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The U-Invariant of the Function Fields of P-Adic Curves

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The U-Invariant of the Function Fields of P-Adic Curves ANNALS OF MATHEMATICS The u-invariant of the function fields of p-adic curves By Raman Parimala and V. Suresh SECOND SERIES, VOL. 172, NO. 2 September, 2010 anmaah Annals of Mathematics, 172 (2010), 1391–1405 The u-invariant of the function fields of p-adic curves By RAMAN 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 nondyadic p-adic curves. Introduction It is an open question ([Lam05, Q. 6.7, Chap XIII]) whether every quadratic form in at least nine variables over the function fields of p-adic curves has a non- trivial 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 nondyadic. In[PS98, 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 ingredi- ¤ ents in the proof were the following: Let K be the function field of a p-adic curve X and p 2. ¤ (1) (Saltman[Sal97, 3.4]). Every element in the Galois cohomology group H 2.K; Z=2Z/ is a sum of at most two symbols. 3 (2) (Kato[Kat86, 5.2]). The unramified cohomology group Hnr.K=ᐄ; Z=2Z.2// is zero for a regular projective model ᐄ of K. If K is as above, we proved ([PS98, 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 2 be a value of a given binary form a; b over K. If, further, given .f / .g/ .h/ h i D 2 H 3.K; Z=2Z/ and a ternary form c; d; e , one can choose g ; h K such that h i 0 0 2 The authors acknowledge partial financial support from NSF DMS-0653382 and UGC/SAP respec- tively . 1391 1392 RAMAN PARIMALA and V. SURESH .f / .g / .h / with g a value of c; d; e , then, one is led to the conclusion that D 0 0 0 h i u.K/ 8 (cf. Proposition 4.3). We in fact prove that such a choice of g ; h K D 0 0 2 is possible by proving the following local global principle: Let k be a p-adic field and K k.X/ the function field of a curve X over k. D For any discrete valuation v of K, let Kv denote the completion of K at v. Let l be a prime not equal to p. Assume that k contains a primitive lth root of unity. 3 2 THEOREM. Let k, K and l be as above. Let H .K; l˝ / and ˛ 2 2 2 H .K; l /. Suppose that ˛ corresponds to a degree l central division algebra over K. If ˛ .hv/ for some hv K , for all discrete valuations v of K, then D [ 2 v there exists h K such that ˛ .h/. In fact, one can restrict the hypothesis 2 D [ to discrete valuations of K centered on codimension-1 points of a regular model ᐄ, projective over the ring of integers ᏻk of k. A key ingredient toward the proof of the theorem is a recent result of Saltman [Sal07] 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-Thel´ ene` 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[CT95] and[Sch85]. Let F be a field and l a prime not equal to the characteristic of F . Let l be th i the group of l roots of unity. For i 1, let l˝ be the Galois module given by n i th the tensor product of i copies of l . For n 0, let H .F; l˝ / be the n Galois i cohomology group with coefficients in l˝ . l 1 We have the Kummer isomorphism F =F H .F; l /. For a F , 1 ' 2 its class in H .F; l / is denoted by .a/. If a1; : : : ; an F , the cup prod- n n 2 uct .a1/ .an/ H .F; l˝ / is called a symbol. We have an isomorphism 2 2 H .F; l / with the l-torsion subgroup l Br.F / of the Brauer group of F . We define the index of an element ˛ H 2.F; / to be the index of the corresponding 2 l central simple algebra in l Br.F /. Suppose F contains all the lth roots of unity. We fix a generator for the cyclic i group l and identify the Galois modules l˝ with Z=lZ. This leads to an identi- n m n n fication of H .F; l˝ / with H .F; Z=lZ/. The element in H .F; Z=lZ/ corre- n n sponding to the symbol .a1/ .an/ H .F; l˝ / through this identification is 2 2 again denoted by .a1/ .an/. In particular, for a; b F , .a/ .b/ H .K; Z=lZ/ 2 2 represents the cyclic algebra .a; b/ defined by the relations xl a, yl b and D D xy yx. D Let v be a discrete valuation of F . The residue field of v is denoted by .v/. Suppose char..v// l. Then there is a residue homomorphism ¤ n m n 1 .m 1/ @v H .F; ˝ / H ..v/; ˝ /: W l ! l THE u-INVARIANT OF THE FUNCTION FIELDS OF p-ADIC CURVES 1393 n m Let ˛ H .F; ˝ /. We say that ˛ is unramified at v if @v.˛/ 0; otherwise it is 2 l D said to be ramified at v. If F is complete with respect to v, then we denote the kernel n m of @v by Hnr.F; l˝ /. Suppose ˛ is unramified at v. Let K be a parameter n 1 .m 1/ 2 n m at v and ˛ ./ H .F; ˝ C /. Let ˛ @v./ H ..v/; ˝ /. D [ 2 C l x D 2 l The element ˛ is independent of the choice of the parameter and is called the x specialization of ˛ at v. We say that ˛ specializes to ˛ at v. The following result x is well known. LEMMA 1.1. Let k be a field and l a prime not equal to the characteristic of k. 3 3 Let K be a complete discrete valuated field with residue field k. If H .k; l˝ / 0, 3 3 2 2 D then Hnr.K; l˝ / 0. Suppose further that every element in H .k; l˝ / is a D 3 3 symbol. Then every element in H .K; l˝ / is a symbol. Proof. Let R be the ring of integers in K. The Gysin exact sequence in etale´ cohomology yields an exact sequence (cf. [C, p. 21, 3.3]) 3 3 3 3 @ 2 2 4 3 H .R; ˝ / H .K; ˝ / H .k; ˝ / H .R; ˝ /: et l ! l ! l ! et l K 3 3 3 3 K Since R is complete, Het.R; l˝ / H .k; l˝ / ([Mil80, p. 224, Cor. 2.7]). 3 3 K ' 3 3 Hence Het.R; l˝ / 0 by the hypothesis. In particular, @ H .K; l˝ / 2 K2 D 3 3 W ! H .k; l˝ / is injective and Hnr.K; l˝ / 0. Let u; v R be units and R D 2 3 23 a parameter. Then we have @..u/ .v/ .// .u/ .v/. Let H .K; l˝ /. 2 2 D x x 2 Since every element in H .k; ˝ / is a symbol, we have @./ .u/ .v/ for some l D x x units u; v R. Since @ is an isomorphism, we have .u/ .v/ ./. Thus every 2 3 3 D element in H .K; l˝ / is a symbol. COROLLARY 1.2. Let k be a p-adic field and K the function field of an in- tegral curve over k. Let l be a prime not equal to p. Let Kv be the completion 3 3 of K at a discrete valuation of K. Then Hnr.Kv; l˝ / 0. Suppose further that th D 3 3 K contains a primitive l root of unity. Then every element in H .Kv; l˝ / is a symbol. Proof. Let v be a discrete valuation of K and Kv the completion of K at v. The residue field .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 .v/ n 3 3 3 is 2 and hence H ..v/; ˝ / 0 for n 3. By Lemma 1.1, H .Kv; ˝ / 0. l D nr l D If .v/ is a local field, by local class field theory, every finite-dimensional central division algebra over .v/ is split by an unramified (cyclic) extension. If .v/ is a function field of a curve over a finite field, then by a classical theorem of Hasse-Brauer-Noether-Albert, every finite-dimensional central division algebra over .v/ is split by a cyclic extension.
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