Some Applications
The theory of weak specialization, developed in §1.3, §1.7 and the end of §2.3, has until now played only an auxiliary role, which we could have done without when dealing with quadratic forms (due to Theorem 2.19). For the first time we now come to indepe
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Some Applications
3.1 Subforms which have Bad Reduction The theory of weak specialization, developed in §1.3, §1.7 and the end of §2.3, has until now played only an auxiliary role, which we could have done without when dealing with quadratic forms (due to Theorem 2.19). For the first time we now come to independent applications of weak specialization. Let λ : K → L ∪ ∞ be a place and ϕ a bilinear or quadratic form over K having good reduction with respect to λ. If ϕ contains subforms which have bad reduction, there are consequences for the form λ∗ (ϕ) which we will discuss now. We will look at bilinear forms first. Theorem 3.1. Let ϕ be a nondegenerate bilinear form over K, having good reduction with respect to λ. Suppose that there exists a nondegenerate subform ∗ b1 , . . . , bm of ϕ with λ(bi c2 ) = 0 or ∞ for every . m i ∈ {1, . . . , m} and every c ∈ K m m Then λ∗ (ϕ) has Witt index ≥ 2 . (As usual, 2 denotes the smallest integer ≥ 2 .) Proof. Let ψ := b1 , . . . , bm . We have a decomposition ϕ ψ ⊥ η, to which we apply the additive map λW : W(K) → W(L). We get λW (ψ) = 0.1 Hence {λ∗ (ϕ)} = class λW (ϕ) = λW (η). Let ρ be the anisotropic form in the Witt λW (η). Then λ∗ (ϕ) ∼ ρ and dim ρ ≤ dim η = dim ϕ − m. Therefore ind λ∗ (ϕ) ≥ m2 . Corollary 3.2. Let ϕ be a nondegenerate bilinear form over K, having good reduction with respect to λ. Suppose that λ∗ (ϕ) is anisotropic. Then every subform of ϕ is nondegenerate and has good reduction with respect to λ. Proof. Clearly ϕ is now also anisotropic. Therefore every subform of ϕ is anisotropic and so definitely nondegenerate. By Theorem 3.1, ϕ does not contain any onedimensional subforms b with λ(bc2 ) = 0 or ∞ for every c ∈ K ∗ . Hence every subform of ϕ has good reduction. 1
We use the shorter notation λW (ψ) instead of λW ({ψ}), cf. §1.7.
M. Knebusch, Specialization of Quadratic and Symmetric Bilinear Forms, Algebra and Applications 11, DOI 10.1007/978-1-84882-242-9 3, c Springer-Verlag London Limited 2010
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3 Some Applications
Remark. If char L = 2, then λ∗ (ϕ) is in general only determined by λ and ϕ up to stable isometry. However, if a representative of λ∗ (ϕ) is anisotropic, then λ∗ (ϕ) is uniquely determined up to genuine isometry, since λ∗ (ϕ) is up to isometry the only anisotropic form in the Witt class λW (ϕ) in this case. Next we can derive the following statement, which is similar to the Substitution Principle in §1.3 (Theorem 1.29). Theorem 3.3 ([32, Prop. 3.3]). Let ( fi j (t)) be an n × n-matrix of polynomials fi j (t) ∈ k[t] over an arbitrary field k, where t = (t1 , . . . , tr ) is a set of indeterminates over k. Let g1 (t), . . . , gm (t) be further polynomials in k[t]. Finally, let c = (c1 , . . . , cr ) be an r-tuple of coordinates in a field extension L of k, such that det( fi j (c)) 0, and such that c is a non-singular zero of every polynomial g p (t) (1 ≤ p ≤ m), i.e. g p (c) = 0, ∂g p ∂tq (c) 0 for an element q ∈ {1, . . . , n}, which depends on p. If g1 (t), . . . , gm (t) is a subform of the bilinear
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