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Comment. Math. Helv. 85 (2010), 337-346 DOI 10.4171/CMH/198 Commentarii Mathematici Helvetici © Swiss Mathematical Society Bounding the symbol length in the Galois cohomology of function fields of p-adic curves Venapally Suresh Dedicated to my teacher Professor R. Parimala on her 60th birthday Abstract. Let K be a function field of a p-adic curve and l a prime not equal to p. Assume that K contains a primitive l th root of unity. We show that every element in the l-torsion subgroup of the Brauer group of K is a tensor product of two cyclic algebras over K. Mathematics Subject Classification (2000). Galois cohomology, central simple algebras, cyclic algebras, symbols. Keywords. Primary 12G05; Secondary 11R58, 11R34. Introduction Let k be a field and l a prime number not equal to the characteristic of k. Let l be the group of l th roots of unity and l .m/ the tensor product of m copies of l . For n 0, let H n .k; l / denote the nth Galois cohomology group with coefficients in l l . Let k D k n f0g. We have an isomorphism k =k ! H 1 .k; l /. For a 2 k , let .a/ denote its image in H 1 .k; l /. For a1 ; : : : ; am 2 k , the cup product gives an element .a1 / .a2 / : : : .am / 2 H n .k; l .m//, which we call a symbol. Assume that k contains a primitive l th root of unity. Fix a primitive l th root of unity 2 k. Then we have isomorphisms l ! l .m/ of Galois groups. Hence we have isomorphisms H n .k; l .m// ! H n .k; l /. A symbol in H n .k; l / is simply the image of a symbol under this map. A classical theorem of Merkurjev ([M]) asserts that every element in H 2 .k; 2 / is a sum of symbols. A deep result of Merkurjev and Suslin ([MS]) says that every element in H 2 .k; l / is a sum of symbols. By a theorem of Voevodsky ([V]), every element in H n .k; 2 / is a sum of symbols. Suppose that k is a p-adic field. Local class field theory tells us that every element in H 2 .k; l / is a symbol and H n .k; l / D 0 for n 3. If k is a global field, then the global class field theory asserts that every element in H n .k; l / is a symbol. 338 V. Suresh CMH Question 1. Do there exist integers Nl .n/.k/ such that every element in H n .k; l / is a sum of at most Nl .n/.k/ symbols? Of course, the answer to the above question is negative in general. It can be shown that for K D k.X1 ; : : : ; Xn ; : : : /, there is no such Nl .n/.K/ for n 2. However we can restrict to some special fields. It is well-known that if Nl .n/.k/ exist for k, then Nl .n/.k..t/// exist. We ask the following Question 2. Suppose that Nl .n/.k/ exist for some field k. Do they exist for k.t /? This is an open question. However we can restrict to fields of arithmetic interest. For example we consider the p-adic fields. The most important result in this direction is the following Theorem (Saltman, [S1], (cf. [S2])). Let k be a p-adic field and K=k.t/ be a finite extension. Suppose that l ¤ p. If A is a central simple algebra over K representing an element in H 2 .K; l /, then ind.A/ divides l 2 . Let K be as in the above theorem. Suppose p ¤ 2. Let ˛ 2 H 2 .K; 2 / and A a central simple algebra over K representing ˛. Then by the above theorem, we have ind.A/ D 1; 2; 4. If ind.A/ D 1, then ˛ is a trivial element. If ind.A/ D 2, then it is well known that ˛ is a symbol. Assume that ind.A/ D 4. By a classical theorem of Albert ([A]), ˛ is a sum of two symbols. For H 3 .K; l /, we have the following Theorem ([PS2], 3.5, (cf. [PS1], 3.9)). Let k be a p-adic field and K=k.t/ be a finite extension. Suppose that l ¤ p. Every element in H 3 .K; l / is a symbol. Let k and K be as above. The field K is of cohomological dimension 3 and H .K; l / D 0 for n 4. By the above theorem, Nl .3/.K/ D 1 and the only case where Nl .n/.K/ is to be determined is for n D 2. It is known that Nl .2/.K/ 2 (cf. [S1], Appendix). In this article we prove the following n Theorem. Let k be a p-adic field and K=k.t/ be a finite extension. Suppose that l ¤ p. Every element in H 2 .K; l / is a sum of at most two symbols; in other words, Nl .2/.K/ D 2. 1. Some preliminaries In this section we recall a few basic facts about Galois cohomology groups and divisors on arithmetic surfaces. We refer the reader to ([C]), ([Li1]), ([Li2]) and ([Se]). Vol. 85 (2010) Bounding the symbol length 339 Let k be a field and l a prime number not equal to the characteristic of k. Assume that k contains a primitive l th root of unity. Let 2 k be a primitive l th root of unity. Let l be the group of l th roots of unity. Since k contains a primitive l th root of unity, the absolute Galois group of k acts trivially on l . For m 1, let l .m/ denote the tensor product of m copies of l . By fixing a primitive l th root of unity in k, we have isomorphisms of Galois modules l .m/ ! l . Throughout this paper we fix a primitive l th root of unity and identify l .m/ with l . Let H n .k; A/ be the nth Galois cohomology group of the absolute Galois group of k with values in a discrete -module A. The identification of l .m/ with l gives an identification of H n .k; l .m// with H n .k; l /. In the rest of this paper we use this identification. Let k D k n f0g. For a; b; c 2 k we have the following relations in H 2 .k; l /. (1) .a/ .bc/ D .a/ .b/ C .a/ .c/; (2) .a/ .b/ D ..b/ .a//; (3) .a/ .b l / D 0; (4) .a/ .a/ D 0. If l 3, we have .a/ .a/ D .a/ ..1/l a/ D .a/ .a/ D 0. Let K be a field and l a prime number not equal to the characteristic of K. Let v be a discrete valuation of K. The residue field of v is denoted by .v/. Suppose char..v// ¤ l. Then there is a residue homomorphism @v W H n .K; l .m// ! H n1 ..v/; l .m 1//. Let ˛ 2 H n .K; l .m//. We say that ˛ is unramified at v if @v .˛/ D 0; otherwise it is said to be ramified at v. Let X be a regular integral scheme of dimension d , with function field K. Let X 1 be the set of points of X of codimension 1. A point x 2 X 1 gives rise to a discrete valuation x on K. The residue field of this discrete valuation ring is denoted by .x/. The corresponding residue homomorphism is denoted by @x . We say that an element 2 H n .K; l .m// is unramified at x if @x ./ D 0; otherwise it is said to P be ramified at x. We define the ramification divisor ramX ./ D x as x runs over points in X 1 where is ramified. 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 @C = @x . If ˛ 2 H n .K; l .m// is unramified at x, then we say that ˛ is unramified at C . We say that ˛ is unramified on X if it is unramified at every point of X 1 . 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 Ok of k. We call such an X a regular projective model of K. Since the generic fibre X of S is connected. Further X is geometrically integral, it follows that the special fibre X if D is a divisor on X, there exists a proper birational morphism X 0 ! X such that the total transform of D on X 0 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 340 V. Suresh CMH reference. If P 2 X is a closed point and f 2 K is a unit at P , then we denote the image of f in the residue field at P by f .P /. 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 number not equal to p. Assume that k contains a primitive l th root of unity. Let ˛ 2 H 2 .K; l /. Let X be a regular projective model of K such that ramX .˛/ D C C E, where C and E are regular curves with normal crossings. We have the following Theorem 1.1 (Saltman [S1]). Let K, ˛, X, C and E be as above and P 2 C [ E. Let R be the local ring at P . Let ; ı 2 R be local equations of C and E respectively at P . (1) If P 2 C n E (or E n C ), then ˛ D ˛ 0 C ./ .u/ (or ˛ D ˛ 0 C .ı/ .u/) for some unit u 2 R, ˛ 0 2 H 2 .K; l / unramified on R. (2) If P 2 C \ E, then either ˛ D ˛ 0 C ./ .u/ C .ı/ .v/ or ˛ D ˛ 0 C ./ .uı i / for some units u; v 2 R, ˛ 0 2 H 2 .K; l / unramified on R. Let P 2 C \ E. Suppose that ˛ D ˛ 0 C ./ .u/ C .ı/ .v/ for some units u; v 2 R, ˛ 0 2 H 2 .K; l / unramified on R and , ı are local equations of C and E respectively. Then u.P / D @C .˛/.P / and v.P / D @E .˛/.P /. Note that u.P / and v.P / are uniquely defined modulo l th powers. Following Saltman ([S3], §2), we say that P is a hot point of ˛ if u.P / and v.P / do not generate the same subgroup of l .P / =.P / . We have the following Theorem 1.2 (Saltman ([S3], 5.2). Let k, K, ˛, X be as above. Then ˛ is a symbol if and only if there are no hot points of ˛. 2. The main theorem Let k be a p-adic field and K=k.t / be a finite extension. Let l 3 be a prime number not equal to p. Assume that k contains a primitive l th root of unity. Let ˇ 2 H 2 .K; l / and X a regular proper model of K. Let W X 0 ! X be a blow-up such that X 0 is a regular proper model of K and ramX 0 .ˇ/ D C 0 C E 0 , where C 0 and E 0 are two regular curves with normal crossings (cf. §1 or [S1], Proof of 2.1). Let Q 2 C 0 \ E 0 . Let C10 C 0 and E10 E 0 be the irreducible curves containing Q. Let R0 D OX 0 ;Q be the regular local ring at Q and mQ its maximal ideal. We have mQ D . 0 ; ı 0 /, where 0 and ı 0 are local equations of C10 and E10 at Q respectively. Let C10 and E10 be the discrete valuations on K at C10 and E10 respectively. Let P D .Q/. Let R be the regular local ring at P and mP its maximal ideal. We have the induced homomorphism W R ! R0 of local rings, which is injective. Let ; ı 2 R be such that mP D .; ı/. Vol. 85 (2010) 341 Bounding the symbol length Lemma 2.1. Suppose that ˇ D ˇ 0 C.f 0 /.g 0 / for some f 0 ; g 0 2 K and ˇ 0 unramified on R0 . Then Q is not a hot point of ˇ. Proof. Since ˇ 0 is unramified on R0 , the ramification data of ˇ on R0 is same as that of .f 0 / .g 0 /. Since .f 0 / .g 0 /, being a symbol, has no hot points ([S3], cf. 1.2), Q is not a hot point of ˇ. Lemma 2.2. Suppose that ˇ D ˇ 0 C .ı/ .gv/ C .f / .g/, where ˇ 0 is unramified on R0 , f 2 R is not divisible by ı and v; g 2 R are units with g.P / D v.P /. Then Q is not a hot point of ˇ. Proof. We have mQ D . 0 ; ı 0 / and ˇ has ramification on R0 only at 0 and ı 0 . Since R=mP ,! R0 =mQ , we have g.Q/ D g.P / D v.P / D v.Q/. If C10 (or E10 ) is the strict transform of a curve on X, then either ı is a local equation of C10 or C10 .ı/ D 0. In fact, if C10 is the strict transform of C1 on X, then C1 .ı/ D C10 .ı/ and ı itself being a prime in R, the assertion follows. Suppose that C10 and E10 are strict transforms of two irreducible curves on X. If ı is not a local equation for either C10 or E10 , we claim that .ı/ .gv/ is unramified on R0 . In fact, since g and v are units in R, .ı/ .gv/ is unramified on R except possibly at ı. Since f is not divisible by ı, .f / .g/ is unramified at ı. Since ˇ is ramified on R0 only at 0 and ı 0 and ı is not one of them, .ı/ .gv/ is unramified on R0 . By (2.1), Q is not a hot point of ˇ. Assume that ı is a local equation for one of them, 0 .f / say C10 . Since ı does not divide f , we have @C10 .ˇ/ D vg and @E10 .ˇ/ D gQ E1 , where bar denotes the image in the residue field of C10 and tilde denotes the image in the residue field of E10 . Since ˇ is ramified at E10 , E10 .f / is not a multiple of l. We have @C10 .ˇ/.Q/ D v.Q/g.Q/ D g.Q/2 and @E10 .ˇ/.Q/ D g.Q/ E 0 .f / 1 . Since E 0 .f / 2 generate the same l ¤ 2 and E10 .f / is not a multiple of l, g.Q/ and g.Q/ 1 th subgroup modulo l powers. Hence Q is not a hot point of ˇ. Suppose that C10 is a strict transform of an irreducible curve on X and E10 is an 0 .ı/ 0 .f / exceptional curve on X 0 . We have @E10 .ˇ/ D .gv/ Q E1 gQ E1 . Since E10 is an exceptional fibre in X 0 , the residue field of R is contained in the residue field at E10 . 0 .ı/ 0 .f / 2 0 .ı/C 0 .f / E1 Hence @E10 .ˇ/ D .g.P /v.P // E1 g.P / E1 D g.P / E1 . Since ˇ is 0 ramified at E1 , 2E10 .ı/ C E10 .f / is not a multiple of l. Suppose ı is a local equation of C10 at Q. Since ı does not divide f and C10 .ı/ D 1, we have @C10 .ˇ/ D gv. Thus @C10 .ˇ/.Q/ D g.P /v.P / D g.P /2 . Since l ¤ 2 and 2E10 .ı/ C E10 .f / is not a 2 0 .ı/C 0 .f / E1 are equal multiple of l, the subgroups generated by g.Q/2 and g.P / E1 th modulo l powers. Hence Q is not a hot point of ˇ. Suppose ı is not a local equation 0 .f / of C10 at Q. We have @C10 .ˇ/ D gN C1 . Since ˇ is ramified at C10 , C10 .f / is not a multiple of l. Thus as above Q is not a hot point of ˇ. 342 V. Suresh CMH The case E10 is a strict transform of an irreducible curve on X and C10 is an exceptional curve in X 0 follows on similar lines. Suppose that both C10 and E10 are exceptional curves in X 0 . Then as above we have 2 0 .ı/C 0 .f / 2 0 .ı/C 0 .f / E1 C1 and @E10 .ˇ/ D g.P / E1 . Since ˇ is [email protected] .ˇ/ D g.P / C1 0 0 ified at C1 and E1 , 2C10 .ı/CC10 .f / and 2E10 .ı/CE10 .f / are not multiples of l. In 2 0 .ı/C 0 .f / C1 and g.P / particular, the subgroups generated by g.P / C1 th are equal modulo the l powers. Thus Q is not a hot point of ˇ. 2E 0 .ı/CE 0 .f / 1 1 Lemma 2.3. Suppose that ˇ D ˇ 0 C ./ .u/ C .ı/ .v/, where ˇ 0 unramified on R0 and u; v 2 R units with u.P / D v.P /. Then Q is not a hot point of ˇ. Proof. Since ˇ is ramified at C10 , either C10 ./ or C10 .ı/ is not divisible by l. In particular their sum C10 .ı/ is non-zero. We have @C10 .ˇ/.Q/ D u.P / C 0 ./ 1 v.P / C 0 .ı/ 1 D u.P / C 0 .ı/ 1 : Suppose that C10 .ı/ is a multiple of l. Since C0 1 .ı/ is non-zero, C10 is an ex 0 .ı/ D 1. ceptional curve. As in the proof of (2.2), we see that @C10 .ˇ/ D u.P / C1 0 Which is a contradiction, as ˇ is ramified at C1 . Hence C10 .ı/ is not a multiple of l. Similarly, we have @E10 .ˇ/.Q/ D u.P / C 0 .ı/ E 0 .ı/ E 0 .ı/ 1 and E10 .ı/ is not a multiple of l. and u.P / 1 generate the same subgroup of .P / modulo Hence u.P / 1 l .P / and Q is not a hot point of ˇ. Theorem 2.4. Let k be a p-adic field and K=k.t/ be a finite extension. Let l be a prime number not equal to p. Suppose that k contains a primitive l th root of unity. Then every element in H 2 .K; l / is a sum of at most two symbols. Proof. If l D 2, then, as we mentioned before, by ([A]), ˛ is a sum of at most two symbols. Assume that l 3. Let ˛ 2 H 2 .K; l /. Let X be a regular proper model of K such that ramX .˛/ D C C E, where C and E are regular curves with normal crossings. Let P 2 C [ E be a closed point of X. Let RP be the regular local ring at P on X and mP be its maximal ideal. Let T be a finite set of closed points of X containing C \E and at least one closed point from each irreducible curve in C and E. Let A be the semi-local ring at T on X. Let 1 ; : : : r ; ı1 ; : : : ; ıs 2 A be prime elements corresponding to irreducible curves in C and E respectively. Let f1 D 1 : : : r ı1 : : : ıs 2 A. Let P 2 C \ E. Then P 2 Ci \ Ej for unique irreducible curves Ci in C and Ej in E. Then D i and ı D ıj are local equations of C and E at P . We have ˛ D ˛ 0 C ./ .uP / C .ı/ .vP / Vol. 85 (2010) Bounding the symbol length 343 or ˛ D ˛ 0 C ./ .uP ı i / for some units, uP ; vP 2 R and ˛ 0 unramified on R ([S1], cf. 1.1). By the choice of f1 , we have f1 D ıwP for some wP 2 A which is a unit at P . Let u 2 A be such that u.P / D wP .P /1 uP .P / for all P 2 C \ E. Let f D f1 u 2 A. Then, we have .f / D C C E C F , where F is a divisor on X which avoids C , E and all the points of C \ E. Further, for each P 2 Ci \ Ej , we have f D i ıj wij for some wij 2 A such that wij .P / D uP .P /. By a similar argument, choose g 2 K satisfying (1) .g/ D C C G, where G is a divisor on X which avoids C , E, F and also avoids the points of C \ E, C \ F , E \ F ; (2) if P 2 E \ F and ˛ D ˛ 0 C .ı/ .v/ for some unit v 2 RP and ˛ 0 is unramified at P , then g.P / D v.P /. Since C \ E \ F D ;, such a g exists. We claim that ˇ D ˛ C .f / .g/ is a symbol. Let W X 0 ! X be a blow up of X such that X 0 is a regular proper model of K and ramX 0 .ˇ/ D C 0 CE 0 , where C 0 and E 0 are regular curves with normal crossings. To show that ˇ is a symbol, it is enough to show that ˇ has no hot points ([S3], cf. 1.2). Let Q 2 C 0 \ E 0 . Let P D .Q/. Then P is a closed point of X, R D OX;P OX 0 ;Q D R0 and the maximal ideal mP of R is contained in the maximal ideal mQ of R0 . Let mQ D . 0 ; ı 0 /, with 0 and ı 0 be local equations of C 0 and E 0 at Q respectively. Suppose that P 62 C [ E. Then ˛ is unramified at P and hence unramified at Q. By (2.1), Q is not a hot point of ˇ. Assume that P 2 C [ E. Suppose that P 2 C \ E. Let and ı be local equations of C and E at P respectively. Then mP D .; ı/. By the choice of f and g, we have f D ıw1 and g D w2 for some units w1 ; w2 2 R. In particular, ˇ is ramified on R only at and ı. Suppose that ˛ D ˛ 0 C ./ .u/ C .ı/ .v/ for some units u; v 2 R and ˛ 0 unramified on R. We have ˇ D ˛ C .f / .g/ D ˛ 0 C ./ .u/ C .ı/ .v/ C .ıw1 / .w2 / D ˛ 0 C ./ .u/ C .ıw1 / .v/ C .w11 / .v/ C ./ .w2 / C .ıw1 / .w2 / D ˛ 0 C .w11 / .v/ C ./ .uw2 / C .ıw1 / .w2 v/ D ˛ 0 C .w11 / .v/ C ./ .uw2 / C .ıw1 / .w2 v/ D ˛ 0 C .w11 / .v/ C .w2 v/ .uw2 / C .w21 v 1 / .uw2 / C .ıw1 / .w2 v/ D ˛ 0 C .w11 / .v/ C .w21 v 1 / .uw2 / C .w2 v/ .uw2 ı 1 w11 /: Since ˛ 0 C .w11 / .v/ C .w21 v 1 / .uw2 / is unramified on R, by (2.1), Q is not a hot point of ˇ. 344 V. Suresh CMH Suppose that ˛ D ˛ 0 C ./ .uı i / for some units, u; v 2 R and ˛ 0 unramified on R. Then we have ˇ D ˛ C .f / .g/ D ˛ 0 C ./ .uı i / C .ıw1 / .w2 / D ˛ 0 C ./ .uı i / C .ıw1 w21 / .w2 / D ˛ 0 C ./ .uı i .ıw1 w21 /1 / C .ıw1 w21 / .w2 / D ˛ 0 C ./ .ı i1 uw11 w2 / C .ıw1 w21 / .w2 /: If i D 1, then ˇ D ˛ 0 C ./ .uw11 w2 / C .ıw1 w21 / .w2 /. Since, by the choice of f , u.P / D w1 .P /, by (2.3), Q is not a hot point of ˇ. Assume that i > 1. Then 1 i 1 < l 1. Let i 0 be the inverse of 1 i modulo l. We have ˇ D ˛ 0 C ./ .ı i1 uw11 w2 / C .ıw1 w21 / .w2 / D ˛ 0 C .ı 1i u1 w1 w21 / ./ C .ıw1 w21 / .w2 / 0 0 D ˛ 0 C ..ı.u1 w1 w21 /i /1i / ./ C .ı.u1 w1 w21 /i / .w2 / 0 C ..u1 w1 w21 /i / .w2 / C .w1 w21 / .w2 / 0 D ˛ 0 C ..u1 w1 w21 /i / .w2 / C .w1 w21 / .w2 / 0 C ..ı.u1 w1 w21 /i /1i / . 1i w2 /: 0 Since ˛ 0 C ..u1 w1 w21 /i / .w2 / C .w1 w21 / .w2 / is unramified on R, by (2.1), Q is not a hot point of ˇ. Suppose that P 2 C n E. We have ˛ D ˛ 0 C ./ .u/ for some unit u in R and ˛ 0 unramified on R. We also have f D f1 for some f1 2 R which is not divisible by . We have ˇ D ˛ C .f / .g/ D ˛ 0 C ./ .u/ C .f1 / .g/ D ˛ 0 C .f11 / .u/ C .f1 / .u/ C .f1 / .g/ D ˛ 0 C .f11 / .u/ C .f1 / .gu/: If f1 is a unit in R, then ˛ 0 C .f11 / .u/ is unramified on R, by (2.1), Q is not a hot point of ˇ. Assume that f1 is not a unit in R. Then P 2 C \ F and g D g1 for some unit g1 2 R. We have ˇ D ˛ C .f / .g/ D ˛ 0 C ./ .u/ C .f1 / .g1 / D ˛ 0 C .g1 / .u/ C .g11 / .u/ C .f1 / .g1 / D ˛ 0 C .g11 / .u/ C .g1 / .u.f1 /1 /: Vol. 85 (2010) Bounding the symbol length 345 Since ˛ 0 C .g11 / .u/ is unramified on R, by (2.1), Q is not a hot point of ˇ. Suppose that P 2 E n C . Then ˛ D ˛ 0 C .ı/ .v/ for some unit v 2 R and f D ıf1 for some f1 2 R which is not divisible by ı. Suppose that f1 is a unit in R. Then, as above, Q is not a hot point of ˇ. Assume that f1 is not a unit in R. Then P 2 E \ F and g is a unit in R. 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