algebras of level and sublevel l · 2^{k}.

We begin our justification of this preference by noting that the dimension of (2^{k} + 1) × ⟨1⟩ ⊥ (l · 2^{k} − 1) × T_{P} is greater than that of ⟨1⟩ ⊥ (l · 2^{k}) × T_{P} for k ≥ 1 in the quaternionic case, and for k ≥ 2 in the octonionic case. Clearly, if we can show that i_{1} ((2^{k} + 1) × ⟨1⟩ ⊥ (l · 2^{k} − 1) × T_{P} ) = 1, then (2^{k} + 1) × ⟨1⟩ ⊥ (l · 2^{k} − 1) × T_{P} will be of greater essential dimension than ⟨1⟩ ⊥ (l·2^{k})×T_{P} for k as above. Theorem 2.3 would then allow for the bounding of s(Q′) and s(O′) in a smaller interval than that in which we can currently place s(Q(n)) and s(O(n)). The suggestion that i_{1} ((2^{k} + 1) × ⟨1⟩ ⊥ (l · 2^{k} − 1) × T_{P} ) = 1 is an entirely reasonable one. Consider the form φ ≃ π ⊗ ρ, with π isometric to an n-fold Pfister form where n ≥ 1. Invoking [EL, Theorem 1.4], we yield i_{1} (φ) ≥ dim π. Theorem 2.3 implies that for φ′ a subform of φ whose codimension is strictly less than dim π, we have i_{1} (φ′) > 1. Hence, we know that the class of forms represented by φ′ have first Witt index strictly greater than 1. We suspect that such forms might actually be the only forms whose first Witt index is strictly greater than 1. The following theorem, which constitutes the main result of this section, shows that the first Witt index of (2^{k} + 1) × ⟨1⟩ ⊥ (l · 2^{k} − 1) × T_{P} does indeed equal 1, as suspected: Theorem 3.19. For Q′ and O′ as above, i_{1} ((2^{k} + 1) × ⟨1⟩ ⊥ (l · 2^{k} − 1) × T_{P} ) = 1.

Proof. For ease of notation, we let ψ_{⟨x⟩} and ψ_{T_{P}} denote (2^{k}+1)×⟨1⟩ ⊥ (l·2^{k}−1)×⟨x⟩ and (2^{k} + 1) × ⟨1⟩ ⊥ (l · 2^{k} − 1) × T_{P} respectively. Let us first consider the quaternion algebra case. Suppose there exist k and l such that i_{1} (ψ_{T_{P}}) > 1. Hence 2^{k} × ⟨1⟩ ⊥ (l · 2^{k} − 1) × T_{P} is isotropic over F (ψ_{T_{P}}), since it is a 1-codimensional subform of ψ_{T_{P}}. Since ψ_{T_{P}} is isotropic over F (ψ_{⟨x⟩}), there exists an F -place from F (ψ_{T_{P}}) to F (ψ_{⟨x⟩}) by [K, Theorem 3.3]. Thus 2^{k} × ⟨1⟩ ⊥ (l·2^{k}−1)×T_{P} is isotropic over F (ψ_{⟨x⟩}) by [K, Proposition 3.1]. Springer’s Theorem implies that either 2^{k} × ⟨1⟩ ⊥ (l · 2^{k} − 1) × ⟨x⟩ or (l · 2^{k} − 1) × ⟨1, −x⟩ is isotropic over F_{0}(x) (ψ_{⟨x⟩}). Suppose (l·2^{k} −1)×⟨1, −x⟩ becomes isotropic over F_{0}(x) (ψ_{⟨x⟩}). Hence 2^{n} ×⟨1, −x⟩ becomes hyperbolic over F_{0}(x) (ψ_{⟨x⟩}), where 2^{n} > l · 2^{k} − 1. The Cassels-Pfister Subform Theorem implies that ψ_{⟨x⟩} is a subform of 2^{n} × ⟨1, −x⟩, which is clearly false. Thus 2^{k} × ⟨1⟩ ⊥ (l · 2^{k} − 1) × ⟨x⟩ must be isotropic over F_{0}(x) (ψ_{⟨x⟩}). Hence, dim_{es} (2^{k} × ⟨1⟩ ⊥ (l · 2^{k} − 1) × ⟨x⟩) = dim_{es} (ψ_{⟨x⟩}) by Theorem 2.3. Consider 2^{k}×⟨1⟩ ⊥ l·2^{k}×⟨x⟩ ≃ 2^{k}×⟨1⟩⊗(⟨1⟩ ⊥ l×⟨x⟩). Now i_{1} (2^{k} × ⟨1⟩ ⊥ l · 2^{k} × ⟨x⟩) ≥ 2^{k} by [EL, Theorem 1.4]. In addition, choosing an ordering P of F_{0}(x) such that x is negative implies that i_{1} (2^{k} × ⟨1⟩ ⊥ l · 2^{k} × ⟨x⟩) ≤ 2^{k} by [ELW, Theorem 3.5 and Remark 3.6]. Thus i_{1} (2^{k} × ⟨1⟩ ⊥ l · 2^{k} × ⟨x⟩) = 2^{k}, implying that 2^{k} × ⟨1⟩ ⊥ (l · 2^{k} − 1) × ⟨x⟩ is isotropic over F_{0} (2^{k} × ⟨1⟩ ⊥ l · 2^{k} × ⟨x⟩) and hence that dim_{es} (2^{k} × ⟨1⟩ ⊥ l · 2^{k} × ⟨x⟩) = dim_{es} (2^{k} × ⟨1⟩ ⊥ (l · 2^{k} − 1) × ⟨x⟩) by Theorem 2.3.

We may therefore conclude that i_{1} (ψ_{⟨x⟩}) = 2^{k}. Consequently (2^{k}+1)×⟨1⟩ ⊥ ((l−1)· 2^{k})×⟨x⟩ is isotropic over F_{0} (ψ_{⟨x⟩}), since it is a subform of ψ_{⟨x⟩} of codimension 2^{k}−1. Invoking Theorem 2.3 yields that i_{1} ((2^{k} + 1) × ⟨1⟩ ⊥ ((l − 1) · 2^{k}) × ⟨x⟩) = 1.

However l is odd, since 2^{k} is the highest 2-power dividing l · 2^{k}, whereby 2^{k+1} divides (l − 1) · 2^{k}. Thus 2^{k+1} × ⟨1⟩ ⊥ ((l − 1) · 2^{k}) × ⟨x⟩ ≃ 2^{k+1} × ⟨1⟩ ⊗ (⟨1⟩ ⊥ \frac{l−1}{2} × ⟨x⟩). Now i_{1} (2^{k+1} × ⟨1⟩ ⊥ ((l − 1) · 2^{k}) × ⟨x⟩) = 2^{k+1} by [EL, Theorem 1.4] and [ELW, Theorem 3.5 and Remark 3.6]. Hence (2^{k} + 1) × ⟨1⟩ ⊥ ((l − 1) · 2^{k}) × ⟨x⟩ becomes isotropic over F_{0} (2^{k+1} × ⟨1⟩ ⊥ ((l − 1) · 2^{k}) × ⟨x⟩), whereby dim_{es} (2^{k+1} × ⟨1⟩ ⊥ ((l − 1) · 2^{k}) × ⟨x⟩) = dim_{es} ((2^{k} + 1) × ⟨1⟩ ⊥ ((l − 1) · 2^{k}) × ⟨x⟩) by Theorem 2.3. Therefore we conclude that i_{1} ((2^{k} + 1) × ⟨1⟩ ⊥ ((l − 1) · 2^{k}) × ⟨x⟩) = 2^{k} + 1, contradicting the supposition that 2^{k} × ⟨1⟩ ⊥ (l · 2^{k} − 1) × ⟨x⟩ is isotropic over F_{0}(x) (ψ_{⟨x⟩}).

Hence 2^{k} × ⟨1⟩ ⊥ (l · 2^{k} − 1) × T_{P} is in fact anisotropic over F (ψ_{T_{P}}), contradicting the supposition that there exist k and l such that i_{1} (ψ_{T_{P}}) > 1.

Next, we prove the result for O′. In this case, we let ψ_{⟨x,y,−xy⟩} denote (2^{k}+1)×⟨1⟩ ⊥ (l · 2^{k} − 1) × ⟨x, y, −xy⟩ and ψ_{T_{P}} be defined as above, with T_{P} denoting the pure trace form of O′ here.

Again we suppose that there exist k and l such that i_{1} (ψ_{T_{P}}) > 1. Hence 2^{k} × ⟨1⟩ ⊥ (l · 2^{k} − 1) × T_{P} is isotropic over F (ψ_{T_{P}}), since it is a 1-codimensional subform of ψ_{T_{P}}. Since ψ_{T_{P}} is isotropic over F (ψ_{⟨x,y,−xy⟩}), there exists an F -place from F (ψ_{T_{P}}) to F (ψ_{⟨x,y,−xy⟩}) by [K, Theorem 3.3]. Thus 2^{k} × ⟨1⟩ ⊥ (l · 2^{k} − 1) × T_{P} is isotropic over F (ψ_{⟨x,y,−xy⟩}) by [K, Proposition 3.1]. Springer’s Theorem implies that either 2^{k} × ⟨1⟩ ⊥ (l · 2^{k} − 1) × ⟨x, y, −xy⟩ or (l · 2^{k} − 1) × ⟨1, −x, −y, xy⟩ is isotropic over F_{0}(x, y) (ψ_{⟨x,y,−xy⟩}).

Suppose (l · 2^{k} − 1) × ⟨1, −x, −y, xy⟩ becomes isotropic over F_{0}(x, y) (ψ_{⟨x,y,−xy⟩}). Hence 2^{n}×⟨1, −x, −y, xy⟩ becomes hyperbolic over F_{0}(x, y) (ψ_{⟨x,y,−xy⟩}), where 2^{n} > l · 2^{k} − 1. The Cassels-Pfister Subform Theorem implies that ψ_{⟨x,y,−xy⟩} is a subform of 2^{n} × ⟨1, −x, −y, xy⟩, which is clearly false.

Thus 2^{k} × ⟨1⟩ ⊥ (l · 2^{k} − 1) × ⟨x, y, −xy⟩ must be isotropic over F_{0}(x, y) (ψ_{⟨x,y,−xy⟩}).

However, we know that i_{1} (ψ_{⟨x,y,−xy⟩}) = 1 (this is what we proved in the first half of the proof), whereby dim_{es} (ψ_{⟨x,y,−xy⟩}) > dim_{es} (2^{k} × ⟨1⟩ ⊥ (l · 2^{k} − 1) × ⟨x, y, −xy⟩). Hence, 2^{k} × ⟨1⟩ ⊥ (l · 2^{k} − 1) × ⟨x, y, −xy⟩ is anisotropic over F_{0}(x, y) (ψ_{⟨x,y,−xy⟩}) by Theorem 2.3, contradicting the supposition that there exist k and l such that i_{1} (ψ_{T_{P}}) > 1.

Since the above argument relies upon the fact that l is an odd number > 1, we cannot hope to apply it to determine the first Witt index of (2^{k} + 1) × ⟨1⟩ ⊥ (2^{k} − 1) × T_{P}. However, we recall that [LM, Proposition 3.4] and [Pu, Proposition 3.5] state that 2^{k}×⟨1⟩ ⊥ (2^{k}−1)×T_{P} remains anisotropic over F ((2^{k} + 1) × ⟨1⟩ ⊥ (2^{k} − 1) × T_{P} ), where T_{P} respectively represents the pure trace form of (\frac{x,y}{F} ) and (\frac{x,y,z}{F} ) and F respectively denotes F_{0}(x, y) and F_{0}(x, y, z). Since 2^{k} × ⟨1⟩ ⊥ (2^{k} − 1) × T_{P} is a 1-codimensional subform of (2^{k} + 1) × ⟨1⟩ ⊥ (2^{k} − 1) × T_{P}, Theorem 2.3 allows us to conclude that i_{1} ((2^{k} + 1) × ⟨1⟩ ⊥ (2^{k} − 1) × T_{P} ) = 1, for all k. Hence, we can incorporate into Theorem 3.19 the case where l is a 2-power, whereby we obtain:

i_{1} ((2^{h} + 1) × ⟨1⟩ ⊥ (l · 2^{k} − 1) × T_{P} ) = 1 for all values of l.

As alluded to earlier, as an immediate consequence of Theorem 3.19 we may increase the lower bounds on the level of our prospective composition algebras of level l · 2^{k}. As dim_{es} ((2^{k} + 1) × ⟨1⟩ ⊥ (l · 2^{k} − 1) × T_{P} ) > dim_{es} (⟨1⟩ ⊥ (l · 2^{k}) × T_{P} ) for k ≥ 1 in the quaternionic case, and for k ≥ 2 in the octonionic case, employing Theorem 2.3 allows us to obtain such a reduction.

For example, we know that \underline{s}(Q(24)) and s (Q(24)) ∈ [18, 24] by Theorem 3.9 and Theorem 3.11, whereas for l = k = 3 we can say that \underline{s}(Q′) and s(Q′) ∈ [20, 24]:

Since 20×T_{Q′} ≃ 4×⟨1⟩⊗(5 × T_{Q′}), i_{1} (20 × T_{Q′}) ≥ 4 by [EL, Theorem 1.4]. Hence, dim_{es} (20 × T_{Q′}) ≤ 76. Theorem 3.19 implies that dim_{es} (9 × ⟨1⟩ ⊥ 23 × T_{P}) = 77. Hence, 20 × T_{Q′} remains anisotropic over F (9 × ⟨1⟩ ⊥ 23 × T_{P}) by Theorem 2.3, whereby \underline{s}(Q′) ∈ [20, 24].

Similarly, while we currently cannot make any stronger statement regarding Q(48) than \underline{s} (Q(48)) and s (Q(48)) ∈ [36, 48], we have that \underline{s} (Q′) and s(Q′) ∈ [40, 48] for l = 3 and k = 4.

As previously noted, Conjecture 3.18 is true in the case where l is a power of 2. As an additional application of Theorem 3.19, we conclude by resolving one further case of this conjecture:

Theorem 3.20. For l = 3 and k = 1, s(O′) = \underline{s}(O′) = l · 2^{k} = 6.

Proof. s(O′) ≤ l ·2^{k} = 6 by construction. Suppose \underline{s}(O′) ≤ 5. Then, as in the proof of Theorem 3.3, either 6 × T_{P} or 6 × (⟨1⟩ ⊥ β) is isotropic over F (3 × ⟨1⟩ ⊥ 5 × T_{P}), where β is some 5-dimensional subform of T_{P}. Hence, either dim_{es}(6 × T_{P}) or dim_{es}(6 × (⟨1⟩ ⊥ β)) ≥ dim_{es}(3 × ⟨1⟩ ⊥ 5 × T_{P}) by Theorem 2.3. However, dim_{es}(6 × T_{P}) = 32 and dim_{es}(6 × (⟨1⟩ ⊥ β)) ≤ 35, whereas dim_{es}(3 × ⟨1⟩ ⊥ 5 × T_{P}) = 37 by Theorem 3.19, yielding a contradiction.

Acknowledgements

I would like to thank Detlev Hoffmann for communicating his recent work, which made me aware of some of the latest techniques and opened up new avenues of approach. I am hugely indebted to the hospitality of Jean-Pierre Tignol and Jan Van Geel for stimulating and enjoyable stays at Université catholique de Louvain and Universiteit Gent, during which most of the above work took place. In addition, I am very grateful to my supervisor, Thomas Unger, for all the valuable suggestions he made regarding this article. Useful comments from Susanne Pumplün were also appreciated. Finally, I gratefully acknowledge the support of my research by the Irish Research Council for Science, Engineering and Technology (scholarship RS Aug/2002/174-7).

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James O’Shea, School of Mathematical Sciences, Science Centre, University College Dublin, Belfield, Dublin 4, Ireland. james.oshea@ucd.ie