The u-invariant of the function fields of p-adic curves

R. Parimala and V. Suresh

Abstract

The u-invariant of a field is the maximum dimension of ansiotropic quadratic forms over the field. It is an open question whether the u-invariant of function fields of p-aidc curves is 8. In this paper, we answer this question in the affirmative for function fields of non-dyadic p-adic curves.

Introduction

It is an open question ([L], Q. 6.7, Chap XIII) whether every quadratic form in at least nine variables over the function fields of p-adic curves has a nontrivial zero. Equivalently, one may ask whether the u-invariant of such a field is 8. The u-invariant of a field F is defined as the maximal dimension of anisotropic quadratic forms over F. In this paper we answer this question in the affirmative if the p-adic field is non-dyadic.

In ([PS], 4.5), we showed that every quadratic form in eleven variables over the function field of a p-adic curve, p ≠ 2, has a nontrivial zero. The main ingredients in the proof were the following: Let K be the function field of a p-adic curve X and p ≠ 2.

1. (Saltman ([S1], 3.4)) Every element in the Galois cohomology group H^{2}(K, Z/2Z) is a sum of at most two symbols.

2. (Kato [K], 5.2) The unramified cohomology group H_{nr}^{3}(K/X , Z/2Z(2)) is zero for a regular projective model X of K.

If K is as above, we proved ([PS], 3.9) that every element in H^{3}(K, Z/2Z) is a symbol of the form (f) · (g) · (h) for some f, g, h ∈ K^{∗} and f may be chosen to be a value of a given binary form <a, b> over K. If further given ζ = (f) · (g) · (h) ∈ H^{3}(K, Z/2Z) and a ternary form < c, d, e >, one can choose g′, h′ ∈ K^{∗} such that ζ = (f) · (g′) · (h′) with g′ a value of <c, d, e>, then, one is led to the conclusion that u(K) = 8 (cf 4.3). We in fact prove that such a choice of g′, h′ ∈ K^{∗} is possible by proving the following local global principle:

Theorem. Let K = k(X) be the function field of a curve X over a p-adic field k. Let l be a prime not equal to p. Assume that k contains a primitive l^{th} root of unity. Given ζ ∈ H^{3}(K, μ_{l}^{⊗2}) and α ∈ H^{2}(K, μ_{l}) corresponding to a degree l central division algebra over K, satisfying ζ = α ∪ (h_{v}) for some (h_{v}) ∈ H^{1}(K_{v}, μ_{l}), for all discrete valuations of K, there exists (h) ∈ H^{1}(K, μ_{l}) such that ζ = α ∪ (h). In fact one can restrict the hypothesis to discrete valuations of K centered on codimension one points of a regular model X, projective over the ring of integers O_{k} of k.

A key ingredient towards the proof of the theorem is a recent result of Saltman ([S3]) where the ramification pattern of prime degree central simple algebras over function fields of p-adic curves is completely described.

We thank J.-L. Colliot-Thélène for helpful discussions during the preparation of this paper and for his critical comments on the text.

1. Some Preliminaries

In this section we recall a few basic facts from the algebraic theory of quadratic forms and Galois cohomology. We refer the reader to ( [C]) and ([Sc]).

Let F be a field and l a prime not equal to the characteristic of F. Let μ_{l} be the group of l^{th} roots of unity. For i ≥ 1, let μ_{l}^{⊗i} be the Galois module given by the tensor product of i copies of μ_{l}. For n ≥ 0, let H^{n}(F, μ_{l}^{⊗i}) be the n^{th} Galois cohomology group with coefficients in μ_{l}^{⊗i}.

We have the Kummer isomorphism F^{∗}/F^{∗^{l}} ≃ H^{1}(F, μ_{l}). For a ∈ F^{∗}, its class in H^{1}(F, μ_{l}) is denoted by (a). If a_{1}, ⋯, a_{n} ∈ F^{∗}, the cup product (a_{1}) ⋯ (a_{n}) ∈ H^{n}(F, μ_{l}^{⊗n}) is called a symbol. We have an isomorphism H^{2}(F, μ_{l}) with the l-torsion subgroup _{l}Br(F) of the Brauer group of F. We define the index of an element α ∈ H^{2}(F, μ_{l}) to be the index of the corresponding central simple algebra in _{l}Br(F).

Suppose F contains all the l^{th} roots of unity. We fix a generator ρ for the cyclic group μ_{l} and identify the Galois modules μ_{l}^{⊗i} with Z/lZ. This leads to an identification of H^{n}(F, μ_{l}^{⊗m}) with H^{n}(F, Z/lZ). The element in H^{n}(F, Z/lZ) corresponding to the symbol (a_{1}) ⋯ (a_{n}) ∈ H^{n}(F, μ_{l}^{⊗n}) through this identification is again denoted by (a_{1}) ⋯ (a_{n}). In particular for a, b ∈ F^{∗},

(a) · (b) ∈ H^{2}(K, Z/lZ) represents the cyclic algebra (a, b) defined by the relations x^{l} = a, y^{l} = b and xy = ρyx.

Let v be a discrete valuation of F. The residue field of v is denoted by κ(v). Suppose char(κ(v)) ≠ l. Then there is a residue homomorphism ∂_{v} : H^{n}(F, μ_{l}^{⊗m}) → H^{n−1}(κ(v), μ_{l}^{⊗(m−1)}). Let α ∈ H^{n}(F, μ_{l}^{⊗m}). We say that α is unramified at v if ∂_{v}(α) = 0; otherwise it is said to be ramified at v. If F is complete with respect to v, we denote the kernel of ∂_{v} by H_{nr}^{n}(F, μ_{l}^{⊗m}). Suppose α is unramified at v. Let π ∈ K^{∗} be a parameter at v and ζ = α ∪ (π) ∈ H^{n+1}(F, μ_{l}^{⊗(m+1)}). Let \overline{α} = ∂_{v}(ζ) ∈ H^{n}(κ(v), μ_{l}^{⊗m}). The element \overline{α} is independent of the choice of the parameter π and is called the specialization of α at v. We say that α specializes to \overline{α} at v. The following result is well known.

Lemma 1.1 Let k be a field and l a prime not equal to the characteristic of k. Let K be a complete discrete valuated field with residue field k. If H^{n}(k, μ_{l}^{⊗3}) = 0 for n ≥ 3, then H_{nr}^{3}(K, μ_{l}^{⊗3}) = 0. Suppose further that every element in H^{2}(k, μ_{l}^{⊗2}) is a symbol. Then every element in H^{3}(K, μ_{l}^{⊗3}) is a symbol.

Proof. Let R be the ring of integers in K. The Gysin exact sequence in étale cohomology yields an exact sequence (cf. [C], p.21, §3.3)

H_{ét}^{3}(R, μ_{l}^{⊗3}) → H^{3}(K, μ_{l}^{⊗3}) →^{∂} H^{2}(k, μ_{l}^{⊗2}) → H_{ét}^{4}(R, μ_{l}^{⊗3})

Since R is complete, H_{ét}^{n}(R, μ_{l}^{⊗3}) ≃ H^{n}(k, μ_{l}^{⊗3}) ([Mi], p.224 Corollary 2.7). Hence H_{ét}^{n}(R, μ_{l}^{⊗3}) = 0 for n ≥ 3, by the hypothesis. In particular ∂ : H^{3}(K, μ_{l}^{⊗3}) → H^{2}(k, μ_{l}^{⊗2}) is an isomorphism and H_{nr}^{3}(K, μ_{l}^{⊗3}) = 0. Let u, v ∈ R be units and π ∈ R a parameter. Then we have ∂((u) · (v) · (π)) = (\overline{u}) · (\overline{v}). Let ζ ∈ H^{3}(K, μ_{l}^{⊗3}). Since every element in H^{2}(k, μ_{l}^{⊗2}) is a symbol, we have ∂(ζ) = (\overline{u}) · (\overline{v}) for some units u, v ∈ R. Since ∂ is an isomorphism, we have ζ = (u) · (v) · (π). Thus every element in H^{3}(K, μ_{l}^{⊗3}) is a symbol. □

Corollary 1.2 Let k be a p-adic field and K the function field of an integral curve over k. Let l be a prime not equal to p. Let K_{v} be the completion of K at a discrete valuation of K. Then H_{nr}^{3}(K_{v}, μ_{l}^{⊗3}) = 0. Suppose further that K contains a primitive l^{th} root of unity. Then every element in H^{3}(K_{v}, μ_{l}^{⊗3}) is a symbol.

Proof. Let v be a discrete valuation of K and K_{v} the completion of K at v. The residue field κ(v) at v is either a p-adic field or a function field of a curve over a finite field of characteristic p. In either case, the cohomological dimension of κ(v) is 2 and hence H^{n}(κ(v), μ_{l}^{⊗3}) = 0 for n ≥ 3. By (1.1), H_{nr}^{3}(K_{v}, μ_{l}^{⊗3}) = 0.

If κ(v) is a local field, by local class field theory, every finite dimension central division algebra over κ(v) is split by an unramified (cyclic) extension. If κ(v) is a function field of a curve over finite field, then by a classical theorem of Hasse-Brauer-Noether-Albert, every finite dimensional central division algebra over κ(v) is split by a cyclic extension. Since κ(v) contains a primitive l^{th} of unity, every element in H^{2}(κ(v), Z/lZ) is a symbol. By (1.1), every element in H^{3}(K_{v}, Z/lZ) is a symbol. □

Let X be a regular integral scheme of dimension d, with field of fractions F. Let X^{1} be the set of points of X of codimension 1. A point x ∈ X^{1} gives rise to a discrete valuation v_{x} on F. The residue field of this discrete valuation ring is denoted by κ(x) or κ(v_{x}). The corresponding residue homomorphism is denoted by ∂_{x}. We say that an element ζ ∈ H^{n}(F, μ_{l}^{⊗m}) is unramified at x if ∂_{x}(ζ) = 0; otherwise it is said to be ramified at x. We define the ramification divisor ramX (ζ) = ∑ x as x runs over X^{1} where ζ is ramified. The unramified cohomology on X, denoted by H_{nr}^{n}(F/X , μ_{l}^{⊗m}), is defined as the intersection of kernels of the residue homomorphisms ∂_{x} : H^{n}(F, μ_{l}^{⊗m}) → H^{n−1}(κ(x), μ_{l}^{⊗(m−1)}), x running over X^{1}. We say that ζ ∈ H^{n}(F, μ_{l}^{⊗m}) is unramified on X if ζ ∈ H_{nr}^{n}(F/X , μ_{l}^{⊗m}). If X = Spec(R), then we also say that ζ is unramified on R if it is unramified on X. Suppose C is an irreducible subscheme of X of codimension 1. Then the generic point x of C belongs to X^{1} and we set ∂_{x} = ∂_{C}. If α ∈ H^{n}(F, μ_{l}^{⊗m}) is unramified at x, then we say that α is unramified at C.

Let k be a p-adic field and K the function field of a smooth, projective, geometrically integral curve X over k. By the resolution of singularities for surfaces (cf. [Li1] and [Li2]), there exists a regular, projective model X of X over the ring of integers O_{k} of k. We call such an X a regular projective model of K. Since the generic fibre X of X is geometrically integral, it follows that the special fibre \overline{X} is connected. Further if D is a divisor on X, there exists a proper birational morphism X′ → X such that the total transform of D on X′ is a divisor with normal crossings (cf. [Sh], Theorem, p.38 and Remark 2, p. 43). We use this result throughout this paper without further reference.

Let k be a p-adic field and K the function field of a smooth, projective, geometrically integral curve over k. Let l be a prime not equal to p. Assume that k contains a primitive l^{th} of unity. Let α ∈ H^{2}(K, μ_{l}). Let X be a regular projective model of K such that the ramification locus ramX (α) is a union of regular curves with normal crossings. Let P be a closed point in the intersection of two regular curves C and E in ramX (α). Suppose that ∂_{C}(α) ∈ H^{1}(κ(C), Z/lZ) and ∂_{E}(α) ∈ H^{1}(κ(E), Z/lZ) are unramified at P. Let u(P), v(P) ∈ H^{1}(κ(P), Z/lZ) be the specialisations at P of ∂_{C}(α) and ∂_{E}(α) respectively. Following Saltman ([S3], §2), we say that P is a cool point if u(P), v(P) are trivial and a chilly point if u(P) and v(P) generate the same subgroup of H^{1}(κ(P), Z/lZ) and neither of them is trivial. If u(P) and v(P) do not generate the same subgroup of H^{1}(κ(P), Z/lZ), then P is said to be a hot point. Let O_{X ,P} be the regular local ring at P and π, δ prime elements in O_{X ,P} which define C and E respectively at P. The condition that ∂_{C}(α) ∈ H^{1}(κ(C), Z/lZ) and ∂_{E}(α) ∈ H^{1}(κ(E), Z/lZ) are unramified at P is equivalent to the condition α = α′ + (u, π) + (v, δ) for some units u, v ∈ O_{X ,P} and α′ unramified on O_{X ,P} ([S2], §2). The specialisations of ∂_{C}(α) and ∂_{E}(α) in H^{1}(κ(P), Z/lZ) ≃ κ(P)^{∗}/κ(P)^{∗^{l}} are given by the images of u and v in κ(P).

Proposition 1.3 ([S3], 2.5) If the index of α is l, then there are no hot points for α.

Suppose P is a chilly point. Then v(P) = u(P)^{s} for some s with 1 ≤ s ≤ l − 1 and s is called the coefficient of P with respect to π. To get some compatibility for these coefficients, Saltman associates to α and X the following graph: The set of vertices is the set of irreducible curves in ramX (α) and there is an edge between two vertices if there is a chilly point in the intersection of the two irreducible curves corresponding to the vertices. A loop in this graph is called a chilly loop.

Proposition 1.4 ([S3], 2.6, 2.9) There exists a projective model X of K such that there are no chilly loops and no cool points on X for α.

Let F be a field of characteristic not equal to 2. The u-invariant of F, denoted by u(F), is defined as follows:

u(F) = sup{rk(q) | q an anisotropic quadratic form over F }.

For a_{1}, ⋯, a_{n} ∈ F^{∗}, we denote the diagonal quadratic form a_{1}X_{1}^{2}+⋯+a_{n}X_{n}^{2} by <a_{1}, ⋯, a_{n} >. Let W (F) be the Witt ring of quadratic forms over F and I(F) be the ideal of W (F) consisting of even dimension forms. Let I^{n}(F) be the n^{th} power of the ideal I(F). For a_{1}, ⋯, a_{n} ∈ F^{∗}, let <<a_{1}, ⋯, a_{n} >> denote the n-fold Pfister form <1, a_{1} > ⊗ ⋯ ⊗ <1, a_{n} >. The abelian group I^{n}(F) is generated by n-fold Pfister forms. The dimension modulo 2 gives an isomorphism e_{0} : W (F)/I(F) → H^{0}(F, Z/2Z). The discriminant gives an isomorphism e_{1} : I(F)/I^{2}(F) → H^{1}(F, Z/2Z). The classical result of Merkurjev ([M]), asserts that the Clifford invariant gives an isomorphism e_{2} : I^{2}(F)/I^{3}(F) → H^{2}(F, Z/2Z).

Let P_{n}(F) be the set of isometry classes of n-fold Pfister forms over F. There is a well-defined map ([A])

e_{n} : P_{n}(F) → H^{n}(F, Z/2Z)

given by e_{n}(<1, a_{1} > ⊗ ⋯ <1, a_{n} >) = (−a_{1}) ⋯ (−a_{n}) ∈ H^{n}(F, Z/2Z).

A quadratic form version of the Milnor conjecture asserts that e_{n} induces a surjective homomorphism I^{n}(F) → H^{n}(F, Z/2Z) with kernel I^{n+1}(F). This conjecture was proved by Voevodsky, Orlov and Vishik. In this paper we are interested in fields of 2-cohomological dimension at most 3. For such fields, the above Milnor’s conjecture is already proved by Arason, Elman and Jacob ([AEJ], Corollary 4 and Theorem 2), using the theorem of Markurjev ([M]).

Let q_{1} and q_{2} be two quadratic forms over F. We write q_{1} = q_{2} if they represent the same class in the Witt group W (F). We write q_{1} ≃ q_{2}, if q_{1} and q_{2} are isometric quadratic forms. We note that if the dimensions of q_{1} and q_{2} are equal and q_{1} = q_{2}, then q_{1} ≃ q_{2}.

2. Divisors on Arithmetic Surfaces

In this section we recall a few results from a paper of Saltman ([S3]) on divisors on arithmetic surfaces.

Let Z be a connected, reduced scheme of finite type over a Noetherian ring. Let OZ^{∗} be the sheaf of units in the structure sheaf OZ. Let P be a finite set of closed points of Z. For each P ∈ P, let κ(P) be the residue field at P and ι_{P} : Spec(κ(P)) → Z be the natural morphism. Consider the sheaf P^{∗} = ⊕_{P ∈P}ι_{P}^{∗} κ(P)^{∗}, where κ(P)^{∗} denotes the group of units in κ(P). Then there is a surjective morphism of sheaves OZ^{∗} → P^{∗} given by the evaluation at each P ∈ P. Let O_{Z,P}^{∗(1)} be its kernel. When there is no

ambiguity we denote O_{Z,P}^{∗(1)} by OP^{∗(1)}. Let K be the sheaf of total quotient rings on Z and K^{∗} be the sheaf of groups given by units in K. Every element γ ∈ H^{0}(Z, K^{∗}/O^{∗}) can be represented by a family {U_{i}, f_{i}}, where U_{i} are open sets in Z, f_{i} ∈ K^{∗}(U_{i}) and f_{i}f_{j}^{−1} ∈ O^{∗}(U_{i} ∩ U_{j}). We say that an element γ = {U_{i}, f_{i}} of H^{0}(Z, K^{∗}/O^{∗}) avoids P if each f_{i} is a unit at P for all P ∈ U_{i} ∩ P. Let HP^{0}(Z, K^{∗}/O^{∗}) be the subgroup of H^{0}(Z, K^{∗}/O^{∗}) consisting of those γ which avoid P. Let K^{∗} = H^{0}(Z, K^{∗}) and KP^{∗} be the subgroup of K^{∗} consisting of those functions which are units at all P ∈ P. We have a natural inclusion KP^{∗} → HP^{0}(Z, K^{∗}/O^{∗}) ⊕ (⊕_{P ∈P}κ(P)^{∗}).

We have the following

Proposition 2.1 ([S3], 1.6) Let Z be a connected, reduced scheme of finite type over a Noetherian ring. Then

H^{1}(Z, OP^{∗(1)}) ≃ \frac{HP^{0}(Z, K^{∗}/O^{∗}) ⊕ (⊕_{P ∈P}κ(P)^{∗})}{KP^{∗}}.

Let k be a p-adic field and O_{k} the ring of integers of k. Let X be a connected regular surface with a projective morphism η : X → Spec(O_{k}). Let \overline{X} be the reduced special fibre of η. Assume that \overline{X} is connected. Note that \overline{X} is connected if the generic fibre is geometrically integral. Let P be a finite set of closed points in X. Since every closed point of X is in \overline{X}, P is also a subset of closed points of \overline{X}. Let m be an integer coprime with p.

Proposition 2.2 ([S3], 1.7) The canonical map H^{1}(X , O_{X ,P}^{∗(1)}) → H^{1}(\overline{X} , O_{\overline{X} ,P}^{∗(1)}) induces an isomorphism

\frac{H^{1}(X , O_{X ,P}^{∗(1)})}{mH^{1}(X , O_{X ,P}^{∗(1)})} ≃ \frac{H^{1}(\overline{X} , O_{\overline{X} ,P}^{∗(1)})}{mH^{1}(\overline{X} , O_{\overline{X} ,P}^{∗(1)})}.

Let X be as above. Suppose that \overline{X} is a union of regular curves F_{1}, ⋯, F_{m} on X with only normal crossings. Let P be a finite set of closed points of X including all the points of F_{i} ∩ F_{j}, i ≠ j and at least one point from each F_{i}. Let E be a divisor on X whose support does not pass through any point of P. In particular, no F_{i} is in the support of E. Hence there are only finitely many closed points Q_{1}, ⋯, Q_{n} on the support of E. For each closed point Q_{i} on the support of E, let D_{i} be a regular geometric curve (i.e. a curve not contained in the special fibre) on X such that Q_{i} is the multiplicity one intersection of D_{i} and \overline{X}. Such a curve exists by ([S3], 1.1). We note that any closed point on X is a point of codimension 2 and there is a unique closed point on any geometric curve on X (cf. §1).

The following is extracted from ([S3], §5).

Proposition 2.3 Let X , P, E, Q_{i}, D_{i} be as above. For each closed point Q_{i}, let m_{i} be the intersection multiplicity of the support of E and the special fibre \overline{X} at Q_{i}. Let l be a prime not equal to p. Then there exist ν ∈ K^{∗} and a divisor E′ on X such that

(ν) = −E + ∑_{i=1}^{n} m_{i}D_{i} + lE′

and ν(P) ∈ κ(P)^{∗^{l}} for each P ∈ P.

Proof. Let F be the divisor on X given by ∑ F_{i}. Let γ ∈ P ic(X) be the line bundle equivalent to the class of the divisor −E and \overline{γ} ∈ P ic(\overline{X}) its image. Since the support of E does not pass through the points of P and P contains all the points of intersection of distinct F_{i}, E and F intersect only at smooth points of \overline{X}. In particular, we have \overline{γ} = − ∑ m_{i}Q_{i}. Let γ′ ∈ H^{1}(X , OP^{∗}) be the element which, under the isomorphism of (2.1), corresponds to the class of the element (−E + ∑ m_{i}D_{i}, 1) in HP^{0}(X , K^{∗}/O^{∗}) ⊕ (⊕_{P ∈P}κ(P)^{∗}). Since the m_{i}’s are intersection multiplicities of E and \overline{X} at Q_{i} and the image of

∑ m_{i}D_{i} in HP^{0}(\overline{X} , K^{∗}/O^{∗}) is ∑ m_{i}Q_{i}, the image \overline{γ′} of γ′ in H^{1}(\overline{X} , OP^{∗}) is zero.

By (2.2), we have γ′ ∈ lH^{1}(X , OP^{∗}). Using (2.1), there exists (E′, (λ_{P})) ∈

HP^{0}(Z, K^{∗}/O^{∗}) ⊕ (⊕_{P ∈P}κ(P)^{∗}) such that (−E + ∑ m_{i}D_{i}, 1) = l(E′, (λ_{P})) =

(lE′, (λ_{P}^{l})) modulo KP^{∗}. Thus there exists ν ∈ KP^{∗} ⊂ K^{∗} such that (ν) = (−E +∑ m_{i}D_{i}, 1)−(lE′, (λ_{P}^{l})). i.e. (ν) = −E +∑ m_{i}D_{i}−lE′ and ν(P) = λ_{P}^{l} for each P ∈ P. □

3. A local-global principle

Let k be a p-adic field, O_{k} be its ring of integers and K the function field of a smooth, projective, geometrically integral curve over k. Let l be a prime not equal to p. Throughout this section, except in 3.6, we assume that k contains a primitive l^{th} root of unity. We fix a generator ρ for μ_{l} and identify μ_{l} with Z/lZ.

Lemma 3.1 Let α ∈ H^{2}(K, μ_{l}). Let X be a regular projective model of K. Assume that the ramification locus ram_{X} (α) is a union of regular curves {C_{1}, ⋯, C_{r}} with only normal crossings. Let T be a finite set of closed points of X including the points of C_{i} ∩ C_{j}, for all i ≠ j. Let D be an irreducible curve on X which is not in the ramification locus of α and does not pass through any point in T. Then D intersects C_{i} at points P where ∂_{C_{i}}(α) is unramified. Suppose further that at such points P, ∂_{C_{i}}(α) specializes to 0 in H^{1}(κ(P), Z/lZ). Then α is unramified at D and specializes to 0 in H^{2}(κ(D), μ_{l}).

Proof. Since k contains a primitive l^{th} of unity, we fix a generator ρ for μ_{l} and identify the Galois modules μ_{l}^{⊗j} with Z/lZ. Let P be a point in the intersection of D and the support of ramX (α). Since D does not pass through the points of T and T contains all the points of intersection of distinct C_{j}, the point P belongs to a unique curve C_{i} in the support of ramX (α). We have ([S1], 1.2) α = α′ + (u, π), where α′ is unramified on O_{X ,P}, u ∈ O_{X ,P} is a unit and π ∈ O_{X ,P} is a prime defining the curve C_{i} at P. Therefore ∂_{C_{i}}(α) = (\overline{u}) ∈ H^{1}(κ(C_{i}), Z/lZ) is unramified at P.

Suppose that ∂_{C_{i}}(α) specialises to zero in H^{1}(κ(P), Z/lZ). Since D is not in the ramification locus of α, α is unramified at D. Let \overline{α} be the specialization of α in H^{2}(κ(D), Z/lZ). Since κ(D) is either a p-adic field or a function field of a curve over a finite field, to show that \overline{α} is zero, by class field theory it is enough to show that \overline{α} is unramified at every discrete valuation of κ(D).

Let v be a discrete valuation of κ(D) and R the corresponding discrete valuation ring. Then there exists a closed point P of D such that R is a localization of the integral closure of the one dimensional local ring O_{D,P} of P on D. The local ring O_{D,P} is a quotient of the local ring O_{X ,P}.

Suppose P is not on the ramification locus of α. Then α is unramified on O_{X ,P} and hence \overline{α} on \overline{O}_{D,P}. In particular \overline{α} is unramified at R.

Suppose P is on the ramification locus of α. As before, we have α = α′ + (u, π), where α′ is unramified on O_{X ,P}, u ∈ O_{X ,P} is a unit and π ∈ O_{X ,P} is a prime defining the curve C_{i} at P. Therefore ∂_{C_{i}}(α) = \overline{u} in κ(C_{i})^{∗}/κ(C_{i})^{∗^{l}}. Since, by the assumption, ∂_{C_{i}}(α) specializes to 0 at P, u(P) ∈ κ(P)^{∗^{l}}. We have \overline{α} = \overline{α′} + (\overline{u}, \overline{π}) ∈ H^{2}(κ(D), Z/lZ). Since α′ is unramified at P, the residue of \overline{α} at R is (u(P))^{ν(\overline{π})}. Since κ(P) is contained in the residue field of the discrete valuation ring R and u(P) is an l^{th} power in κ(P), it follows that \overline{α} is unramified at R. □

Proposition 3.2 Let K and l be as above. Let α ∈ H^{2}(K, μ_{l}) with index l. Let X be a regular projective model of K such that the ramification locus ram_{X} (α) and the special fibre of X are a union of regular curves with only normal crossings and α has no cool points and no chilly loops on X (cf. 1.4). Let s_{i} be the corresponding coefficients (cf. §1). Let F_{1}, ⋯, F_{r} be irreducible regular curves on X which are not in ram_{X} (α) = {C_{1}, ⋯, C_{n}} and such that {F_{1}, ⋯, F_{r}} ∪ ram_{X} (α) have only normal crossings. Let m_{1}, ⋯, m_{r} be integers. Then there exists f ∈ K^{∗} such that

div_{X} (f) = ∑ s_{i}C_{i} + ∑ m_{s}F_{s} + ∑ n_{j}D_{j} + lE′,

where D_{1}, ⋯, D_{t} are irreducible curves which are not equal to C_{i} and F_{s} for all i and s and α specializes to zero at D_{j} for all j and (n_{j}, l) = 1.

Proof. Let T be a finite set of closed points of X containing all the points of intersection of distinct C_{i} and F_{s} and at least one point from each C_{i} and F_{s}. By a semilocal argument, we choose g ∈ K^{∗} such that div_{X} (g) = ∑ s_{i}C_{i} + ∑ m_{s}F_{s} + G where G is a divisor on X whose support does not contain any of C_{i} or F_{s} and does not intersect T.

Since α has no cool points and no chilly loops on X, by ([S3], Prop. 4.5), there exists u ∈ K^{∗} such that div_{X} (ug) = ∑ s_{i}C_{i} + ∑ m_{s}F_{s} + E, where E is a divisor on X whose support does not contain any C_{i} or F_{s}, does not pass through the points in T and either E intersect C_{i} at a point P where the specialization of ∂_{C_{i}}(α) is 0 or the intersection multiplicity (E · C_{i})_{P} is a multiple of l.

Suppose C_{i} for some i is a geometric curve on X. Since every closed point of X is on the special fibre \overline{X}, the closed point of C_{i} is in T. Since the support of E avoids all the points in T, the support of E does not intersect C_{i}. Thus the support of E intersects only those C_{i} which are in the special fibre \overline{X}. Let Q_{1}, ⋯, Q_{t} be the points of intersection of the support of the divisor E and the special fibre with intersection multiplicity n_{j} at Q_{j} coprime with l, for 1 ≤ j ≤ r. In particular we have ∂_{C_{j}}(α) = 0 for 1 ≤ j ≤ r. For each Q_{j}, let D_{j} be a regular geometric curve on X such that Q_{j} is the multiplicity one intersection of D_{j} and \overline{X} (cf. paragraph after 2.2). Then by (2.3) there exists ν ∈ K^{∗} such that div_{X} (ν) = −E + ∑ n_{j}D_{j} + lE′ and ν(P) ∈ κ(P)^{∗^{l}} for all P ∈ T. Let f = ugν ∈ K^{∗}. Then we have

div_{X} (f) = ∑ s_{i}C_{i} + ∑ m_{s}F_{s} + ∑ n_{j}D_{j} + lE′.

Since each Q_{j} is the only closed point on D_{j} and ∂_{C_{i}}(α) specialises to zero at Q_{j}, by (3.1), the α specializes to 0 at D_{j}. Thus f has all the required properties. □

Lemma 3.3 Let α ∈ H^{2}(K, μ_{l}) and let v be a discrete valuation of K. Let u ∈ K^{∗} be a unit at v such that \overline{u} ∈ κ(v)^{∗} \ κ(v)^{∗^{l}}. Suppose further that if α is ramified at v, ∂_{v}(α) = [L] ∈ H^{1}(κ(v), Z/lZ), where L = K(u^{\frac{1}{l}}). Then, for any g ∈ L^{∗}, the image of (N_{L/K}(g)) · α ∈ H^{3}(K_{v}, μ_{l}^{⊗2}) is zero.

Proof. We identify the Galois modules μ_{l}^{⊗j} with Z/lZ as before. Since u is a unit at v and \overline{u} ∉ κ(v)^{∗^{l}}, there is a unique discrete valuation ṽ of L extending the valuation v of K, which is unramified with residual degree l. In particular v(N_{L/K}(g)) is a multiple of l. Thus if α′ ∈ H^{2}(K_{v}, Z/lZ) is unramified at v, then (N_{L/K}(g)) · α′ ∈ H^{3}(K_{v}, Z/lZ) is unramified. Since H_{nr}^{3}(K_{v}, Z/lZ) = 0 (cf. 1.2), we have (N_{L/K}(g)) · α′ = 0 for any α′ ∈ H^{2}(K_{v}, Z/lZ) which is unramified at v. In particular, if α is unramified at v, then α·(N_{L/K}(g)) = 0.

Suppose that α is ramified at v. Then by the choice of u, we have α = α′ + (u) · (π_{v}), where π_{v} is a parameter at v and α′ ∈ H^{2}(K_{v}, Z/lZ) is unramified at v. Thus we have (N_{L/K}(g)) · α = (N_{L/K}(g)) · α′ + (N_{L/K}(g)) · (u)·(π_{v}) = (N_{L/K}(g))·(u)·(π_{v}) ∈ H^{3}(K_{v}, Z/lZ). Since L_{v} = K_{v}(u^{\frac{1}{l}}), we have ((N_{L/K}(g))·(u) = 0 ∈ H^{2}(K_{v}, Z/lZ) and (N_{L/K}(g))·α = 0 in H^{3}(K_{v}, Z/lZ). □

Theorem 3.4 Let K and l be as above. Let α ∈ H^{2}(K, μ_{l}) and ζ ∈ H^{3}(K, μ_{l}^{⊗2}). Assume that the index of α is l. Let X be a regular projective model of K. Suppose that for each x ∈ X^{1}, there exists f_{x} ∈ K_{x}^{∗} such that ζ = α ∪ (f_{x}) ∈ H^{3}(K_{x}, μ_{l}^{⊗2}), where K_{x} is the completion of K at the discrete valuation given by x. Then there exists f ∈ K^{∗} such that ζ = α ∪ (f) ∈ H^{3}(K, μ_{l}^{⊗2}).

Proof. We identify the Galois modules μ_{l}^{⊗j} with Z/lZ as before. By weak approximation, we may find f ∈ K^{∗} such that (f) = (f_{v}) ∈ H^{1}(K_{v}, Z/lZ) for all the discrete valuations corresponding to the irreducible curves in ram_{X} (α) ∪ ram_{X} (ζ). Let

div_{X} (f) = C′ + ∑ m_{i}F_{i} + lE,

where C′ is a divisor with support contained in ram_{X} (α) ∪ ram_{X} (ζ), F_{i}’s are distinct irreducible curves which are not in ram_{X} (α) ∪ ram_{X} (ζ), m_{i} coprime with l and E some divisor on X.

For any C_{j} ∈ ram_{X} (ζ) \ ram_{X} (α), let λ_{j} ∈ κ(C_{j})^{∗} \ κ(C_{j})^{∗^{l}}. By weak approximation, we choose u ∈ K^{∗} with \overline{u} = ∂_{C_{i}}(α) ∈ H^{1}(κ(C_{i}), Z/lZ) for all C_{i} ∈ ram_{X} (α), ν_{F_{i}}(u) = m_{i}, where ν_{F_{i}} is the discrete valuation at F_{i} and \overline{u} = λ_{j} for any C_{j} ∈ ram_{X} (ζ) \ ram_{X} (α). In particular u is a unit at the generic point of C_{j} and \overline{u} ∉ κ(C_{j})^{∗^{l}} for any C_{j} ∈ ram_{X} (ζ) \ ram_{X} (α).

Let L = K(u^{\frac{1}{l}}). Let η : Y → X be the normalization of X in L. Since ν_{F_{i}}(u) = m_{i} and m_{i} is coprime with l, η : Y → X is ramified at F_{i}. In particular there is a unique irreducible curve F̃_{i} in Y such that η(F̃_{i}) = F_{i} and κ(F_{i}) = κ(F̃_{i}).

Let π : Ỹ → Y be a proper birational morphism such that the ramification locus ramỸ(α_{L}) of α_{L} on Ỹ and the strict transform of the curves F̃_{i} on Ỹ is a union of regular curves with only normal crossings and there are no cool points and no chilly loops for α_{L} on Ỹ (cf 1.4). We denote the strict transforms of F̃_{i} by F̃_{i} again. By (3.2), there exists g ∈ L^{∗} such that

divỸ(g) = C + ∑ −m_{i}F̃_{i} + ∑ n_{j}D_{j} + lD,

where the support of C is contained in ramỸ(α_{L}) and D_{j}’s are irreducible curves which are not in ramỸ(α_{L}) and α_{L} specializes to zero at all D_{j}’s.

We now claim that ζ = α∪(fN_{L/K}(g)). Since the group H_{nr}^{3}(K/X , Z/lZ) = 0 ([K], 5.2), it is enough to show that ζ − α ∪ (fN_{L/K}(g)) is unramified on X. Let S be an irreducible curve on X. Since the residue map ∂_{S} factors through the completion K_{S}, it suffices to show that ζ − α ∪ (fN_{L/K}(g)) = 0 over K_{S}.

Suppose S is not in ram_{X} (α) ∪ ram_{X} (ζ) ∪ Supp(fN_{L/K}(g)). Then each of ζ and α · (fN_{L/K}(g)) is unramified at S.

Suppose that S is in ram_{X} (α) ∪ ram_{X} (ζ). Then by the choice of f we have (f) = (f_{v}) ∈ H^{1}(K_{v}, Z/lZ) where v is the discrete valuation associated to S. Hence ζ = α ∪ (f) over the completion K_{S} of K at the discrete valuation given by S. It follows from (3.3) that (N_{L/K}(g)) ∪ α = 0 over K_{S} and ζ = α ∪ (fN_{L/K}(g)) over K_{S}.

Suppose that S is in the support of div_{X} (fN_{L/K}(g)) and not in ram_{X} (α)∪ ram_{X} (ζ). Then α is unramified at S. We show that in this case α ∪ (fN_{L/K}(g)) = 0 over K_{S}. We have

div_{X} (fN_{L/K}(g)) = div_{X} (f) + div_{X} (N_{L/K}(g))
= C′ + ∑ m_{i}F_{i} + lE + η_{∗}π_{∗}(C + ∑ −m_{i}F̃_{i} + ∑ n_{j}D_{j} + lD)
= C′ + η_{∗}π_{∗}(C) + ∑ n_{j}η_{∗}π_{∗}(D_{j}) + lη_{∗}π_{∗}(E).

We note that if D_{j} maps to a point, then η_{∗}π_{∗}(D_{j}) = 0. Since the support of C is contained in ramỸ(α_{L}), the support of η_{∗}π_{∗}(C) is contained in ram_{X} (α). Thus S is in the support of η_{∗}π_{∗}(D_{j}) for some j or S is in the support of lη_{∗}π_{∗}(E). In the later case, clearly α · (fN_{L/K}(g)) is unramified at S and hence α · (fN_{L/K}(g)) = 0 over K_{S}. Suppose S is in the support of η_{∗}π_{∗}(D_{j}) for some j. In this case, if D_{j} lies over an inert curve, then η_{∗}π_{∗}(D_{j}) is a multiple of l and we are done. Suppose that D_{j} lies over a split curve. Since α_{L} specializes to zero at D_{j}, it follows that α specializes to zero at η_{∗}π_{∗}(D_{j}) and we are done. □

Theorem 3.5. Let k be a p-adic field and K a function field of a curve over k. Let l be a prime not equal to p. Suppose that all the l^{th} roots of unity are in K. Then every element in H^{3}(K, μ_{l}^{⊗3}) is a symbol.

Proof. We again identify the Galois modules μ_{l}^{⊗j} with Z/lZ.

Let v be a discrete valuation of K and K_{v} the completion of K at v. By (1.2), every element in H^{3}(K_{v}, Z/lZ) is a symbol.

Let ζ ∈ H^{3}(K, Z/lZ) and X be a regular projective model of K. Let v be a discrete valuation of K corresponding to an irreducible curve in ramX (ζ). Then we have ζ = (f_{v}) · (g_{v}) · (h_{v}) for some f_{v}, g_{v}, h_{v} ∈ K_{v}^{∗}. By weak approximation, we can find f, g ∈ K^{∗} such that (f) = (f_{v}) and (g) = (g_{v}) in H^{1}(K_{v}, Z/lZ) for all discrete valuations v corresponding to the irreducible curves in ramX (ζ). Let v be a discrete valuation of K corresponding to an irreducible curve C in X. If C is in ramX (ζ), then by the choice of f and g we have ζ = (f) · (g) · (h_{v}) ∈ H^{3}(K_{v}, Z/lZ). If C is not in the ramX (ζ), then ζ ∈ H_{nr}^{3}(K_{v}, Z/lZ) ≃ H^{3}(κ(v), Z/lZ) = 0. In particular we have ζ = (f) · (g) · (1) ∈ H^{3}(K_{v}, Z/lZ). Let α = (f) · (g) ∈ H^{2}(K, Z/lZ). Then we have ζ = α·(h′_{v}) ∈ H^{3}(K_{v}, Z/lZ) for some h′_{v} ∈ K_{v}^{∗} for each discrete valuation v of K associated to any point of X^{1}. By (3.4), there exists h ∈ K^{∗} such that ζ = α · (h) = (f) · (g) · (h) ∈ H^{3}(K, Z/lZ). □

Remark 3.6. We remark that all the results of this section can be extended to the situation where k does not necessarily contain a primitive l^{th} root of unity. This can be achieved by going to the extension k′ of k obtained by adjoining a primitive l^{th} of unity to k and noting that the extension k′/k is unramified of degree l − 1. We do not use this remark in the sequel.

4. The u-invariant

In (4.1) and (4.2) below, we give some necessary conditions for a field k to have the u-invariant less than or equal to 8. If K is the function field of a curve over a p-adic field and K_{v} is the completion of K at a discrete valuation v of K, then the residue field κ(v) of K_{v}, which is either a global field of positive characteristic or a p-adic field, has u-invariant 4. By a theorem of Springer, u(K_{v}) = 8 and we use (4.1) and (4.2) for K_{v}.

Proposition 4.1 Let K be a field of characteristic not equal to 2. Suppose that u(K) ≤ 8. Then I^{4}(K) = 0 and every element in I^{3}(K) is a 3-fold Pfister form. Further if ϕ is a 3-fold Pfister form and q_{2} a rank 2 quadratic form over K, then there exists f, g, h ∈ K^{∗} such that f is a value of q_{2} and ϕ =<1, f ><1, g ><1, h>.

Proof. Suppose that u(K) = 8. Then every 4-fold Pfister form is isotropic and hence hyperbolic; in particular, I^{4}(K) = 0. Let ϕ_{1} =<1, f_{1} ><1, g_{1} >< 1, h_{1} > and ϕ_{2} =< 1, f_{2} >< 1, g_{2} >< 1, h_{2} > be two anisotropic 3-fold Pfister forms. Since u(K) ≤ 8, the Witt index of ϕ_{1} − ϕ_{2} is at least 4. In particular ϕ_{1} −ϕ_{2} is isotropic, i.e. ϕ_{1} and ϕ_{2} represent a common value a ∈ K. Since ϕ_{1} and ϕ_{2} are anisotropic, a ≠ 0. We have ϕ_{1} =< a >⊥ ϕ′_{1} and ϕ_{2} =< a >⊥ ϕ′_{2} for some quadratic form ϕ_{1} and ϕ_{2} over K ([Sc], p.7, Lemma 3.4). Since the Witt index of ϕ_{1} − ϕ_{2} is at least 4, the Witt index of ϕ′_{1} − ϕ′_{2} is at least 3. Repeating this process, we see that there exists a quadratic form < a, b, c, d > over K which is a subform of both ϕ_{1} and ϕ_{2} . Let C be a conic given by the quadratic form abc < a, b, c >=< bc, ac, ab >. Since abc < a, b, c > is isotropic over the function field K(C) of C (cf. [Sc], p.154, Remark 5.2(iv)) and abc < a, b, c > is a subform of abcϕ_{1} and abcϕ_{2}, ϕ_{1} and ϕ_{2} are isotropic over K(C). Since ϕ_{1} and ϕ_{2} are Pfister forms, they are hyperbolic over k(C) ([Sc], p.144, Cor. 1.5). Let ψ =< 1, bc >< 1, ac > and k(ψ) be the function field of the quadratic form ψ. Then k(C) and k(ψ) are birational (cf. [Sc], p.154) and hence ϕ_{1} and ϕ_{2} are hyperbolic over k(ψ). Therefore ϕ_{1} ≃<1, bc><1, ac><a_{1}, b_{1} > and ϕ_{2} ≃<1, bc><1, ac><a_{2}, b_{2} > for some a_{1}, a_{2}, b_{1}, b_{2} ∈ K^{∗} (cf. [Sc], p.155, Th.5.4). Since I^{4}(K) = 0, we have ϕ_{1} ≃< 1, bc >< 1, ac >< 1, a_{1}b_{1} > and ϕ_{2} ≃< 1, bc >< 1, ac >< 1, a_{2}b_{2} >. We have

ϕ_{1} + ϕ_{2} = ϕ_{1} − ϕ_{2}
= <1, bc><1, ac><1, a_{1}b_{1}, −1, −a_{2}b_{2} >
= <1, bc><1, ac><a_{1}b_{1}, −a_{2}b_{2} >
= a_{1}b_{1} <1, bc><1, ac ><1, −a_{1}a_{2}b_{1}b_{2} >
= <1, bc><1, ac><1, −a_{1}a_{2}b_{1}b_{2} > .

Thus the sum of any two 3-fold Pfister forms is a Pfister form in I^{3}(K) and every element in I^{3}(K) is the class of a 3-fold Pfister form.

Let ϕ =<1, a><1, b><1, c> be a 3-fold Pfister form and ϕ′ be its pure subform. Let q_{2} be a quadratic form over K of dimension 2. Since dim(ϕ′) = 7 and u(K) ≤ 8, the quadratic form ϕ′ − q_{2} is isotropic. Therefore there exists f ∈ K^{∗} which is a value of q_{2} and ϕ′ ≃< f > +ϕ′′ for some quadratic form ϕ′′ over K. Hence by ([Sc], p.143), ϕ =< 1, f >< 1, b′ >< 1, c′ > for some b′, c′ ∈ K^{∗}. □

Proposition 4.2. Let K be a field of characteristic not equal to 2. Suppose that u(K) ≤ 8. Let ϕ =<1, f ><1, a><1, b> be a 3-fold Pfister form over K and q_{3} a quadratic form over K of dimension 3. Then there exist g, h ∈ K^{∗} such that g is a value of q_{3} and ϕ =<1, f ><1, g ><1, h>.

Proof. Let ψ =< 1, f >< a, b, ab >. Since u(K) ≤ 8, the quadratic form ψ − q_{3} is isotropic. Hence there exists g ∈ K^{∗} which is a common value of q_{3} and ψ. Thus, ψ ≃< g > +ψ_{1} for some quadratic form ψ_{1} over K. Since ψ is hyperbolic over K(\sqrt{−f}), ψ_{1} ≃< 1, f >< a_{1}, b_{1} > + < g_{1} > for some a_{1}, b_{1}, g_{1} ∈ K^{∗}. By comparing the determinants, we get g_{1} = gf modulo squares. Hence ψ =< 1, f >< g, a_{1}, b_{1} > and ϕ =< 1, f > +ψ =< 1, f >< 1, g, a_{1}, b_{1} >. The form ϕ is isotropic and hence hyperbolic over the function field of the conic given by < f, g, fg >. Hence, as in 4.1, ϕ = λ < 1, f >< 1, g >< 1, h > for some λ, h ∈ K^{∗}. Since I^{4}(K) = 0, ϕ =< 1, f >< 1, g >< 1, h> with g a value of q_{3}. □

Proposition 4.3. Let K be a field of characteristic not equal to 2. Assume the following:

1. Every element in H^{2}(K, μ_{2}) is a sum of at most 2 symbols.

2. Every element in I^{3}(K) is equal to a 3-fold Pfister form.

3. If ϕ is a 3-fold Pfister form and q_{2} is a quadratic form over K of dimension 2, then ϕ =<1, f ><1, g ><1, h> for some f, g, h ∈ K^{∗} with f a value of q_{2}.

4. If ϕ =<1, f ><1, a><1, b> is a 3-fold Pfister form and q_{3} a quadratic form over K of dimension 3, then ϕ =<1, f ><1, g ><1, h> for some g, h ∈ K^{∗} with g a value of q_{3}.

5. I^{4}(K) = 0.

Then u(K) ≤ 8.

Proof. Let q be a quadratic form over K of dimension 9. Since every element in H^{2}(K, μ_{2}) is a sum of at most 2 symbols, as in ([PS], proof of 4.5 ), we find a quadratic form q_{5} = λ < 1, a_{1}, a_{2}, a_{3}, a_{4} > over K such that ϕ = q + q_{5} ∈ I^{3}(K). By the assumptions 2, 3 and 4, there exist f, g, h ∈ K^{∗} such that ϕ =< 1, f >< 1, g >< 1, h > and f is a value of < a_{1}, a_{2} > and g is a value of < fa_{1}a_{2}, a_{3}, a_{4} >. We have < a_{1}, a_{2} >≃< f, fa_{1}a_{2} > and < fa_{1}a_{2}, a_{2}, a_{3} >≃< g, g_{1}, g_{2} > for some g_{1}, g_{2} ∈ K^{∗}. Since I^{4}(K) = 0, we have λϕ = ϕ and

λq = λq + λq_{5} − λq_{5}
= λϕ − λq_{5}
= ϕ − λq_{5}
= <1, f ><1, g ><1, h> − <1, a_{1}, a_{2}, a_{3}, a_{4} >
= <1, f ><1, g ><1, h> − <1, f, g, g_{1}, g_{2} >
= <gf > + <1, f ><h, gh> − <g_{1}, g_{2} > .

Since the dimension of λq is 9 and the dimension of <gf >⊥<1, f ><h, gh> − <g_{1}, g_{2} > is 7, it follows that λq and hence q is isotropic over K. □

Proposition 4.4. Let k be a p-adic field, p ≠ 2 and K a function field of a curve over k. Let ϕ be a 3-fold Pfister form over K and q_{2} a quadratic form over K of dimension 2. Then there exist f, a, b ∈ K^{∗} such that f is a value of q_{2} and ϕ =<1, f ><1, a><1, b>.

Proof. Let ζ = e_{3}(ϕ) ∈ H^{3}(K, μ_{2}). Let X be a projective regular model of K. Let C be an irreducible curve on X and v the discrete valuation given by C. Let K_{v} be the completion of K at v. Since the residue field κ(v) = κ(C) is either a p-adic field or a function field of a curve over a finite field, u(κ(v)) = 4 and u(K_{v}) = 8 ([Sc], p.209). By (4.1), there exist f_{v}, a_{v}, b_{v} ∈ K_{v}^{∗} such that f_{v} is a value of q_{2} over K_{v} and ϕ =< 1, f_{v} >< 1, a_{v} >< 1, b_{v} > over K_{v}. By weak approximation, we can find f, a ∈ K^{∗} such that f is a value of q_{2} over K and f = f_{v}, a = a_{v} modulo K_{v}^{∗^{2}} for all discrete valuations v corresponding to the irreducible curves C in the support of ramX (ζ). Let C be any irreducible curve on X and v the discrete valuation of K given by C. If C is in the support of ramX (ζ), then by the choice of f and a, we have ζ = e_{3}(ϕ) = (−f) · (−a) · (−b_{v}) over K_{v}. If C is not in the support of ramX (ζ), then ζ ∈ H_{nr}^{3}(K_{v}, μ_{2}) ≃ H^{3}(κ(v), μ_{2}) = 0. In particular we have ζ = (−f) · (−a) · (1) over K_{v}. Let α = (−f) · (−a) ∈ H^{2}(K, μ_{2}). By (3.4), there exists b ∈ K^{∗} such that ζ = α · (−b) ∈ H^{3}(K, μ_{2}). Since e_{3} : I^{3}(K) → H^{3}(K, μ_{2}) is an isomorphism, we have ϕ =< 1, f >< 1, a >< 1, b > as required. □.

There is a different proof of the Proposition 4.4 in ([PS], 4.4)!

Proposition 4.5 Let k be a p-adic field, p ≠ 2 and K a function field of a curve over k. Let ϕ =<1, f ><1, a><1, b> be a 3-fold Pfister form over K and q_{3} a quadratic form over K of dimension 3. Then there exist g, h ∈ K^{∗} such that g is a value of q_{3} and ϕ =<1, f ><1, g ><1, h>.

Proof. Let ζ = e_{3}(ϕ) = (−f) · (−a) · (−b) ∈ H^{3}(K, μ_{2}). Let X be a projective regular model of K. Let C be an irreducible curve on X and v the discrete valuation of K given by C. Let K_{v} be the completion of K at v. Then as in the proof of (4.4), we have u(K_{v}) = 8. Thus by (4.2), there exist g_{v}, h_{v} ∈ K_{v}^{∗} such that g_{v} is a value of the quadratic form q_{3} and ϕ =< 1, f >< 1, g_{v} >< 1, h_{v} > over K_{v}. By weak approximation, we can find g ∈ K^{∗} such that g is a value of q_{3} over K and g = g_{v} modulo K_{v}^{∗^{2}} for all discrete valuations v of K given by the irreducible curves C in ramX (ζ). Let C be an irreducible curve on X and v the discrete valuation of K given by C. By the choice of g it is clear that ζ = e_{3}(ϕ) = (−f) · (−g) · (−h_{v}) for all the discrete valuations v of K given by the irreducible curves C in the support of ramX (ζ). If C is not in the support of ramX (ζ), then as is the proof of (4.4), we have ζ = (−f) · (−g) · (1) over K_{v}. Let α = (−f) · (−g) ∈ H^{2}(K, μ_{2}). By (3.4), there exists h ∈ K^{∗} such that ζ = α ∪ (−h) = (−f) · (−g) · (−h). Since e_{3} : I^{3}(K) → H^{3}(K, μ_{2}) is an isomorphism, ϕ =<1, f ><1, g ><1, h>.

Theorem 4.6 Let K be a function field of a curve over a p-adic field k. If p ≠ 2, then u(K) = 8.

Proof. Let K be a function field of a curve over a p-adic field k. Assume that p ≠ 2. By a theorem of Saltman ([S1], 3.4, cf. [S2]), every element in H^{2}(K, μ_{2}) is a sum of at most 2 symbols. Since the cohomological dimension of K is 3, we also have I^{4}(K) ≃ H^{4}(K, μ_{2}) = 0 ([AEJ]). Now the theorem follows from (4.4), (4.5) and (4.3). □.

References

[A] Arason, J.K., Cohomologische Invarianten quadratischer Formen, J. Algbera 36 (1975), 448-491.

[AEJ] Arason, J.K., Elman, R. and Jacob, B., Fields of cohomological 2-dimension three, Math. Ann. 274 (1986), 649-657 .

[C] Colliot-Thélène, J.-L., Birational invariants, purity, and the Gresten conjecture, Proceedings of Symposia in Pure Math. 55, Part 1, 1-64.

[K] Kato, K., A Hasse principle for two-dimensional global fields, J. reine Angew. Math. 366 (1986), 142-181.

[L] Lam, T.Y., Introduction to quadratic forms over fields, GSM 67, American Mathematical Society, 2004.

[Li1] Lipman, J., Introduction to resolution of singularities, Proc. Symp. Pure Math. 29 (1975), 187-230.

[Li2] Lipman, J., Desingularization of two-dimensional schemes, Ann. Math. 107 (1978), 151-207.

[M] Merkurjev, A.S., On the norm residue symbol of degree 2, Dokl. Akad. Nauk. SSSR 261 (1981), 542-547.

[Mi] Milne, J.S.. Étale Cohomology, Princeton University Press, Princeton, New Jersey 1980.

[PS] Parimala, R. and Suresh, V., Isotropy of quadratic forms over function fields in one variable over p-adic fields, Publ. de I.H.É.S. 88 (1998), 129-150.

[S1] Saltman, D.J., Division Algebras over p-adic curves, J. Ramanujan Math. Soc. 12 (1997), 25-47.

[S2] Saltman, D.J., Correction to Division algebras over p-adic curves, J. Ramanujan Math. Soc. 13 (1998), 125-130.

[S3] Saltman, D.J., Cyclic Algebras over p-adic curves (to appear).

[Sc] Scharlau, W., Quadratic and Hermitian Forms, Grundlehren der Math. Wiss., Vol. 270, Berlin, Heidelberg, New York 1985.

[Sh] Shafarevich, I. R. Lectures on Minimal Models and Birational Transformations of two Dimensional Schemes, Tata Institute of Fundamental Research, (1966).

Department of Mathematics and Computer Science Emory University 400 Dowman Drive Atlanta, Georgia 30322 USA E-mail: parimala@mathcs.emory.edu

Department of Mathematics and Statistics University of Hyderabad Gahcibowli Hyderabad - 500046 India E-mail: vssm@uohyd.ernet.in