ANNALS OF

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 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 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 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 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 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 of F . Let l be th i the group of l roots of unity. For i 1, let l˝ be the 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 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 l Br.F / of the 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 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-INVARIANTOFTHEFUNCTIONFIELDSOF 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 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 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. Since .v/ contains a primitive lth root of unity, every element in H 2..v/; Z=lZ/ is a symbol. By Lemma 1.1, every 3 element in H .Kv; Z=lZ/ is a symbol.  Let ᐄ be a regular integral scheme of dimension d, with field of fractions F . Let ᐄ1 be the set of points of ᐄ of codimension-1. A point x ᐄ1 gives rise 2 1394 RAMAN PARIMALA and V. SURESH to a discrete valuation vx on F . The residue field of this is denoted by .x/ or .vx/. The corresponding residue homomorphism is de- n m noted by @x. We say that an element  H .F; ˝ / is unramified at x if 2 l @x./ 0; otherwise it is said to be ramified at x. We define the ramification D P 1 divisor ramᐄ./ x as x runs over ᐄ where  is ramified. The unramified D n m cohomology on ᐄ, denoted by Hnr.F=ᐄ; l˝ /, is defined as the intersection of kernels of the residue homomorphisms n m n 1 .m 1/ @x H .F; ˝ / H ..x/; ˝ /; W l ! l 1 n m with x running over ᐄ . We say that  H .F; l˝ / is unramified on ᐄ if n m 2  H .F=ᐄ; ˝ /. If ᐄ Spec.R/, then we also say that  is unramified 2 nr l D on R if it is unramified on ᐄ. Suppose C is an irreducible subscheme of ᐄ of 1 codimension-1. Then the generic point x of C belongs to ᐄ and we set @x = @C . n m If ˛ H .F; ˝ / is unramified at x, then we say that ˛ is unramified at C . 2 l Let k be a p-adic field and K the function field of a smooth, projective, geo- metrically integral curve X over k. By the resolution of singularities for surfaces (cf.[Lip75] and[Lip78]), there exists a regular, projective model ᐄ of X over the ring of integers ᏻk of k. We call such an ᐄ a regular projective model of K. Since the generic fibre X of ᐄ is geometrically integral, it follows that the special fibre is connected. Further if D is a divisor on , there exists a proper birational ᐄx ᐄ morphism ᐄ ᐄ such that the total transform of D on ᐄ is a divisor with nor- 0 ! 0 mal crossings (cf.[Sha66, Thm., p. 38 and Rem. 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, geo- metrically integral curve over k. Let l be a prime not equal to p. Assume that k con- tains a primitive lth root of unity. Let ˛ H 2.K;  /. Let ᐄ be a regular projective 2 l model of K such that the ramification locus ramᐄ.˛/ is a union of regular curves with normal crossings. Let P be a closed point in the intersection of two regular 1 curves C and E in ram .˛/. Suppose that @C .˛/ H ..C /; Z=lZ/ and @E .˛/ ᐄ 2 2 H 1..E/; Z=lZ/ are unramified at P . Let u.P /; v.P / H 1..P /; Z=lZ/ be the 2 specializations at P of @C .˛/ and @E .˛/ respectively. Following Saltman ([Sal07, 2]), we say that P is a cool point if u.P / and v.P / are trivial, a chilli point if u.P / and v.P / both are nontrivial, and a hot point if one of them is trivial and the other one nontrivial. Note that if u.P / is nontrivial, then u.P / gener- 1 ates H ..P /; Z=lZ/. Let ᏻᐄ;P be the regular at P and , ı prime elements in ᏻᐄ;P which define C and E respectively at P . The condition that 1 1 @C .˛/ H ..C /; Z=lZ/ and @E .˛/ H ..E/; Z=lZ/ are unramified at P is 2 2 equivalent to the condition ˛ ˛ .u; / .v; ı/ for some units u; v ᏻ ;P and D 0 C C 2 ᐄ ˛0 unramified on ᏻᐄ;P ([Sal98, 2]). The specializations of @C .˛/ and @E .˛/ in l H 1..P /; Z=lZ/ .P / =.P / are given by the images of u and v in .P /. '   Let P be a closed point of a regular curve C in ramᐄ.˛/ which is not on any other regular curve in ram .˛/. We have ˛ ˛ .u; /, where ˛ is unramified ᐄ D 0 C 0 THE u-INVARIANTOFTHEFUNCTIONFIELDSOF p-ADIC CURVES 1395 on ᏻᐄ;P , u ᏻᐄ;P is a unit and  ᏻᐄ;P is a prime defining the curve C at P ; 2 2 1 see[Sal97, 1.2]. Therefore @C .˛/ .u/ H ..C /; Z=lZ/ is unramified at P . D x 2 PROPOSITION 1.3 ([Sal07, 2.5]). If the index of ˛ is l, then there are no hot points for ˛. Suppose P is a chilli point. Then v.P / u.P /s for some s with 1 s l 1 D Ä Ä and s is called the coefficient of P ([Sal97, p. 830]) with respect to . To get some compatibility for these coefficients, Saltman associates to ˛ and ᐄ the following graph: The set of vertices is the set of irreducible curves in ramᐄ.˛/ and there is an edge between two vertices if there is a chilli point in the intersection of the two irreducible curves corresponding to the vertices. A loop in this graph is called a chilli loop.

PROPOSITION 1.4 ([Sal07, 2.6, 2.9]). There exists a projective model ᐄ of K such that there are no chilli loops and no cool points on ᐄ for ˛. 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 : D f j g 2 2 For a1; : : : ; an F , we denote the diagonal quadratic form a1X anX 2  1 C    C n by a1; : : : ; an . Let W.F / be the Witt ring of quadratic forms over F and I.F / h i be the ideal of W.F / consisting of even dimension forms. Let I n.F / be the nth power of the ideal I.F /. For a1; : : : ; an F , let a1; : : : ; an denote the n-fold 2 hh n ii Pfister form 1; a1 1; an . The I .F / is generated by n-fold h i˝  ˝h i Pfister forms. The dimension modulo 2 gives an isomorphism e0 W.F /=I.F / W ! H 0.F; Z=2Z/. The gives an isomorphism 2 1 e1 I.F /=I .F / H .F; Z=2Z/: W ! The classical result of Merkurjev[Mer81], asserts that the Clifford invariant gives 2 3 2 an isomorphism e2 I .F /=I .F / H .F; Z=2Z/. W ! Let Pn.F / be the set of isometry classes of n-fold Pfister forms over F . There is a well-defined map ([Ara75]) n en Pn.F / H .F; Z=2Z/ W ! n given by en. 1; a1 1; an / . a1/ . an/ H .F; Z=2Z/. h i ˝    ˝ h i D    2 A quadratic form version of the Milnor conjecture asserts that en induces a surjective homomorphism I n.F / H n.F; Z=2Z/ with kernel I n 1.F /. This ! C conjecture was proved by Voevodsky, Orlov and Vishik. In this paper we are inter- ested in fields of 2-cohomological dimension at most 3. For such fields, Milnor’s conjecture above has already been proved by Arason, Elman and Jacob[AEJ86, Cor. 4 and Th. 2], using the theorem of Merkurjev[Mer81]. Let q1 and q2 be two quadratic forms over F . We write q1 q2 if they D represent the same class in the W.F /. We write q1 q2, if q1 and ' 1396 RAMAN PARIMALA and V. SURESH q2 are isometric quadratic forms. We note that if the dimensions of q1 and q2 are equal and q1 q2, then q1 q2. D ' 2. Divisors on arithmetic surfaces In this section we recall a few results from a paper of Saltman[Sal07] on divisors on arithmetic surfaces. Let ᐆ be a connected, reduced scheme of finite type over a .

Let ᏻᐆ be the sheaf of units in the structure sheaf ᏻᐆ. Let ᏼ be a finite set of closed points of ᐆ. For each P ᏼ, let .P / be the residue field at P and 2 P Spec..P // ᐆ be the natural morphism. Consider the sheaf W ! ᏼ P ᏼP .P /; D ˚ 2 where .P / denotes the group of units in .P /. Then there is a surjective mor- .1/ phism of sheaves ᏻᐆ ᏼ given by the evaluation at each P ᏼ. Let ᏻᐆ;ᏼ be ! .1/ 2.1/ its kernel. When there is no ambiguity we denote ᏻᐆ;ᏼ by ᏻᏼ . Let ᏷ be the sheaf of total quotient rings on ᐆ and ᏷ be the sheaf of groups given by units 0 in ᏷. Every element H .ᐆ; ᏷=ᏻ/ can be represented by a family Ui ; fi , 2 1 f g where Ui are open sets in ᐆ, fi ᏷.Ui / and fi fj ᏻ.Ui Uj /. We say that 0 2 2 \ an element Ui ; fi of H .ᐆ; ᏷=ᏻ/ avoids ᏼ if each fi is a unit at P for D f 0g 0 all P Ui ᏼ. Let H .ᐆ; ᏷ =ᏻ / be the subgroup of H .ᐆ; ᏷ =ᏻ / consisting 2 \ ᏼ     of those which avoid ᏼ. Let K H 0.ᐆ; ᏷ / and K be the subgroup of K  D  ᏼ  consisting of those functions which are units at all P ᏼ. We have a natural 0 2 inclusion Kᏼ Hᏼ.ᐆ; ᏷=ᏻ/ . P ᏼ.P //. ! ˚ ˚ 2 Now, we have PROPOSITION 2.1 ([Sal07, 1.6]). Let ᐆ be a connected, reduced scheme of finite type over a Noetherian ring. Then 0 1 .1/ Hᏼ.ᐆ; ᏷=ᏻ/ . P ᏼ.P // H .ᐆ; ᏻᏼ / ˚ ˚ 2 : ' Kᏼ

Let k be a p-adic field and ᏻk the ring of integers of k. Let ᐄ be a connected regular surface with a projective morphism  ᐄ Spec.ᏻ /. Let ᐄ be the reduced W ! k x special fibre of . Assume that is connected. Note that is connected if the ᐄx ᐄx generic fibre is geometrically integral. Let ᏼ be a finite set of closed points in ᐄ. Since every closed point of is in , is also a subset of closed points of . Let ᐄ ᐄx ᏼ ᐄx m be an integer coprime with p.

PROPOSITION 2.2 ([Sal07, 1.7]). The canonical map 1 .1/ 1 .1/ H .ᐄ; ᏻ / H .ᐄ; ᏻ / ᐄ;ᏼ x ᐄ;ᏼ ! x induces an isomorphism .1/ 1 .1/ 1 H .ᐄ; ᏻ / H .ᐄ; ᏻᐄ;ᏼ / x ᐄ;ᏼ x : 1 .1/ 1 .1/ mH .ᐄ; ᏻ / ' mH .ᐄ; ᏻ / ᐄ;ᏼ x ᐄ;ᏼ x THE u-INVARIANTOFTHEFUNCTIONFIELDSOF p-ADIC CURVES 1397

Let be as above. Suppose that is a union of regular curves F ;:::;F on ᐄ ᐄx 1 m ᐄ with only normal crossings. Let ᏼ be a finite set of closed points of ᐄ including all the points of Fi Fj , i j and at least one point from each Fi . Let E be a \ ¤ divisor on ᐄ whose support does not pass through any point of ᏼ. In particular, no Fi is in the support of E. Hence there are only finitely many closed points Q1;:::;Qn on the support of E. For each closed point Qi on the support of E, let Di be a regular curve on ᐄ not contained in the special fiber of ᐄ such that Qi is the multiplicity one intersection of D and . Such a curve exists by ([Sal07, i ᐄx 1.1]). We note that any closed point on ᐄ is a point of codimension-2 and there is a unique closed point on any geometric curve on ᐄ (cf. 1). The following is extracted from[Sal07, 5].

PROPOSITION 2.3. Let ᐄ; ᏼ;E;Qi ;Di be as above. For each closed point Qi , let mi be the intersection multiplicity of the support of E and the special fibre ᐄ at Qi . Let l be a prime not equal to p. Then there exist  K and a divisor E x 2  0 on ᐄ such that n X ./ E mi Di lE D C C 0 i 1 l D and .P / .P / for each P ᏼ. 2  2 P Proof. Let F be the divisor on ᐄ given by Fi . Let Pic.ᐄ/ be the line bun- 2 dle equivalent to the class of the divisor E and Pic.ᐄ/ its image. Since the sup- x 2 x port of E does not pass through the points of ᏼ and ᏼ contains all the points of in- tersection of distinct Fi , E and F intersect only at smooth points of ᐄ. In particular, P 1 x mi Qi . Let 0 H .ᐄ; ᏻ / be the element which, under the isomorphism x D 2 ᏼ P of Proposition 2.1, corresponds to the class of the element . E mi Di ; 1/ in 0 C Hᏼ.ᐄ; ᏷=ᏻ/ . P ᏼ.P //. Since the mi ’s are intersection multiplicities ˚ ˚ 2 of E and ᐄ at Q and the image of P m D in H 0.ᐄ; ᏷ =ᏻ / is P m Q , the x i i i ᏼ x   i i 1 1 image 0 of 0 in H .ᐄ; ᏻᏼ / is zero. By Proposition 2.2, we have 0 lH .ᐄ; ᏻᏼ /. x 0 2 Using Proposition 2.1, there exists .E0; .P // Hᏼ.ᐆ; ᏷=ᏻ/ . P ᏼ.P // P 2 l ˚ ˚ 2 such that . E mi Di ; 1/ l.E0; .P // .lE0; .P // modulo Kᏼ. Thus C D D P l there exists  Kᏼ K such that ./ . E mi Di ; 1/ .lE0; .P //.That 2 P  D l C is, ./ E mi Di lE and .P /  for each P ᏼ.  D C 0 D P 2 3. A local-global principle

Let k be a p-adic field, ᏻk 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 Remark 3.6, we assume that k contains a th primitive l root of unity. We fix a generator  for l and identify l with Z=lZ. LEMMA 3.1. Let ˛ H 2.K;  /. Let ᐄ be a regular projective model of K. 2 l Assume that the ramification locus ram .˛/ is a union of regular curves C1;:::;Cr ᐄ f g with only normal crossings. Let T be a finite set of closed points of ᐄ including the 1398 RAMAN PARIMALA and V. SURESH points of Ci Cj , for all i j . Let D be an irreducible curve on ᐄ which is not \ ¤ in the ramification locus of ˛ and does not pass through any point in T . Then D intersects Ci at points P where @Ci .˛/ is unramified. Suppose further that at such 1 points P , @Ci .˛/ specializes to 0 in H ..P /; Z=lZ/. Then ˛, which is unramified 2 at D, specializes to 0 in H ..D/; l /. Proof. Since k contains a primitive lth root of unity, we fix a generator  for j l and identify the Galois modules l˝ with Z=lZ. Let P be a point in the intersection of D and the support of ramᐄ.˛/. Since D does not pass through the points of T and T contains all the points of intersection of distinct Cj , the point P belongs to a unique curve Ci in the support of ramᐄ.˛/. 1 Thus @Ci .˛/ .u/ H ..Ci /; Z=lZ/ is unramified at P (cf. 1). D x 2 1 Suppose that @Ci .˛/ specializes to zero in H ..P /; Z=lZ/. Since D is not in the ramification locus of ˛, ˛ is unramified at D. Let ˛ be the specialization of x ˛ in H 2..D/; Z=lZ/. Since .D/ is either a p-adic field or a function field of a curve over a finite field, to show that ˛ is zero, by class field theory it is enough to x show that ˛ is unramified at every discrete valuation of .D/. x Let v be a discrete valuation of .D/ and R the corresponding discrete valu- ation 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 ᏻD;P of P on D. The local ring ᏻD;P is a quotient of the local ring ᏻᐄ;P . Suppose P is not on the ramification locus of ˛. Then ˛ is unramified on ᏻ ;P and hence ˛ on ᏻD;P . In particular, ˛ is unramified at R. ᐄ x x Suppose P is on the ramification locus of ˛. As before, we have ˛ ˛ D 0 C .u; /, where ˛0 is unramified on ᏻᐄ;P , u ᏻᐄ;P is a unit and  ᏻᐄ;P is a prime 2 2 l defining the curve Ci at P . Therefore @Ci .˛/ u in .Ci /=.Ci / . Since, D x l by the assumption, @Ci .˛/ specializes to 0 at P , u.P / .P / . We have ˛ 2 2 x D ˛0 .u; / H ..D/; Z=lZ/. Since ˛0 is unramified at P , the residue of ˛ at R C x x./2 x is .u.P // x . Since .P / is contained in the residue field of the discrete valuation ring R and u.P / is an lth power in .P /, it follows that ˛ is unramified at R.  x PROPOSITION 3.2. Let K and l be as above. Let ˛ H 2.K;  / with index l. 2 l Let ᐄ be a regular projective model of K such that the ramification locus ramᐄ.˛/ and the special fibre of ᐄ are a union of regular curves with only normal crossings and ˛ has no cool points and no chilli loops on ᐄ (cf. Proposition 1.4). Let si be the corresponding coefficients (cf. §1). Let F1;:::;Fr be irreducible regular curves on ᐄ which are not in ram .˛/ C1;:::;Cn and such that F1;:::;Fr ram .˛/ ᐄ D f g f g[ ᐄ have only normal crossings. Let m1; : : : ; mr be integers. Then there exists f K 2  such that X X X div .f / si Ci msFs nj Dj lE ; ᐄ D C C C 0 where D1;:::;Dt are irreducible curves which are not equal to Ci and Fs for all i and s and ˛ specializes to zero at Dj for all j and .nj ; l/ 1. D THE u-INVARIANTOFTHEFUNCTIONFIELDSOF p-ADIC CURVES 1399

Proof. Let T be a finite set of closed points of ᐄ containing all the points of intersection of distinct Ci and Fs, and at least one point from each Ci and Fs. By a P P semilocal argument, we choose g K such that div .g/ si Ci msFs G 2  ᐄ D C C where G is a divisor on ᐄ whose support does not contain any of Ci or Fs and does not intersect T . Since ˛ has no cool points and no chilli loops on ᐄ, by[Sal07, Prop. 4.6], P P there exists u K such that div .ug/ si Ci msFs E, where E is a 2  ᐄ D C C divisor of ᐄ whose support does not contain any Ci or Fs, does not pass through the points in T and either E intersects Ci at a point P where the specialization of @C .˛/ is 0 or the intersection multiplicity .E Ci /P is a multiple of l. i  Suppose Ci for some i is a geometric curve on ᐄ. Then the closed point of Ci is in T . Since the support of E avoids all the points in T , the support of E does not intersect Ci . Thus the support of E intersects only those Ci which are in the special fibre . Let Q ;:::;Q be the points of intersection of the ᐄx 1 t support of the divisor E and the special fibre with intersection multiplicity nj at Qj coprime with l. For each Qj , let Dj be a regular geometric curve on ᐄ such that Q is the multiplicity one intersection of D and (cf. paragraph after j j ᐄx Proposition 2.2). Then by Proposition 2.3 there exists  K such that divᐄ./ P l 2 D E nj Dj lE and .P / .P / for all P T . Let f ug K . Then C C 0 2  2 D 2  X X X div .f / si Ci msFs nj Dj lE : ᐄ D C C C 0

Since each Qj is the only closed point on Dj and @Ci .˛/ specializes to zero at Qj , by Lemma 3.1, the ˛ specializes to 0 at Dj . Thus f has all the required properties.  2 LEMMA 3.3. Let ˛ H .K; l / and let v be a discrete valuation of K. Let 2 l u K be a unit at v such that u .v/ .v/ . Suppose further that if ˛ is 2 1 x 2 n 1 ramified at v, @v.˛/ ŒL H ..v/; Z=lZ/, where L K.u l /. Then, for any D 2 3 2 D g L , the image of ˛ .N .g// H .Kv; ˝ / is zero. 2  [ L=K 2 l j Proof. We identify the Galois modules l˝ with Z=lZ as before. Since u is l a unit at v and u .v/ , there is a unique discrete valuation v of L extending x 62  Q the valuation v of K, which is unramified with residual degree l. In particular, 2 v.NL=K .g// is a multiple of l. Thus if ˛0 H .Kv; Z=lZ/ is unramified at v, 3 2 3 then ˛0 .NL=K .g// H .Kv; Z=lZ/ is unramified. Since Hnr.Kv; Z=lZ/ 0 [ 2 2 D (cf. Corollary 1.2), we have ˛ .N .g// 0 for any ˛ H .Kv; Z=lZ/ which 0 [ L=K D 0 2 is unramified at v. In particular, if ˛ is unramified at v, then ˛ .N .g// 0. [ L=K D Suppose that ˛ is ramified at v. Then by the choice of u, we have ˛ 2 D ˛ .u/ .v/, where v is a parameter at v and ˛ H .Kv; Z=lZ/ is unramified 0 C  0 2 at v. Thus we have

˛ .N .g// ˛ .N .g// .N .g// .u/ .v/ [ L=K D 0 [ L=K C L=K   3 .N .g// .u/ .v/ H .Kv; Z=lZ/: D L=K   2 1400 RAMAN PARIMALA and V. SURESH

1 2 Since Lv Kv.u l /, we have ..NL=K .g// .u/ 0 H .Kv; Z=lZ/ and ˛ D 3  D 2 [ .N .g// 0 in H .Kv; Z=lZ/.  L=K D 2 THEOREM 3.4. Let K and l be as above. Let ˛ H .K; l / and  3 2 2 2 H .K; l˝ /. Assume that the index of ˛ is l. Let ᐄ be a regular projective 1 model of K. Suppose that for each x ᐄ , there exists fx Kx such that 3 2 2 2  ˛ .fx/ H .Kx; l˝ /, where Kx is the completion of K at the discrete val- D [ 2 3 2 uation given by x. Then there exists f K such that  ˛ .f / H .K; ˝ /. 2  D [ 2 l j Proof. We identify the Galois modules l˝ with Z=lZ as before. By weak 1 approximation, we may find f K such that .f / .fv/ H .Kv; Z=lZ/ for 2  D 2 all the discrete valuations corresponding to the irreducible curves in ram .˛/ ᐄ [ ramᐄ./. Let X div .f / C mi Fi lE; ᐄ D 0 C C where C is a divisor with support contained in ram .˛/ ram ./, Fi ’s are distinct 0 ᐄ [ ᐄ irreducible curves which are not in ram .˛/ ram ./, mi is coprime with l and ᐄ [ ᐄ E is some divisor on ᐄ. l For any Cj ramᐄ./ ramᐄ.˛/, let j .Cj / .Cj / . By weak ap- 2 n 2 n1 proximation, we choose u K with u @C .˛/ H ..Ci /; Z=lZ/ for all 2  x D i 2 Ci ram .˛/, F .u/ mi , where F is the discrete valuation at Fi and u j 2 ᐄ i D i x D for any Cj ramᐄ./ ramᐄ.˛/. In particular, u is a unit at the generic point of 2 l n Cj and u .Cj / for any Cj ramᐄ./ ramᐄ.˛/. x 62 1 2 n Let L K.u l /. Let  ᐅ ᐄ be the normalization of ᐄ in L. Since F .u/ D W ! i D mi and mi is coprime with l,  ᐅ ᐄ ramified at Fi . In particular, there is a W ! unique irreducible curve Fei in ᐅ such that .Fei / Fi and .Fi / .Fei /. D D Let  ᐅ ᐅ be a proper birational morphism such that the ramification locus W z ! ram .˛ / of ˛ on ᐅ and the strict transform of the curves F on ᐅ is a union ᐅ L L z ei z of regularz curves with only normal crossings and there are no cool points and no chilli loops for ˛ on (cf. Proposition 1.4). We denote the strict transforms of L ᐅz Fei by Fei again. By Proposition 3.2, there exists g L such that 2  X X divᐅ.g/ C mi Fei nj Dj lD; z D C C C where the support of C is contained in ramᐅ.˛L/ and Dj ’s are irreducible curves Q which are not in ramᐅ.˛L/ and ˛L specializes to zero at all Dj ’s. We now claim thatz  ˛ .f N .g//. Since the group H 3 .K=ᐄ; Z=lZ/ 0 D [ L=K nr D ([K, 5.2]), it is enough to show that  ˛ .f N .g// is unramified on ᐄ. Let [ L=K S be an irreducible curve on ᐄ. Since the residue map @S factors through the completion KS , it suffices to show that  ˛ .f N .g// 0 over KS . [ L=K D Suppose S is not in ram .˛/ ram ./ Supp.f N .g//. Then each of  ᐄ [ ᐄ [ L=K and ˛ .f N .g// is unramified at S. [ L=K Suppose that S is in ramᐄ.˛/ ramᐄ./. Then by the choice of f we have 1 [ .f / .fv/ H .Kv; Z=lZ/ where v is the discrete valuation associated to S. D 2 THE u-INVARIANTOFTHEFUNCTIONFIELDSOF p-ADIC CURVES 1401

Hence  ˛ .f / over the completion KS of K at the discrete valuation given D [ by S. It follows from Lemma 3.3 that .N .g// ˛ 0 over KS and  L=K [ D D ˛ .f N .g// over KS . [ L=K Suppose that S is in the support of div .f N .g// and not in ram .˛/ ᐄ L=K ᐄ [ ram ./. Then ˛ and  are unramified at S. We show that in this case ˛ ᐄ [ .f N .g//  0 over KS . Now, L=K D D div .f N .g// div .f / div .N .g// ᐄ L=K D ᐄ C ᐄ L=K X  X X Á C 0 mi Fi lE   C mi Fei nj Dj lD D C C C   C C C X C 0   .C / nj   .Dj / lE0 D C   C   C for some E0. We note that if Dj maps to a point, then   .Dj / 0. Since the   D support of C is contained in ramᐅ.˛L/, the support of   .C / is contained in z   ramᐄ.˛/. Thus S is in the support of   .Dj / for some j or S is in the support   of l  .E/. In the later case, clearly ˛ .f NL=K .g// is unramified at S and   [ hence ˛ .f NL=K .g// 0 over KS . Suppose S is in the support of   .Dj / for [ D   some j . In this case, if Dj lies over an inert curve, then   .Dj / is a multiple of   l and we are done. Suppose that Dj lies over a split curve. Since ˛L specializes to zero at Dj , it follows that ˛ specializes to zero at   .Dj / 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 lth roots of unity are 3 3 in K. Then every element in H .K; l˝ / is a symbol. j Proof. We again identify the Galois modules l˝ with Z=lZ. Let v be a discrete valuation of K and Kv the completion of K at v. By 3 Corollary 1.2, every element in H .Kv; Z=lZ/ is a symbol. Let  H 3.K; Z=lZ/ and ᐄ be a regular projective model of K. Let v be a 2 discrete valuation of K corresponding to an irreducible curve in ramᐄ./. Then we have  .fv/ .gv/ .hv/ for some fv; gv; hv Kv. By weak approximation, we D   2 1 can find f; g K such that .f / .fv/ and .g/ .gv/ in H .Kv; Z=lZ/ for all 2  D D discrete valuations v corresponding to the irreducible curves in ramᐄ./. Let v be a discrete valuation of K corresponding to an irreducible curve C in ᐄ. If C is in 3 ramᐄ./, then by the choice of f and g we have  .f / .g/ .hv/ H .Kv; Z=lZ/. 3 D  3 2 If C is not in ramᐄ./, then  Hnr.Kv; Z=lZ/ H ..v/; Z=lZ/ 0. In 2 3 ' D particular, we have  .f / .g/ .1/ H .Kv; Z=lZ/. Let ˛ .f / .g/ 2 D   2 3 D  2 H .K; Z=lZ/. Then we have  ˛ .h / H .Kv; Z=lZ/ for some h K D [ v0 2 v0 2 v for each discrete valuation v of K associated to any point of ᐄ1. By Theorem 3.4, there exists h K such that  ˛ .h/ .f / .g/ .h/ H 3.K; Z=lZ/:  2  D [ D   2 Remark 3.6. We note that all the results of this section can be extended to the situation where k does not necessarily contain a primitive lth root of unity. This can be achieved by going to the extension k0 of k obtained by adjoining a primitive 1402 RAMAN PARIMALA and V. SURESH

th l root of unity to k and noting that the extension k0=k is unramified of degree l 1. We do not use this remark in the sequel. 4. The u-invariant

In Proposition 4.1 and Proposition 4.2 below, we give some necessary con- ditions 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 Kv is the completion of K at a discrete valuation v of K, then the residue field .v/ of Kv, which is either a global field of positive characteristic or a p-adic field, has u-invariant 4. By a theorem of Springer, u.Kv/ 8 and we use Propositions 4.1 and 4.2 for Kv. D 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. Ä D Further, if  is a 3-fold Pfister form and q2 a rank 2 quadratic form over K, then there exists f; g; h K such that f is a value of q2 and  1; f 1; g 1; h . 2  D h ih ih i Proof. Suppose that u.K/ 8. Then every 4-fold Pfister form is isotropic D and hence hyperbolic; in particular, I 4.K/ 0. Let  be an anisotropic quadratic D form representing an element in I 3.K/. Since u.K/ 8, the rank of  is 8 (cf. Ä [Sch85, p. 156, Th. 5.6]). Then  is a scalar multiple of a 3-fold Pfister form (cf. [Lam05, Ch. X, Th. 5.6]). Since I 4.K/ 0,  is a 3-fold Pfister form. D Let  1; a 1; b 1; c be a 3-fold Pfister form and  be its pure subform. D h ih ih i 0 Let q2 be a quadratic form over K of dimension 2. Since dim. / 7 and u.K/ 8, 0 D Ä the quadratic form  q2 is isotropic. Therefore there exists f K which is 0 2  a value of q2 and  f  for some quadratic form  over K. Hence by 0 ' h i C 00 00 [Sch85, p. 143],  1; f 1; b 1; c for some b ; c K .  D h ih 0ih 0i 0 0 2  PROPOSITION 4.2. Let K be a field of characteristic not equal to 2. Suppose that u.K/ 8. Let  1; f 1; a 1; b be a 3-fold Pfister form over K and q3 Ä D h ih ih i a quadratic form over K of dimension 3. Then there exist g; h K such that g is 2  a value of q3 and  1; f 1; g 1; h . D h ih ih i Proof. Let 1; f a; b; ab . Since u.K/ 8, the quadratic form q3 is D h ih i Ä isotropic. Hence there exists g K which is a common value of q3 and . Thus, 2  g 1 for some quadratic form 1 over K. Since is hyperbolic over 'p h i C K. f /, 1 1; f a1; b1 g1 for some a1; b1; g1 K . By comparing ' h ih i C h i 2  the determinants, we get g1 gf modulo squares. Hence 1; f g; a1; b1 D D h ih i and  1; f 1; f 1; g; a1; b1 . The form  is isotropic and hence D h i C D h ih i hyperbolic over the function field of the conic given by f; g; fg . Hence, as in h i Proposition 4.1,   1; f 1; g 1; h for some ; h K . Since I 4.K/ 0, D h ih ih i 2  D  1; f 1; g 1; h with g a value of q3.  D h ih ih i PROPOSITION 4.3. Let K be a field of characteristic not equal to 2. Assume the following: (1) Every element in H 2.K; Z=2Z/ is a sum of at most two symbols. THE u-INVARIANTOFTHEFUNCTIONFIELDSOF p-ADIC CURVES 1403

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

(3) If  is a 3-fold Pfister form and q2 is a quadratic form over K of dimension 2, then  1; f 1; g 1; h for some f; g; h K with f a value of q2. D h ih ih i 2  (4) If  1; f 1; a 1; b is a 3-fold Pfister form and q3 a quadratic form over D h ih ih i K of dimension 3, then  1; f 1; g 1; h for some g; h K with g a D h ih ih i 2  value of q3. (5) I 4.K/ 0: D Then u.K/ 8. Ä Proof. Let q be a quadratic form over K of dimension 9. Since every element in H 2.K; Z=2Z/ is a sum of at most two symbols, as in[PS98, proof of 4.5], we find a quadratic form q5  1; a1; a2; a3; a4 over K such that  q q5 D h i D C 2 I 3.K/. By assumptions (2), (3) and (4), there exist f; g; h K such that  2  D 1; f 1; g 1; h and f is a value of a1; a2 and g is a value of fa1a2; a3; a4 . h ih ih i h i h i We have a1; a2 f; fa1a2 and fa1a2; a2; a3 g; g1; g2 for some g1; g2 h i ' h i h i ' h i 2 K . Since I 4.K/ 0, we have   and  D D q q q5 q5 D C  q5 D  q5 D 1; f 1; g 1; h 1; a1; a2; a3; a4 D h ih ih i h i 1; f 1; g 1; h 1; f; g; g1; g2 D h ih ih i h i gf 1; f h; gh g1; g2 : D h i C h ih i h i The above equalities are in the Witt group of K. Since the dimension of q is 9 and the dimension of gf 1; f h; gh g1; g2 is 7, it follows that q, and h i ? h ih i h i 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  be a 3-fold Pfister form over K and q2 a quadratic form over K of dimension 2. Then there exist f; a; b K such that f is a value of q2 and 2   1; f 1; a 1; b . D h ih ih i 3 Proof. Let  e3./ H .K; Z=2Z/. Let ᐄ be a projective regular model D 2 of K. Let C be an irreducible curve on ᐄ and v be the discrete valuation given by C . Let Kv be the completion of K at v. Since the residue field .v/ .C / is D either a p-adic field or a function field of a curve over a finite field, u..v// 4 D and u.Kv/ 8 ([Sch85, p. 209]). By Proposition 4.1, there exist fv; av; bv K D 2 v such that fv is a value of q2 over Kv and  1; fv 1; av 1; bv over Kv. By D h ih ih i weak approximation, we can find f; a K such that f is a value of q2 over K 2 2 and f fv; a av modulo K for all discrete valuations v corresponding to the D D v irreducible curves C in the support of ramᐄ./. Let C be any irreducible curve on ᐄ and v be the discrete valuation of K given by C . If C is in the support of ram ./, then by the choice of f and a, we have  e3./ . f / . a/ . bv/ ᐄ D D   1404 RAMAN PARIMALA and V. SURESH

3 over Kv. If C is not in the support of ramᐄ./, then  Hnr.Kv; Z=2Z/ 3 2 ' H ..v/; Z=2Z/ 0. In particular, we have  . f / . a/ .1/ over Kv. Let D 2 D   ˛ . f / . a/ H .K; Z=2Z/. By Theorem 3.4, there exists b K such D  2 3 3 3 2 that  ˛ . b/ H .K; Z=2Z/. Since e3 I .K/ H .K; Z=2Z/ is an D [ 2 W ! isomorphism, we have  1; f 1; a 1; b as required.  D h ih ih i There is a different proof of Proposition 4.4 in[PS98, 4.4]! PROPOSITION 4.5. Let k be a p-adic field, p 2 and K be a function field ¤ of a curve over k. Let  1; f 1; a 1; b be a 3-fold Pfister form over K and D h ih ih i q3 a quadratic form over K of dimension 3. Then there exist g; h K such that 2  g is a value of q3 and  1; f 1; g 1; h . D h ih ih i 3 Proof. Let  e3./ . f / . a/ . b/ H .K; Z=2Z/. Let ᐄ be a D D   2 projective regular model of K. Let C be an irreducible curve on ᐄ and v be the discrete valuation of K given by C . Let Kv be the completion of K at v. Then as in the proof of Proposition 4.4, we have u.Kv/ 8. Thus by Proposition 4.2, D there exist gv; hv K such that gv is a value of the quadratic form q3 and 2 v  1; f 1; gv 1; hv over Kv. By weak approximation, we can find g K D h ih ih i 2 2 such that g is a value of q3 over K and g gv modulo K for all discrete D v valuations v of K given by the irreducible curves C in ramᐄ./. Let C be an irreducible curve on ᐄ and v be the discrete valuation of K given by C . By the choice of g it is clear that  e3./ . f / . g/ . hv/ for all the discrete D D   valuations v of K given by the irreducible curves C in the support of ramᐄ./. If C is not in the support of ramᐄ./, then as in the proof of Proposition 4.4, we 2 have  . f / . g/ .1/ over Kv. Let ˛ . f / . g/ H .K; Z=2Z/. By D   D  2 Theorem 3.4, there exists h K such that  ˛ . h/ . f / . g/ . h/. 3 3 2 D [ D   Since e3 I .K/ H .K; Z=2Z/ is an isomorphism,  1; f 1; g 1; h .  W ! D h ih ih i 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. ¤ D 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 ([Sal97, 3.4]; cf.[Sal98]), every element in ¤ H 2.K; Z=2Z/ is a sum of at most two symbols. Since the cohomological dimen- sion of K is 3, we also have I 4.K/ H 4.K; Z=2Z/ 0 ([AEJ86]). Now the ' D theorem follows from Propositions 4.3, 4.4 and 4.5. 

References

[Ara75] J.K.ARASON, Cohomologische invarianten quadratischer Formen, J. Algebra 36 (1975), 448–491. MR 52 #10592 Zbl 0314.12104 [AEJ86] J.K.ARASON, R.ELMAN, and B.JACOB, Fields of cohomological 2-dimension three, Math. Ann. 274 (1986), 649–657. MR 87m:12006 Zbl 0576.12025 [CT95] J.-L.COLLIOT-THÉLÈNE, Birational invariants, purity and the Gersten conjecture, in K- Theory and Algebraic Geometry: Connections with Quadratic Forms and Division Alge- bras (Santa Barbara, CA, 1992), Proc. Sympos. Pure Math. 58, Amer. Math. Soc., Provi- dence, RI, 1995, pp. 1–64. MR 96c:14016 Zbl 0834.14009 THE u-INVARIANTOFTHEFUNCTIONFIELDSOF p-ADIC CURVES 1405

[Kat86] K.KATO, A for two-dimensional global fields, J. Reine Angew. Math. 366 (1986), 142–183. MR 88b:11036 Zbl 0576.12012 [Lam05] T. Y. LAM, Introduction to Quadratic Forms over Fields, Grad. Stud. Math. 67, Amer. Math. Soc., Providence, RI, 2005. MR 2005h:11075 Zbl 1068.11023 [Lip75] J.LIPMAN, Introduction to resolution of singularities, in Algebraic Geometry (Humboldt State Univ., Arcata, Calif., 1974), Proc. Sympos. Pure Math. 29, Amer. Math. Soc., Provi- dence, R.I., 1975, pp. 187–230. MR 52 #10730 Zbl 0306.14007 [Lip78] , Desingularization of two-dimensional schemes, Ann. Math. 107 (1978), 151–207. MR 58 #10924 Zbl 0349.14004

[Mer81] A.S.MERKUR0EV, On the norm residue symbol of degree 2, Dokl. Akad. Nauk SSSR 261 (1981), 542–547. MR 83h:12015 Zbl 0496.16020 [Mil80] J.S.MILNE, Étale Cohomology, Princeton Mathematical Series 33, Princeton Univ. Press, Princeton, N.J., 1980. MR 81j:14002 Zbl 0433.14012 [PS98] R.PARIMALA and V. SURESH, Isotropy of quadratic forms over function fields of p-adic curves, Inst. Hautes Études Sci. Publ. Math. (1998), 129–150 (1999). MR 2000m:11034 Zbl 0972.11020 [Sal97] D.J.SALTMAN, Division algebras over p-adic curves, J. Ramanujan Math. Soc. 12 (1997), 25–47. MR 98d:16032 Zbl 0902.16021 [Sal98] , Correction to: “Division algebras over p-adic curves” [J. Ramanujan Math. Soc. 12 (1997), no. 1, 25–47; MR1462850 (98d:16032)], J. Ramanujan Math. Soc. 13 (1998), 125–129. MR 99k:16036 Zbl 0920.16008 [Sal07] , Cyclic algebras over p-adic curves, J. Algebra 314 (2007), 817–843. MR 2008i: 16018 Zbl 1129.16014 [Sch85] W. SCHARLAU, Quadratic and Hermitian Forms, Grundl. Math. Wissen. 270, Springer- Verlag, New York, 1985. MR 86k:11022 Zbl 0584.10010 [Sha66] I.R.SHAFAREVICH, Lectures on Minimal Models and Birational Transformations of Two Dimensional Schemes, Tata Inst. Fund. Res. Lect. Math. Phys. 37, Tata Institute of Funda- mental Research, Bombay, 1966. MR 36 #163 Zbl 0164.51704

(Received August 23, 2007) (Revised October 19, 2008)

E-mail address: [email protected] DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE,EMORY UNIVERSITY, 400 DOWMAN DRIVE,ATLANTA,GEORGIA 30322, UNITED STATES http://www.mathcs.emory.edu/~parimala/ E-mail address: [email protected] DEPARTMENT OF MATHEMATICS AND STATISTICS,UNIVERSITYOF HYDERABAD, GAHCIBOWLI,HYDERABAD - 500046, INDIA ISSN 0003-486X ANNALSOF MATHEMATICS

This periodical is published bimonthly by the Department of Mathematics at Princeton University with the cooperation of the Institute for Advanced Study. Annals is typeset in TEX by Sarah R. Warren and produced by Mathematical Sciences Publishers. The six numbers each year are divided into two volumes of three numbers each. Editorial correspondence Papers submitted for publication and editorial correspondence should be addressed to Maureen Schupsky, Annals of Mathematics, Fine Hall-Washington Road, Princeton University, Princeton, NJ, 08544-1000 U.S.A. The e-mail address is [email protected]. Preparing and submitting papers The Annals requests that all papers include an abstract of about 150 words which explains to the nonspecialist mathematician what the paper is about. It should not make any reference to the bibliography. Authors are encouraged to initially submit their papers electronically and in PDF format. Please send the file to: [email protected] or to the Mathematics e-print arXiv: front.math.ucdavis.edu/submissions. If a paper is submitted through the arXiv, then please e-mail us with the arXiv number of the paper. Proofs A PDF file of the galley proof will be sent to the corresponding author for correction. If requested, a paper copy will also be sent to the author. Offprints Authors of single-authored papers will receive 30 offprints. (Authors of papers with one co-author will receive 15 offprints, and authors of papers with two or more co-authors will receive 10 offprints.) Extra offprints may be purchased through the editorial office. Subscriptions The price for a print and online subscription, or an online-only subscription, is $390 per year for institutions. In addition, there is a postage surcharge of $40 for print subscriptions that are mailed to countries outside of the United States. Individuals interested in subscriptions for their own personal use should contact the publisher at the address below. Subscriptions and changes of address should be sent to Mathematical Sciences Publishers, Department of Mathematics, University of California, Berkeley, CA 94720-3840 (e-mail: [email protected]; phone: 1-510-643-8638; fax: 1-510- 295-2608). (Checks should be made payable to “Mathematical Sciences Publishers”.) Back issues and reprints Orders for missing issues and back issues should be sent to Mathematical Sciences Publishers at the above address. Claims for missing issues must be made within 12 months of the publication date. Online versions of papers published five or more years ago are available through JSTOR (www.jstor.org). Microfilm Beginning with Volume 1, microfilm may be purchased from NA Publishing, Inc., 4750 Venture Drive, Suite 400, PO Box 998, Ann Arbor, MI 48106-0998; phone: 1-800-420-6272 or 734-302- 6500; email: [email protected], website: www.napubco.com/contact.html. ALLRIGHTSRESERVEDUNDERTHE BERNE CONVENTIONAND THE UNIVERSAL COPYRIGHT CONVENTION

Copyright © 2010 by Princeton University (Mathematics Department) Printed in U.S.A. by Sheridan Printing Company, Inc., Alpha, NJ table of contents o.12 o 2 No. 172, Vol.

Gonzalo Contreras. Geodesic flows with positive topological entropy, twist maps and hyperbolicity ...... 761–808 Eckart Viehweg. Compactifications of smooth families and of moduli spaces of polarized manifolds ...... 809–910 Chang-Shou Lin and Chin-Lung Wang. Elliptic functions, Green functions and the mean field equations on tori ...... 911–954 Yichao Tian. Canonical of Barsotti-Tate groups...... 955–988 Akshay Venkatesh. Sparse equidistribution problems, period bounds and subconvexity ...... 989–1094 Thomas Geisser. via cycle complexes ...... 1095–1126 Viktor L. Ginzburg. The Conley conjecture ...... 1127–1180 Christopher Voll. Functional equations for zeta functions of groups and rings...... 1181–1218 mathematics of annals Monika Ludwig and Matthias Reitzner. A classification of SL.n/ invariant valuations ...... 1219–1267 Isaac Goldbring. Hilbert’s fifth problem for local groups...... 1269–1314 Robert M. Guralnick and Michael E. Zieve. Polynomials with PSL.2/ monodromy...... 1315–1359 Robert M. Guralnick, Joel Rosenberg and Michael E. Zieve.A new family of exceptional polynomials in characteristic two ...... 1361–1390 Raman Parimala and V. Suresh. The u-invariant of the function fields of p-adic curves ...... 1391–1405 Avraham Aizenbud, Dmitry Gourevitch, Stephen Rallis and Gerard´ Schiffmann. Multiplicity one theorems ...... 1407–1434 Stanislav Smirnov. Conformal invariance in random cluster models. I. Holmorphic fermions in the Ising model ...... 1435–1467 Kannan Soundararajan. Weak subconvexity for central values of L-functions ...... 1469–1498 Roman Holowinsky. Sieving for mass equidistribution ...... 1499–1516 Roman Holowinsky and Kannan Soundararajan. Mass equidistribution for Hecke eigenforms ...... 1517–1528 Kannan Soundararajan. Quantum unique ergodicity for

SL2.ޚ/ ވ ...... 1529–1538 2010 September, n