ON QUADRATIC FORMS ISOTROPIC OVER THE FUNCTION FIELD OF A CONIC

MARKUS ROST

Consider a quadratic extension L = F (\sqrt{a}) of a field F (Char F ≠ 2). The behaviour of quadratic forms over F under base extension φ → φ_{L} is well understood, since any anisotropic form φ is isomorphic to ψ ⊥ ⟨1, −a⟩ϱ with forms ψ, ϱ over F such that ψ_{L} is anisotropic [S, 2. Sect. 5]. This implies that the anisotropic part (φ_{L})_{an} of φ_{L} is isomorphic to ψ_{L} and therefore already defined over F and that if φ is anisotropic and φ_{L} is hyperbolic then φ is a multiple of ⟨1, −a⟩.

Now let K be the function field of a conic, i.e. K is for some a, b ∈ F^{∗} isomorphic to the fraction field of R = F [s, t]/(s^{2} − at^{2} − b). Then K is the universal splitting field of the form ⟨1, −a, −b⟩ and in view of the decomposition mentioned above it is natural to ask whether a form φ which becomes isotropic over K contains a subform similar to ⟨1, −a, −b⟩. This however is not true in general; see [L, Sect. 6] for further information. The purpose of this note is to prove

Proposition. Let φ be a form over F. Then there exist a number p, forms φ_{i}, ψ_{i} (i = 0, …, p) and elements c_{i} ∈ F^{∗} (i = 0, …, p − 1) such that φ = φ_{0} and

i) φ_{i} ≃ c_{i}⟨1, −a⟩ ⊥ ψ_{i}, i = 0, …, p − 1; ii) φ_{i+1} ≃ c_{i}b⟨1, −a⟩ ⊥ ψ_{i}, i = 0, …, p − 1; iii) ((φ_{p})_{K})_{an} ≅ ((φ_{p})_{an})_{K}.

This proposition shows that the extension K|F has similar splitting properties as the quadratic extension L|F:

Corollary. Let φ be a form over F. Then there exists a form ψ over F such that (φ_{K})_{an} is isomorphic to ψ_{K}. If φ is anisotropic and φ_{K} is hyperbolic then φ is a multiple of ⟨1, −a, −b, ab⟩.

The first statement of the corollary has been proved by Arason in [ELW, Appendix II]; it follows from the proposition by taking ψ = (φ_{p})_{an}, since all φ_{i} are isomorphic over K. The second statement is well known, see e.g. [A, Sect. 2] or [S, 4. Sect. 5]; it is a consequence of the proposition and Witt cancellation. The method of proof presented here is direct and constructive and might indicate a way to handle the extension K|F also for other questions.

Note that R = F [t] ⊕ sF [t] as F -vector space. Define d: R → ℕ ∪ {−∞} by

d(P + sQ) = max{deg P, 1 + deg Q} for P, Q ∈ F [t]

(here deg 0 = −∞). Moreover let R_{n} = { r ∈ R | d(r) ≤ n }. R_{n} is a F -vector subspace of R and one has R_{0} = F and R_{n} · R_{m} ⊂ R_{n+m}.

Lemma. Let φ: V → F be an anisotropic form and suppose that for some n ≥ 1 there exist

v ∈ (V ⊗_{F} R_{n}) \ (V ⊗_{F} R_{n−1})

The text appeared as: Math. Ann. 288 (1990), no. 3, 511–513.

such that φ(v) = 0 ∈ R.

Then there exists a subspace L ⊂ V of dimension 2 such that

1) φ|L ≃ c⟨1, −a⟩ for some c ∈ F^{∗},

2) there exists a nonzero ṽ ∈ V ⊗_{F} R_{n−1} such that φ̃(ṽ) = 0, where φ̃ = b(φ|L) ⊥ (φ|W) and W = L^{⊥}.

Proof of the proposition. We use induction on dim φ_{an}. It is clear that we may assume that φ is anisotropic and φ_{K} is isotropic.

Since K is the fraction field of R there exist n ≥ 0 and a nonzero v ∈ V ⊗_{F} R_{n} such that φ(v) = 0. We proceed by induction on n. If n = 0, then v ∈ V and φ would be isotropic over F; hence n ≥ 1. We may assume v ∉ V ⊗_{F} R_{n−1} and we take φ_{1} = φ̃ where φ̃ is the form in the lemma. If φ̃ is anisotropic we apply the induction hypothesis for n − 1 and if φ̃ is isotropic we apply the induction hypothesis for dim φ̃_{an} < dim φ. In any case we find forms φ̃ = φ̃_{0}, φ̃_{1}, …, φ̃_{p} as in the proposition and φ = φ_{0}, φ_{i} = φ̃_{i−1} (i = 1, …, p + 1) is a sequence as required.

In order to prove the lemma we write

v = v_{0} + ∑_{i=1}^{n} v_{i}st^{i−1} + w_{i}t^{i}; v_{i}, w_{i} ∈ V.

Claim. ⟨v_{n}, w_{n}⟩_{φ} = 0 and φ(w_{n}) = −aφ(v_{n}).

Proof of the claim:

0 = φ(v) mod R_{2n−1}
= φ(v_{n})s^{2}t^{2(n−1)} + 2⟨v_{n}, w_{n}⟩_{φ}st^{2n−1} + φ(w_{n})t^{2n} mod R_{2n−1}
= (φ(v_{n})a + φ(w_{n}))t^{2n} + 2⟨v_{n}, w_{n}⟩_{φ}st^{2n−1} mod R_{2n−1}.

The claim follows since t^{2n} and st^{2n−1} define F -independent vectors of R/R_{2n−1}.

Note that v_{n} ≠ 0 and w_{n} ≠ 0 since v ∉ V ⊗_{F} R_{n−1} and φ is anisotropic. Let

L = F [z]/(z^{2} − a)

and let α ∈ L^{∗} be the class of z. We identify L with ⟨v_{n}, w_{n}⟩_{F} ⊂ V by 1 → v_{n} and α → w_{n}. Then the claim shows that φ|L = cN_{L|F} with c = φ(v_{n}) and N_{L|F} : L → F, e + αf → e^{2} − af^{2} the norm form.

Now write v = x + y with x ∈ L ⊗_{F} R and y ∈ W ⊗_{F} R, W = L^{⊥}. Then x ∈ (s + tα)t^{n−1} + L ⊗_{F} R_{n−1} and y ∈ W ⊗_{F} R_{n−1}.

Put ṽ = b^{−1}(s − tα)x + y. Then ṽ is a zero of the form φ̃ = b(φ|L) ⊥ (φ|W), since

bφ(b^{−1}(s − tα)x) = bcN_{L|F} (b^{−1}(s − tα)x) = bcb^{−2}(s^{2} − at^{2})N_{L|F} (x)
= cN_{L|F} (x) = φ(x).

It remains to show that ṽ ∈ V ⊗_{F} R_{n−1}. In order to do this we have to show that (s − tα)x ∈ L ⊗_{F} R_{n−1}.

Case I: n ≥ 2. Then there exist μ, λ ∈ L and x̃ ∈ L ⊗_{F} R_{n−2} such that

x = (s + tα)t^{n−1} + (s + tα)t^{n−2}μ + t^{n−1}λ + x̃

We have with ω = t^{n−1} + t^{n−2}μ and λ̄ the conjugate of λ under α → −α:

0 = φ(v) mod R_{2n−2} = φ(x) mod R_{2n−2}
= cN_{L|F} ((s + tα)ω + t^{n−1}λ) mod R_{2n−2}
= c[N_{L|F} ((s + tα)ω) + tr_{L|F} ((s + tα)ωt^{n−1}λ̄) + N_{L|F} (t^{n−1}λ)] mod R_{2n−2}
= c[b · N_{L|F} (ω) + tr_{L|F} ((s + tα)t^{2n−2}λ̄) + 0] mod R_{2n−2}
= c[0 + st^{2n−2} tr_{L|F} λ̄ + t^{2n−1} tr_{L|F} (αλ̄)] mod R_{2n−2}

Hence the traces of λ̄ and αλ̄ are zero and therefore λ = 0. Finally:

(s − tα)x = bω + (s − tα)x̃ ∈ L ⊗_{F} R_{n−1}.

Case II: n = 1. Then x = s + tα + λ for some λ ∈ L and it suffices to show λ = 0. However

0 = φ(v) = b + s tr_{L|F} λ̄ + t tr_{L|F} αλ̄ + N_{L|F} (λ) + φ(y)

and therefore again λ = 0 since φ(y) ∈ F.

References

[A] Arason, J. Kr.: Cohomologische Invarianten quadratischer Formen. J. Algebra 36, 448–491 (1975).

[ELW] Elman, R., Lam, T. Y., Wadsworth, A. R.: Amenable fields and Pfister extensions. Conference on quadratic forms 1976, Queen’s pap. Pure Appl. Math. 46, 445–491 (1977).

[L] Lam, T. Y.: Fields of u-invariant 6 after A. Merkuriev. In: “Ring theory 1989” in honour of S. A. Amitsur, Isr. Conf. Proc., vol. 1, pp. 12–30.

[S] Scharlau, W.: Quadratic and Hermitian forms. Berlin-Heidelberg-New York: Springer 1985.

NWF I - Mathematik, Universität Regensburg, D-93040 Regensburg, Germany

   E-mail address: markus.rost@mathematik.uni-regensburg.de
   URL: http://www.physik.uni-regensburg.de/~rom03516