CHAPTER 1 Local Theory 1.1. Some lemmas Our basic notations will be standard if not explained (see [0M1], [Sa] and [Ril]). In this chapter, F denotes a finite field of characteristic 2, F = F((TT)) the formal Laurent series field over F with uniformizer TT and o = -F[[7r]] the ring of formal power series. Let p be the unique non-zero prime ideal of o and u the group of units in o. Write 0 and p the fixed representatives of F/p(F) where the mapping p(x) = x2 + x for any x G F is a homomorphism of the additive group of F. Let ord be the ordinal function in F. A quadratic space V over F means a finite dimensional vector space V over F with a map Q : V — F such that Q(ax) = a2Q(x) MaeF and (x, y) = Q(x + y) + Q(x) + Q(y) is bilinear. Therefore a quadratic space has a sympletic structure as well in char- acteristic 2. We always assume that all quadratic spaces are non-defective which means that x = 0 £ (x,V)=0. For any a G F, Va means the quadratic space of the same vector space with the map Qa scaling by a. For a non-defective quadratic space V, there is a sympletic basis (see [Ar]) of V such that (e», fj) = Sij, (ei, ej) = 0 and (/», /,) = 0. This implies that the dimension of a non-defective quadratic space is always even. One can define the Arf invariant A(V) of V (see [Ar]) as n A(V) = Yl^e^(^ in F MF) which is well-defined and independent of the choice of the sympletic basis (see [Ar]). Let Ord(A(V)) = max{ ord(x) : x e A(V)}. By [Sa, Lemma 1.1], one has Ord(A(V)) = oc,0 or a negative odd integer. A lattice L in V means a finitely generated o-module in V such that FL is a non-defective quadratic space. A vector x G L is called primitive if x can be extended as a basis of L. A lattice ox + oy of rank two is denoted by (a, b)r if Q(x) = a, Q(y) = b and (x, y) = 7rr. l

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