FULL SIGNATURE INVARIANTS FOR L0(F (t))

Let F/Q be a number field closed under complex conjugation. Denote by L0(F (t)) the Witt group of hermitian forms over F (t). We find full invariants for detecting non–zero elements in L0(F (t))⊗Q, this group plays an important role in topology in the work done by Casson and Gordon. 1. L-groups and signatures Let R be a ring with (possibly trivial) involution. An –hermitian ( = ±1) form is a sesquilinear map θ : V × V → R over a finitely generated free R–module with the properties that θ(ra, b) = r̄θ(a, b), θ(a, rb) = θ(a, b)r and θ(a, b) = θ(b, a) for all a, b ∈ V, r ∈ R. It is called non–singular if the map V → Hom(V,R), a 7→ (b 7→ θ(b, a)) is an isomorphism. We denote by L0(R, ), = ±1, the Witt group of –hermitian non– singular forms (cf. [L93] and [R98]). More precisely denote by M the groupoid under direct sum of –hermitian non–singular forms. Let ∼ be the equivalence relation generated by setting any form (V, θ) to zero which has a submodule of half–rank on which θ vanishes. Then define L0(R, ) := M/ ∼, this is a group (cf. [L93]) under the direct sum operation. We’ll abbreviate L0(R) for L0(R,+1). Let F ⊂ C be a subfield, closed under complex conjugation. We will always equip the rings F [t, t−1], F (t) with the involution given by the complex involution on F and t̄ := t−1. For τ = [(V, θ)] ∈ L0(F, ) we define sign(τ) := dim(V )− dim(V −) where V + (resp. V −) denotes the maximal positive (resp. negative) subspace of √ θ. This number is independent of the choice of representative (V, θ). If F ⊂ C is a subfield such that all positive elements are squares, then by Sylvester’s theorem sign : L0(F, ) → Z (V, θ) 7→ sign(V, θ) is an isomorphism. In particular L0(C,±1) = L0(Q̄,±1) ∼= Z via the signature map, where we denote by Q̄ ⊂ C the algebraic closure of Q. Since we are interested in studying to which degree signatures determine forms we’ll work in this paper with L̃0(R, ) := L0(R, ) ⊗ Q, i.e. we ignore the torsion part of L0(R, ). Note that the above maps extend to an isomorphism sign : L0(F, )⊗Q→ Q. Date: May 13, 2003. 1