Diagonal forms of higher degree over function fields of $p$-adic curves

We investigate diagonal forms of degree $d$ over the function field $F$ of a smooth projective $p$-adic curve: if a form is isotropic over the completion of $F$ with respect to each discrete valuation of $F$, then it is isotropic over certain fields $F_U$, $F_P$ and $F_p$. These fields appear naturally when applying the methodology of patching; $F$ is the inverse limit of the finite inverse system of fields $\{F_U,F_P,F_p\}$. Our observations complement some known bounds on the higher $u$-invariant of diagonal forms of degree $d$. We only consider diagonal forms of degree $d$ over fields of characteristic not dividing $d!$.


Introduction
The fact that Springer's Theorem holds for diagonal forms of higher degree over fields of characteristic not dividing d! [9] guarantees that on occasion diagonal forms of higher degree defined over function fields behave similarly to quadratic forms. For a survey on the behaviour of (diagonal) forms of higher degree in general the reader is referred to [12].
In this note we consider diagonal forms of degree d over function fields F = K(X) where X is a smooth, projective, geometrically integral curve over K and K is the fraction field of a complete discrete valuation ring with a residue field k of characteristic not dividing d!. Let v be a rank one discrete valuation of F , and F v the completion of F with respect to v. It was shown by Colliot-Thélène, Parimala and Suresh [2, Theorem 3.1] that a quadratic form which is isotropic over F v for each v is already isotropic over F , using the methodology of patching developed by Habater and Hartmann [4], i.e. viewing F as the inverse limit of a finite inverse system of certain fields {F U , F P , F p }.
Given a nondegenerate diagonal form ϕ over F of degree d > 2 and dimension > 2, it is not clear, however, whether the isotropy of ϕ over F v for each v implies that ϕ is isotropic.
Our main result proves that the isotropy of a nondegenerate diagonal form ϕ over F v for each v implies that at least over the field extensions F U , F P and F p of F , ϕ is isotropic as well (Theorem 6). These fields depend on the choice of the form ϕ = a 1 , . . . , a n , more precisely on the choice of the regular proper model X (over the complete discrete valuation ring A) of the curve X over K, which depends on ϕ: X is selected such that there exists a reduced divisor D with strict normal crossings, which contains both the support of the divisor of all the entries a i , 1 ≤ i ≤ n, and the components of the special fibre of X/A. Since nondegenerate diagonal forms of degree d ≥ 3 have finite automorphism groups [6, p. 137], we are not able to apply [5,Theorem 3.7] to conclude that the isotropy of ϕ over the F U 's and F P 's implies that ϕ is also isotropic over F , however. This is only possible for d = 2.
After collecting the terminology and some basic results in Section 1, in particular defining diagonal u-invariants of degree d over k, we consider diagonal forms of higher degree over valued fields in Section 2 and then study diagonal forms of higher degree over function fields of p-adic curves using some of the ideas of [2] in Section 3. Recall that a p-adic field is a finite field extension of Q p .
As a consequence of Springer's Theorem for diagonal forms, any diagonal form of degree d and dimension > d 3 + 1 over a function field in one variable F = K(t), where K is a p-adic field with residue field k, char(k) ∤ d!, is isotropic over F v for every discrete valuation v with residue field either a function field in one variable over k or a finite extension of K. Moreover, it is isotropic over F U for each reduced, irreducible component U ⊂ Y of the complement of S in the special fibre Y = X × A k of X/A, and isotropic over F P for each P ∈ S (Corollary 9), and thus isotropic over F p for each p = (U, P ). Here, S is the inverse image under a finite A-morphism f : X → P 1 A of the point at infinity of the special fibre P 1 k .

Preliminaries
Let k be a field such that char(k) does not divide d!.
1.1. Forms of higher degree. Let V be a finite-dimensional vector space over k of dimension n. A d-linear form over k is a k-multilinear map θ : is a d-linear form over k. By fixing a basis {e 1 , . . . , e n } of V , any form ϕ of degree d can be viewed as a homogeneous polynomial of degree d in n = dim V variables x 1 , . . . , x n via ϕ(x 1 , . . . , x n ) = ϕ(x 1 e 1 + · · · + x n e n ) and, vice versa, any homogeneous polynomial of degree d in n variables over k is a form of degree d and dimension n over k. Any d-linear form θ : V × · · · × V → k induces a form ϕ : V → k of degree d via ϕ(v) = θ(v, . . . , v). We can hence identify d-linear forms and forms of degree d.
we use the notation ϕ = a 1 , . . . , a n and call ϕ diagonal. A diagonal form ϕ = a 1 , . . . , a n over k is nondegenerate if and only if a i ∈ k × for all 1 ≤ i ≤ n.
If d ≥ 3, a i , b j ∈ k × , then a 1 , . . . , a n ∼ = b 1 , . . . , b n if and only if there is a permutation π ∈ S n such that b i ∼ = a π(i) for every i. This is a special case of [6,Theorem 2.3].
Note that for quadratic forms (d = 2), the automorphism group of ϕ is infinite, whereas for d ≥ 3, the automorphism group of ϕ usually is finite, for instance if ϕ is is nonsingular in the sense of algebraic geometry [13]. In particular, nondegenerate diagonal forms of degree d ≥ 3 have finite automorphism groups [6, p. 137], which creates a problem when trying to imitate patching arguments as it is not possible to apply [5, Theorem 3.7].

1.2.
Higher degree u-invariants. The u-invariant (of degree d) of k is defined as u(d, k) = sup{dim k ϕ}, where ϕ ranges over all the anisotropic forms of degree d over k. The diagonal u-invariant (of degree d) of k is defined as u diag (d, k) = sup{dim ϕ}, where ϕ ranges over all the anisotropic diagonal forms over k.
Thus the diagonal u-invariant u diag (d, k) is the smallest integer n such that all diagonal forms of degree d over k of dimension greater than n are isotropic, and the u-invariant u(d, k) is the smallest integer n such that all forms of degree d over k of dimension greater than n are isotropic. Obviously, u diag (d, k) ≤ u(d, k). If u = u(d, k) then each anisotropic form of degree d over k of dimension u is universal. If u = u diag (d, k) then each diagonal anisotropic form of degree d over k of dimension u is universal. We have u diag (d, k) ≤ min{n | all forms of degree d over k of dimension ≥ n are universal} with the understanding that the "minimum" of an empty set of integers is ∞, cf. [12].
For an algebraically closed field k, |k × /k ×d | = 1 and hence u diag (d, k) = u(d, k) = 1. For a formally real field k, the diagonal u-invariant is infinite for even d: , is the smallest real number n such that (1) every finite field extension E/k satisfies u diag (d, E) ≤ n, and (2) every finitely generated field extension E/k of transcendence degree one sat- Analogously as observed in [5] for d = 2, u diag, s (d, k) ≤ n if and only if every finitely generated field extension E/k of transcendence degree l ≥ 1 satisfies 1.3. C 0 r fields. Let r ≥ 1 be an integer. A field F is a C r -field if for all d ≥ 1 and n > d r , every homogeneous form of degree d in n variables over F has a non-trivial solution in F . In particular, then F satisfies u(d, F ) ≤ d r . Moreover, every finite extension of F is a C r -field, and every one-variable function field over F a C r+1 -field [14,II.4.5]. Hence u diag, s (d, F ) ≤ d r for a C r -field F .
A field F is a C 0 r -field if the following holds: For any finite field extension F ′ of F and any integers d ≥ 1 and n > d r , for any homogeneous form over F ′ of degree d in n variables, the greatest common divisor of the degrees of finite field extensions F ′′ /F ′ over which the form acquires a nontrivial zero is one. This amounts to requiring that the F ′ -hypersurface defined by the form has a zero-cycle of degree 1 over F ′ .
Assume char(F ) = 0. For each prime l, let F l be the fixed field of a pro-l-Sylow subgroup of the absolute Galois group of F . Any finite subextension of F l /F is of degree coprime to l. The field F is C 0 r if and only if each of the fields Remark 1. Assume that p-adic fields have the C 0 2 -property. Let K(X) be any function field of transcendence degree r over a p-adic field K (here we do not need to assume p = 2, 3). Suppose that there is ℓ = 2 such that there exists a finite subextension of K ℓ (X)/K(X) of degree 2. Then any cubic form over K(X) in strictly more than 3 2+r variables has a nontrivial zero: If the p-adic field K is C 0 3 , then the function field E = K(X) in r variables over K is C 0 3+r [2, Lemma 2.1]. Thus a cubic form over E = K(X) in strictly more than 3 2+r has a nontrivial zero in each of the fields K ℓ (X), l a prime, hence in a finite extension of K(X) of degree coprime to ℓ, for each ℓ prime. Pick ℓ = 2, then [K ℓ (X) : K(X)] is even. Moreover, pick l = 2 such that there exists a finite subextension of E ℓ /K(X) of degree 2 then the cubic form has a zero over it. By Springer's Theorem for cubic forms and their behaviour under quadratic field extensions [8,VII], thus the cubic form has a nontrivial zero in K(X). This is the analogue of [2, Proposition 2.2].

Diagonal forms over Henselian valued fields
2.1. Let K be a valued field with valuation v, valuation ring R and maximal ideal m. Let Γ be the value group. Assume that d! is not divisible by the characteristic of the residue field k = R/m. For u ∈ R, denote byū the image of u in k. For a polynomial f ∈ R[X], f = a n x n + · · · + a 1 x + a 0 , define the polynomial f =ā n x n + · · · +ā 1 x +ā 0 over k. If ϕ = a 1 , . . . , a n is a nondegenerate diagonal form with entries a i ∈ R, define the diagonal form ϕ = ā 1 , . . . ,ā n over k. ϕ is called a unit form, if ϕ is nondegenerate. Choose a set {π γ ∈ R | γ ∈ I} such that the values of the π γ 's represent the distinct cosets in Γ/dΓ. We may decompose a diagonal form ϕ as ϕ =⊥ ϕ ′ γ by taking ϕ ′ γ to be the diagonal form whose entries comprise all a i with v(a i ) = v(π γ ) mod d Γ. By altering the slots by d-powers if necessary, we may then write ϕ ′ γ = π γ ϕ γ with each ϕ γ a diagonal unit form. There are only finitely many non-trivial ϕ γ [9]. If Γ = Z, the set {π γ | γ ∈ I} can be chosen to be If R satisfies Hensel's Lemma then (K, v) is called a Henselian valued field and R a Henselian valuation ring. Every complete discretely valued field is Henselian.
Let ϕ be a diagonal form over a Henselian valued field (K, v). Write ϕ = π 1 ϕ 1 ⊥ · · · ⊥ π r ϕ r with each ϕ i a diagonal unit form and the π i having distinct values in Γ/dΓ. Then ϕ is isotropic if and only if some ϕ i is isotropic [9, Proposition 3.1]. This is because for a diagonal unit form ϕ over a Henselian valued field (K, v), ϕ is isotropic if and only if ϕ is isotropic [9, Lemma 2.3].
Theorem 2. ( [9] or [12, Theorem 4, Corollary 2]) Suppose that char(k) ∤ d!.   (iv) Let F be a field extension of finite type over k of transcendence degree n. Then u diag (d, F ) ≥ d n u diag (d, k ′ ) for a suitable finite field extension k ′ /k.

The (in)equalities in (i), (iii), (iv) also hold when the values are infinite.
For d = 2, (ii) is Springer's Theorem for quadratic forms over Henselian valued fields [15]. Springer's Theorem does not hold for non-diagonal forms of higher degree than 2 [9, 2.7]. Theorem 2 is a major ingredient in our proofs, for instance we can show: Proposition 3. Let A be a discrete valuation ring with fraction field K and residue field k such that char(k) ∤ d!.
The first assertions of (i) as well as (ii) and (iii) follow from Theorem 2. The proof of the second claim in (i) is analogous to the one of [5, 4.9], employing Theorem 2 instead of Springer's Theorem.

A field
K is called an m-local field with residue field k if there is a sequence of fields k 0 , . . . , k m with k 0 = k and k m = K, and such that k i is the fraction field of an excellent Henselian discrete valuation ring with residue field k i−1 for i = 1, . . . , m.
Recall that a discrete valuation ring R is called excellent, if the field extension K/K is separable, where K denotes the quotient field of R and K is its completion. (This condition is trivially satisfied if K has characteristic 0 or R is complete.) Proposition 3 implies (compare the next two results with [5, Corollary 4.13, 4.14] for quadratic forms): Corollary 4. Suppose that K is an m-local field whose residue field k is a C r -field with char(k) ∤ d!. Let F be a function field over K in one variable.
Moreover, if some normal K-curve with function field F has a K-point, then Proof. (i) Since k is a C r -field, u diag (d, k) = d r , thus u diag (d, k) ≤ u diag,s (d, k) ≤ d r and the first two equations follow. Applying Proposition 3 and induction yields that u diag (d, K) ≥ d m u diag (d, k) = d r+m . Let X be a normal K-curve with function field F and let ξ be a K-point on X. The local ring at ξ has fraction field F and residue field K. So Proposition 3 implies that u diag (d, K) = d m u diag (d, k) and (ii) Choose a normal or equivalently a regular K-curve X with function field F , and a closed point ξ on X. Let R be the local ring of X at ξ with residue field κ(ξ). Then the fraction field of R is F , and κ(ξ) is a finite extension of K. Hence κ(ξ) is an m-local field whose residue field k ′ is a finite extension of k. By assumption, u diag (d, k ′ ) = d r and k ′ is a C r -field since k is. So applying part (ii) to k ′ and κ(ξ), it follows that u diag (d, κ(ξ)) ≥ d r+m . Proposition 3 yields u diag (d, F ) ≥ d r+m+1 .
Corollary 5. (i) Let F be a one-variable function field over an m-local field K with residue field k such that char(k) ∤ d! and k is algebraically closed. Then Proof. (i) This is a special case of Corollary 4 using that an algebraically closed field k is C 0 , satisfies u diag (d, k) = 1, and has no non-trivial finite extensions. In general, for any finite field k = F q we obviously do not have 3. The behaviour of diagonal forms of higher degree over function fields of p-adic curves Whenever we write 'discrete valuation ring' and 'discrete valuation' we mean a discrete valuation ring of rank one and a valuation with value group Z.

Let
A be a complete discrete valuation ring with fraction field K and residue field k with char(k) ∤ d!. Let X be a smooth, projective, geometrically integral curve over K and F = K(X) be the function field of X. Let t denote a uniformizing parameter for A. For each (rank one) discrete valuation v of F , let F v denote the completion of F with respect to v.
We will adapt some ideas from [2] to diagonal forms of higher degree: take a nondegenerate form ϕ = a 1 , . . . , a n of degree d over F . Then choose a regular proper model X /A of X/K, such that there exists a reduced divisor D with strict normal crossings which contains both the support of the divisor of all the entries a i , 1 ≤ i ≤ n, and the components of the special fibre of X/A. (Note that this implies that the regular proper model X /A depends on the form ϕ, and thus so do Y, Y i , S 0 , S, F P , F U , . . . as defined in the following.) Let Y = X × A k be the special fibre of X/A. Let x i be the generic point of an irreducible component Y i of Y . Then there is an affine Zariski neighbourhood W i ⊂ X of x i , such that the restriction of Y i to W i is a principal divisor. Let S 0 be a finite set of closed points of Y containing all singular points of D, and all the points that lie on some Y i , but not in the corresponding W i .
Choose a finite A-morphism f : X → P 1 A as in [4,Proposition 6.6]. Let S be the inverse image under f of the point at infinity of the special fibre P 1 k . Then the set S 0 is contained in S. All the intersection points of two components Y i are in S. Each component Y i contains at least one point of S. Let U ⊂ Y run through the reduced irreducible components of the complement of S in Y . Then each U is a regular affine irreducible curve over k and we define k[U ] to be its ring of regular functions and k(U ) to be its fraction field. k[U ] is a Dedekind domain and U = Spec k[U ]. Each U is contained in an open affine subscheme Spec R U of X and is a principal effective divisor in Spec R U . Moreover, R U is the ring of elements in F which are regular on U and also a regular ring, since it is the direct limit of regular rings. The ring R U is a localisation of R U and so U is a principal effective divisor on Spec R U given by the vanishing of an element s ∈ R U . The t-adic completion R U of R U is a domain and coincides with the s-adic completion of R U , since t = us r for some integer r ≥ 1 and a unit u ∈ R × U . By definition, F U is the field of fractions of R U . We have k[U ] = R U /s = R U /s. For P ∈ S, the completion R P of the local ring R P of X at P is a domain and F P is the field of fractions of R P . Let p = (U, P ) be a pair with P ∈ S in the closure of an irreducible component U of the complement of S in Y . Then let R p be the complete discrete valuation ring which is the completion of the localisation of R P at the height one prime ideal corresponding to U . Then F p is the field of fractions of R p and F is the inverse limit of the finite inverse system of fields {F U , F P , F p } by [4, Proposition 6.3].
The following can be seen as a weak generalization of [2, Theorem 3.1] to diagonal forms of higher degree. Here we are not able to conclude that under the given assumptions, ϕ is isotropic over F , only over the F U 's and F P 's: Theorem 6. Let ϕ be a nondegenerate diagonal form of degree d over F . If ϕ is isotropic over the completion F v of F with respect to each discrete valuation v of F with residue field either a function field in one variable over k or a finite extension of K, then: Proof. Suppose ϕ = a 1 , . . . , a n . (i) Each entry a i of ϕ is supported only along U in Spec R U , thus has the form us j where u ∈ R × U . We sort the entries a i = u i s j by the power j of s and use them to define new diagonal forms ρ j which have all the u i 's belonging to those a i where s occurred in the jth power as their diagonal entries. Hence ϕ is isomorphic to the diagonal form over F , where the ρ i are nondegenerate diagonal forms of degree d over R U . Note that if for some j ∈ {0, 1, . . . , d − 1} there is no a i with a i = u i s j , then there is no corresponding form ρ j and a ρ j does not appear as a component in the sum.
By hypothesis, ϕ is isotropic over the field of fractions of the completed local ring of X at the generic point of U . By Theorem 2, this implies that the image of at least one of the forms ρ 0 , ρ 1 or ρ d−1 under the composite homomorphism Since the variety is smooth over R U , a k[U ]-point lifts to an R U -point (cf. the discussion after [5,Lemma 4.5]). Thus ϕ has a nontrivial zero over F U . (ii) Let P ∈ S. The local ring R P of X at P is regular. Its maximal ideal is generated by two elements (x, y) with the property that any a i is the product of a unit, a power of x and a power of y. Thus over F , the fraction field of R P , ϕ is isomorphic to where each ϕ i is a nondegenerate diagonal form over R P . Let R y be the localization of R P at the prime ideal (y). R y is a discrete valuation ring with fraction field F . The residue field E of R y is the field of fractions of the discrete valuation ring R P /(y). By hypothesis, the form is isotropic over the field of fractions of the completion of R y . By Theorem 2, the reduction of one of the forms is isotropic over E. Since x is a uniformizing parameter for R P /(y), by Theorem 2 this implies that over the residue field of R P /(y), the reduction of one of the forms ϕ 1 , ϕ 2 , ϕ 3 . . . , ϕ d 2 is isotropic. But then one of these forms is isotropic over R P , hence over the field F P which is the fraction field of R P .
Remark 7. (i) In the proof of Theorem 6, one of the forms is isotropic over R P , and since R p is the complete discrete valuation ring which is the completion of the localisation of R P at the height one prime ideal corresponding to U when p = (U, P ), this form is also isotropic over R p and therefore over the field of fractions F p of R p . This implies that if ϕ is isotropic over the completion of F with respect to each discrete valuation of F , then ϕ is isotropic over F U for each reduced irreducible component U ⊂ Y of the complement of S in Y , over F P for each P ∈ S and over R p for each p = (U, P ). Since F is the inverse limit of the finite inverse system of fields {F U , F P , F p }, ϕ is isotropic over all overfields used in the inverse limit.
(ii) The discrete valuation rings used in the above proof are the local rings at a point of codimension 1 on a suitable regular proper model X of X determined by the choice of ϕ (analogously as noted in [2, Remark 3.2]).
Given a nondegenerate diagonal form ϕ of degree d and dimension greater than two over F , it is not clear whether the isotropy of ϕ over F v for each v (respectively, of ϕ over all F U , F P and F p ) implies that ϕ is isotropic (the fact that dim ϕ > 2 is necessary: it is easy to adjust the example in [2, Appendix] to two-dimensional diagonal forms of even degree).
Corollary 8. Let r ≥ 1 be an integer and d ≥ 3. Assume that any diagonal form in strictly more than dr variables over any function field in one variable over k is isotropic. Then: (i) Any diagonal form of degree d and dimension > d 2 r over the function field F = K(X) of a curve X/K is isotropic over F v , for every discrete valuation v with residue field either a function field in one variable over k or a finite extension of K.
(ii) Any diagonal form of degree d and dimension > d 2 r over the function field F = K(X) of a curve X/K is isotropic over F U for each reduced, irreducible component U ⊂ Y of the complement of S in Y and is isotropic over F P for each P ∈ S.
Note that Y and S depend on ϕ.
Proof. (i) Let L be a finite field extension of K. This is a complete discretely valued field with residue field a finite extension ℓ of k. The assumption made on diagonal forms of degree d over functions fields in one variable over k, in particular diagonal forms of degree d over the field ℓ(t), and Theorem 2 applied to ℓ((t)) show that any diagonal form of dimension > r over ℓ has a zero. A second application of Theorem 2 yields that any diagonal form of degree d of dimension > dr over L is isotropic. Let ϕ be a diagonal form of dimension n over F with n > d 2 r. By the assumption and Theorem 2, ϕ is isotropic over F v for every discrete valuation v with residue field either a function field in one variable over k or a finite extension of K.
This shows that trying to extend [2, Corollary 3.4] from quadratic to diagonal forms of higher degree results in a much weaker version.
3.2. Let K be a p-adic field with residue field k such that char(k) ∤ d!.
Corollary 9. Any diagonal form of degree d and dimension > d 3 +1 over a function field in one variable F = K(t) is (i) isotropic over F v , for every discrete valuation v with residue field either a function field in one variable over k or a finite extension of K; (ii) isotropic over F U for each reduced, irreducible component U ⊂ Y of the complement of S in Y and isotropic over F P for each P ∈ S.
Proof. Every finite field k is C 1 and so every diagonal form of degree d and dimension > d 2 over any function field in one variable over k (which is C 2 ) is isotropic. Assertion (i) is a direct consequence of Theorem 2 and (ii) follows from Corollary 8 (ii).
So if ϕ is a diagonal form of degree d in at least d 3 + 1 variables over Q(t) then ϕ is isotropic over (Q p (t)) U for any p ∤ d!, for each reduced, irreducible component U ⊂ Y of the complement of S in Y , and isotropic over (Q p (t)) P for each P ∈ S.
Remark 10. Let us compare Corollary 9 with the Ax-Kochen-Ersov Transfer Theorem [1]: given a degree d, for almost all primes p, a form of degree d over Q p of dimension greater than or equal to d 2 + 1 is isotropic [G, (7.4)]. Moreover, for any form ϕ of degree d ≥ 2 and dimension greater than d 3 over Q(t), for almost all primes p the form ϕ is isotropic over Q p (t) ( [16] for d = 2, [12] for d ≥ 3). The model-theoretic proofs of both results do not allow for a more concrete observation on which primes exactly are included here, nor can they be extended to other base fields.
Stronger upper bounds on u diag (d, F q (t)) will yield stronger results on its dimension, since we only used the upper bound in the well known inequality d · u diag (d, F q ) ≤ u diag (d, F q (t)) ≤ d 2 to prove Corollary 9, for instance we obtain: Corollary 11. Assume that u diag (d, k(t)) = dr < d 2 for some r ∈ {1, . . . , d − 1}. Let ϕ be a diagonal form of degree d and dimension > d 2 r + 1 over a function field in one variable F = K(t). Then: (i) ϕ is isotropic over F v , for every discrete valuation v with residue field either a function field in one variable over k or a finite extension of K; (ii) ϕ is isotropic over F U for each reduced, irreducible component U ⊂ Y of the complement of S in Y and over F P for each P ∈ S.
It is well known that u diag (d, K) ≤ d 2 for a p-adic field K with residue field k = F q [3]. Indeed, u diag (d, K) = d u diag (d, F q ) by Theorem 2, assuming that char F q = p ∤ d! as before, which shows that clearly u diag (d, K) can be smaller than d 2 . On the other hand, Artin's conjecture that Q p is a C 2 -field is false for instance for forms of degree 4.