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Divisors on division algebras and error correctingcodesPatrick J. Morandi a & B.A. Sethuraman ba Department of Mathematical Sciences , New Mexico State University , Las Cruces,NM, 88003 E-mail:b Department of Mathematics , California State University , Northridge, Northridge,CA, 91330 E-mail:Published online: 05 Jul 2007.

To cite this article: Patrick J. Morandi & B.A. Sethuraman (1998) Divisors on division algebras and error correctingcodes, Communications in Algebra, 26:10, 3211-3221, DOI: 10.1080/00927879808826337

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COMMUNICATIONS IN ALGEBRA. 26(10). 32 11-3221 (1998)

Divisors on Division Algebras arid Error Cor*re<:tirig Codes

Patrick J . Morandi Department of Matherriatical Sciences

New Mexico State University Las Cruces, NM 88003 prnorandi@nmsu.edu

B. A. Sethuraman' Department of Mathematics

California State University, Northridge Northridge, CA 91330

al.sethuraman@csun.edu

1. Introduction

Recall that a (linear) code is just a k-subspace C of kn, wlwre k is some finite field. The eleirlents of C are referred to as codewords, and the ditnenszon of thr code, dinl(C), is just the dimension of C as a k-space. The length of the code is tlie tiitncnsion of the ambient space k"; that is, the length of C is n. The minimum dzslance, d(n, b) brtwee~i two codewords a - ( a l , . . . ,%) and b = (bl,. . . ,b,,) is defined by d(a,b) = j { i 1 u, f 6 , } / , and the mznzmuin dastance of the code, d(C), is defined by d(C) = min{d(a,h) I r ~ , 6 E C , u + 6 ) . The wezght, u:(n), of a codeword a = (a,, . . . ,a,) is defined by ui(a) = I{, / u, # 011. 'Thr liilearity of thc code ensures that d(G) is also equal to rnin{w(a) I a E C, a # 0).

A major goal in coding theory is to construct codes whose ~liinimuin drstance and di- rnension are large relative to their length. Goppa Codes were introduced by Col)pa in 131 (in a form dual to the one we consider here), and are effective at aclrrevirig this goal. One starts with a smooth geometrically irreducible project,ive curve X / k ( i d thus, a function field F lk ) , and a divisor E = PI +. . . + Pn, where the P, are prinw rlivisors of degree 1. One fixes a divisor G' such that the support of E and G' are disjoint (this guilrantres that any f E L(G) is auton~atically in each valuation ring Opt). 011c thr.11 coi~sidtv t l ~ r rvaluat,ion

'Supported in part by a grant from the N S.F.

Copyr~ght 8 1998 by Marcel Dekkcr, Inc

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32 12 MORANDI AND SETHURAMAN

map ev : C(G) -. k." given by 1 I- PI), . . . , fie,)), where by J(P , ) , wc mean the residue of J with rcspect to t l ~ e valuation at P,. Since ev is k-linear, the Inlage is a k-subspace of k"; this is the Goppa code associated to the curvc X / k and the tlivisors E and G.

We provide in this paper an analogous construction of codes using division algebra 1) defined over the function field Flk . The structure of such a I) is of course well-understood Crorn class field t.heory; we cxploit this structure in our constructlun. I'laces of the field F/k are now replaced by maximal ordrrs, selected to form 8 shenJ of ~ ; L X I I I I A ~ ordc:rs over X . By work of various aulhors (see I!)], [ I I ] , and (12]), one has an a~lalogous definition of divisors i r ~ this situaLion, as well as il dehiLiw of Lire spaces L(G) for. an) clivis~o~~ algebra divisor G, and a noncommutative Riemann-Roch Theorem.

Our code is actually an I-code, where 1 is a finite extension of 6. that xises naturally as the residue of D. The function field 1F is a maximal subfield of uur d~vision algebra; we prove that under certain conditions our 1-code is the same as the co~nn~utative Coppa code obtained by restricting our division algebra divisors in a suitable mallner 1.0 l F .

2. Maximal Orders and Divisors

As mentioned in the introduction, we will use maximal orders i r ~ our co~~struction of codes LVc recall here the basic properties of maximal orders over discrete vuluat.iur~ rings. A good reference for maximal orders is Reiner's book [5]. Let A be an integral domain with quotient, field F and let S be an F-r:entml simple algebra. A subring B of S wit11 A > A i s M I

A-order if B is a finitely-generated A-module and if BF = S. If B is maxitnal with respect to inclusion among all A-orders in S , then B is said to be a inminc~l wrder over A.

If A is a discrete valuat.ion ring with quotient field F, then the following properties hold for ~naximal A-orders in an F-central simple algebra S:

I. Maximal A-orders exist dncl are unique up to mnjugation.

2. I f B is a maxinlal A-order in S, then the Jacobson radical J ( B ) is tlw unique mmimal ideal of B , and every ideal of B is a power of J ( B ) .

3. There is a b E R with J ( B ) = B6 = 6B.

4 . Thc residue ring = B / J ( B ) is simple and finite d~mens~onal ovi,~ 3 A/.I(A)

I'roofs of these properties can be found in 15, Theorems 18.3, 18.71. In this paper we need to make use of value functions. If A is a discrete valuation rlng

and if B is a maximal A-order In 5, then we can define a functiou u! : S - { O ) - Z in the following way: For b E B , we set w(b) = n if BbB = J ( B ) ^ . More generally, if s E S, write s =- bn-I with 6 E B and cr E A, and set w(s) = w(b) - w ( a ) . It is not I~ard to see that u! is well defined and that U J satisfies the following properties:

2. st) > w ( s ) 1 r t : ( l ) , and w(st) = w(s) + w(t ) i f s E Z(S);

3. LI = { s E S I w ( s ) > 0 } U {0) and J ( B ) = {s E S 1 w(s) > O } u ( 0 )

Moreover, ui is uniquely rlet~rmirlcd by R. We call w the value ~ I I I I ( . I I O I I O I I .S (:orr.esponCI- 111:: t o 8. We will der~olc. by 1'13 thc uulue group of B; that 15, r B I I I I ( W ~ , ) . LL. We refer t l ~ c rcarl~r to [ G , Src. 21 for more inforn~ation about value functio~~s

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DIVISORS ON DIVISION ALGEBRAS 3213

Suppose that X is a smootlr gtumet.rically irreducible project.ive curve defined over a field k, and let. F be t,he function field of A'. Let S be a central simple F-algebra, and suppose that we have a maximal Or-ordcr Bp In S for each P E S. Let wp hc the value function on S corresponding to Bp. Not.e that wp depends on both tlle point P and t,he choice of maximal order Rp. Let ep be tlrc index [Z! : wp(F*)]. We will call e p the mmijication index of S/F' wit11 respect to 1'.

We will be working wit.11 sheaves of maximal orders on X . We cirll a sheaf A of rings on X a sheaf of maximal orders 111 S i f A(U) is a maximal O(C1)-order in S for every proper open subset Cl of S. Let S be t.lre cunstarrt sheaf S (U) - S on X. \Ve recall that any sheaf of maxintal orders on S is isoinoiphic to a subsheaf of S. Here is a sketch of the argument. Let ( be the generic point of X . I f A is a sheaf of Ox-maximal orders in S , then thc structure maps pl, : A([/) - A( extend by bilinearity to a map 6" : A(U)%o(IilC)~ + A<. Now Oi - F, and for every proper open set (1 , A([/) is a maximal O(U) order, so the map A(U) i30(l~l F -+ S induced by the inclusion A((/) 2 S is an isomorphism. For every proper open set. I J , we thus have a map Oh : S --. A<. 8; is an injection as S is simple. Since any z E Ai is in pv(A(V)) for some open V, we find z E &(A(V) Bo(vr O1). We may assunle V C_ U. 'The structure map puv : A(U) -+ A(V) induces an isomorphism A(U) @o(") O( 2 A(V) @ q v 1 U( (as each side is isomorphic to S ) , so we find z E &(A(U) BO(") O() as well. Thus, A< S. We thus have injective maps ipc, : A(U) - Ac Z S for each proper open set 11 X . Usmg the sheaf property of A(X) and the fact that the structure map ipx : A(X) -. A( factors through A(U) for every proper open set U, it, is easy to see that 9 x is injective as well. Since pu = p v p u ~ for open sets V C U, we have an injective sheaf map from A to the constallt sheaf S.

Lemma 1. Suppose for eaeq P E X that there is a maximal Op-order Bp zn S, and let .wp be the value Junction on S corresponding to Bp. Then, for each s E S , we have wp(s) 2 0 for all but finitely many P zf and only if there is a sheaf A of mazimul Ox-orders on X such that for each P E A', the stalk Ap zs equal to Bp.

Proof: Suppose there is a sheaf A 2 S of maximal orders such that A p -- Bp for each P E X. Let s E S. If GI is any proper open subset of X, we may write s - bn-I with b E A(U) and a E O(U). There is a nonempty open subset V of U such that a-I E O(V). so s E A(V). Thus, for cach P E V we have w p ( s ) 2 0. Since the complement of V in S is finite, wp(s) 2 0 for all but finitely many P.

Conversely, suppose that we have a maximal Op-order Bp for each P E S, and that i f wp is the value function associated to Bp, then, for any s E S, we have wp(s) 2 0 for a11 but finitely many P. We define a sheaf A on A' by

for all open subsets li of A' witl~ strlict.ure maps being the obvious inclusions. I t is proved in (7; Ex. 3.41 that A([/) is a 111i~xi11la1 O(l1)-order for each proper open subset U , and so A C S is a sheaf of maximal orders o11 S. hlorcover, in 17, Ex. 3.41, it is shown that A(U)Op . - 13p

i f P E U. This shows t.lrizt A p -- Bp for each P. I

We now describe the divisor group on S and state a geileralizaticin of the R~e~lrairn-Rocli theorem. If S is a central s i i~~ple F-algebra, let A be a sheaf of nlaximal Ox-orders 011 A . For each P let wp be the value function on S corresponding to the maxinlal Op-odrr A!'. The dzznsor group div(.S) is thc: Irtv abelian group on the set {AP}p,x. \?;hilt> lhis groi~i) depends on the cho~ce of slivnf A , we will not deal with more thair olir slie,tf o f mnx~ind orders a t a tiine, so wc w~ll not i ~ ~ d to worry about this dependence. For E = Cb,npi1p. u,r define deg(E) - C p n p [ x p : kI arid supp(E) = {Ap I np # 0). Each s E S defines a divisor

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DIVISORS O N DIVISION ALGEBRAS 3215

some cr E k[t] - (0). Then d E A p for all P such that P f Ib, and rr-' E Up.) With this sheaf, we verify that the divisor of 1 t j is equal to -P,, so ( 1 -4 j) 11,s negative degree. By Proposition 9 ahead (with z = j ) , we find that I and J form a stro~iglp orthogonal basis for D / k ( t ) ( & ) , so wp(l + j ) = min(wp(l), wpb) ) for all P E S. Since 1 I S a unit in O p for all P except those that correspond to t or lit, and since I - ' = 1 - ' j. we f i l d that J conjugates A p to itself for all such P. By (6. Prop. 2.61, wp(j) - 0 for all sucl~ P . As for Po and P,, the maximal orders at these points arr valuation rings, and ulp,,(j) - 1 . a ~ ~ d ,wp_(j) = -1. Thus, (1 + j ) - -P, as claimed.

We define a map Gs : div(F) -. div(S) by # s ( C p n p P ) - C, , (n t epAp) The main property of this map that we need is given in the following lemma.

Lemma 3. I f E E div(F), then deg(ds(E)) I- IS : F] tlegf(b')

Proof: If E = Cp n p P , then we have

I

If E , E' are divisors on S, we write E 2 E' if E - E' = C npAr with each np 2 0. Let L (E) = {s E S 1 (s) + E 2 01, a k-subspace of S . As a consequence of Lemma 2, we point out that if deg(E) < 0, then L(E) = 0. To see this, i f f E L(E) is nonzero, then (1) + E 2 0, so deg(( f ) + E ) = deg((f))+deg(E) 5 deg(E) < 0, a contradiction to the condition ( f ) + E 2 0. Let C be a canonical divisor on F , and set K = ds(C) + C p ( e p - l)Ap E div(S). We now state the generalization of the Riemann-Roch theorem to this sett,ing.

Theorem 4 (Riemann-Roch) Let g = max~,d,,~s){deg(E) t 1 - dim(L(E))}. Then g ezisls, and for all divisors E , we have dinl(C(E)) = deg(E) t 1 - g t dim(K - E) .

A proof of this theorem c a ~ be found in 112, Sat,z 121 or [ I 1. 'rheore~ri 3.171. We would like to thank J.-L Colliot-ThClkne for telling us about Wit t . '~ paprr. Analogous to the commutative case, we have deg(K) .= 29 - 2. By the definition of K and [ ~ n i m a 3, if g~ is the genus of F, we see that

In the next section, in order to construct sheaves of ntaximnl ortlers. we will use t,he fact that. i f IJ is a proper open subset of S, t.1ic11 Op is H I)t:tlek~~~tl domain of F. This is a st,andard result, but we give a proof here for the conveniencr of tlw reader.

Lemma 5. If U zs a proper open subset of X, then Of. 1 5 tr 1)rdoLmd doirmn o j F

Proof: Let X- U = {QI. . . . . Q,,} , let E - C, n,Q, 6 tliv(1;'). ,u~rI I d I;, ~- E-Q,, where the n, are positive integers. I f we c110ost: the n, large crrougl~. ~ I I < , I I l)y ( I I V Riemann-hch theorem, dim(C(E)) is larger than ~itclt tli~~l(L(b;,)) sincc tlcp,(I:') > tltag((.:,) for. each r . Each of these are finite-dimensional k-vector sprc,es, so C(E) propc.1.l~ c.olllrnlls the union U, L(E,).

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3216 MORANDI A N D SETHURAMAN

C'hoose L L. L ( E ) such that x I$ Lit.:,) for each i. The~t (x) -t I;' 0 hut ( J ) i E, 2j 0 for r x h 1. 'I'lris forces vp,(x) = -71, for each i , and vp(x) 2 0 for well 1' E ( 1 . C'or~sequentl~, t l ~ e valuation rings of F that contain x are precisely the Okb for /' E ( 1 . Note t.lrat x @ k, so F IS algebraic over k(x). A valuation ring of F/k conhins 2 if and only i f i t . contains k[x], so we see that the integral closure of k[z] in F is ope, Op. IIOWCVCI.. S I I I C ~ kl:rj is a Dedekind domain of k(x), the ring npEI, Up is a Dedekind t1o111ai11 of F by 15. 'Tlicwrenr 4.41. 1

3. The Construction of the Code

Ixt F/k be a function field in one variable over the fi~titc. fivltl k . Oiic 11;~s thr following scqucnce from class field theory (see the discussion in 15, pages 277 2781 for il~sta~rce)

where the sum in the middle term is over all the places of the field F'. Fir ,irry D E Br(F), D aF 1,'p is trivial almost everywhere; thus the second n ~ a p in thr sequence above, which is simply the sum of the various restriction maps, is well-defined. It IS well-known that for each P, there is a canonical map Inv : Br(Fp) - Q/Z which is an isonlorphisn~, and the third map above is just the sum of the Inv maps for each P. 'I'he ntap I I I V can I x explicitly described as follows: any division algebra D/Fp of degree T is isoniorphic to the cyclic algebra (W/Fp,n , T ' ) , where W is the unique unraniified field extension of Fp of d e g r t ~ rr, n is a uniformizer for Fp, u is a generator of the Galois group of the cyclic field extension W/Fp chosen so as to induce the Frobenius automorphism at the residue level, and 1 < i < r with gcd(i,r) = I . With this isomorphism at hand, Inv(D) is defined to be ilr. (See tShe discussion in 15, pages 263-2661 for the description of division dgebras over local fields.)

D is said t,o be mmijed at a place P if D g p F p is not split. Moreover, i f l n ~ ( U @ ~ P p ) = i / s with gcd(i,s) = 1, then s is said to be the local indrz of 1) at P. a ~ l d ind(D) is simply the lemt, common multiple of t.he various local indices (see 15. 'Tl~c:ore~~r 3'2. IS ] ) .

Now let X/k be a smooth geometrically irreducible projective curve, and 12/k the associ- at,cd fl~nctiorl field. Let P i , . . . , P,, be plwes of F/k of degree I. (:hoose Intrgers r, j l . . . . , j, such that 1 < ji < r , gcd(j,,r) = I , and CiCj,/r) = 0 in :$/L 'l'lius, ~ h c exact sequence (1) shows that there is a division algebra D / F ramified at exactly tltc placcs P,, and at each P,, D has local index T. It follows that ind(D) = T as well. Also, the residur field of Fp, is just k since the P, have degree 1, and if Ilk is the unique esteusion or k of degree r , then lFp, is the unique unraniified extension of Fpa of degree r 'Thus. 1,. t.l~e discussion in the previous paragraph, D @ p Fpa is isomorphic to (lFp,/Fp,,o,, K"), wl~crc r r , is a genrrator of Gal(1 Fp, IFp,) chosen so as to induce the Frobenius autolnorpllisrn 0 1 1 I l k .

1% first show that L = 1 F is a maximal subfield of D. K J ~ ally place Q of L, let I' be the corresponding place of F induced by Q. The completio~i of 1, i l t (2 is isonlorphi(. to tlle compositum lFp. If P is not one of the selected P,, the11 L) ::;F 1:p is already split, so ( D @p L) @[, Lq = (D @, Fp) @pP IFp is split. If P is olle of tht. selected P,, then, iw re~narked in the last paragraph, lFp is a maximal subfield of D :.;r f i n , so once again. (1) 'i3p L) OL Lg = ( D @F Fp) @ p p lFP is split. It follows ~ . l ~ a t D :)I. I, is split c~verywhere, so by our exact sequence ( I ) , L splits 0. Since [L : I.'] - r ( i~s k is iilget)raically closed i l l

F , this follows from the geometric irreducibility of S l k ) . I , 1s iittlcc,(l it ~~t i iu i rn i r l st~bficld of U / F .

We construct a sheaf of niaxinral orders A on our curw S L\S Iollows. I,(:( 4 - n Oy. wlterc the intersection runs across nll pl,wes Q of F en.cepl t 1 1 ( , ( I I O W I I ~ I < L ( ( . > 1.:. As described in I,erntuil 5 , A is a Dedekind donlain. Since 1 C D . 1A is a frw f i ~ ~ ~ t ( . l L R(t~~t~~, i r ted A rnodule.

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DlVlSORS ON DIVISION ALGEBRAS 3217

Choose a basis {Dl = 1 , b2 , . . . ,b,) of D / L , and consider l l ~ e frec A ~ ~ ~ c ~ t l u l c .bl = @:LI(lA)b,. Then M F = D , and 1A 01(M), where 01(M) = {x E D / LA^ C Al). As discussed in 15, page 1091, Ol(M) is all A-order in D, and is hence contained in a ~ ~ r a x ~ ~ n i l l A-order; call this rnax~~ual order B. Then, for each place Q of F (Q $ {PI, . . . , t',)). 13 i;?, (7y is a maximal OQ order. \Ve set Aq -- R aCiA Oy.

As for t,l~e places PI , . . . . e,, there is a unique rnaxinlal or-dcr ovc,r (w11 Oq which IS in facl a valuat,ior~ nng. ('l'llis follows, for instance, from 15, ' l ' l ~ e o w ~ ~ l ~ 12 8 d11c1 11.51. WC set ApE to be Lhis n~axi~nal Op, order.

Now let wy be the value funct.ion associated with this cholcc. of I I I H X I I I I ~ orders <uld consider the sheaf of rings A given by the assignment U u ny,, /Iq. ICxact.l,y as in the argument in the proof of Lenlnta I , any s E D can be written as bn-' for sonw b E A ( [ [ ) , where U = A' - { P I , . . . . P,), and for all but finitely many points C) E 11, a-I E A y . I t follows that utQ(s) 2 0 for all hut finitely many points Q E I i . S ~ n w the coi~~ple~neul of U contains only fiuitely many points, Lemma 1 shows that A is intleetl a. sheaf of n~axi~nal orders, whose stalk at each point Q E X is AQ.

We let P be the D-div~sor Ap, + . . . + Ap,, and we let G be any I)-divisor whose support, is disjoint from the P,, and such that L(G) # {O). The condition 011 t l~e supports guarantees that any f E L(G) is autonlatically in each Ap,. Since each Ap8 is just the valuation rmg corresponding to the P,-i~dic valuation, the factor ring Ap,/J(AE). where J(Ap,) is the maximal ideal of Ap,, is just the residue field of D with respect to the & - d i c valuation. The residue of D is the same as the residue of the completion D @ F Fp,, which is just 1. We thus have, exactly as in the commutative case, a map

where we have written J(P,) for the residue of f modulo the maximal ideal J (AE) of Ap,. By our choice of maximal orders above, the field 1 is contained I I I cb\wy maxi~nal order

AQ for Q $ { P I , . . . , P,,). Since each Apt is a valuation ring, and since ally valuation on 1 [nust be trivial, 1 must be contained in each Ap, as well. It follows that f (~r trny Q, wQ(x) 2 0 for all x E 1'. Hence, for any Q, any x E I * , and any f E L(G),

so x f E L(G). Thus, besides being a k-space, L(G) is also an 1-space. Lloreover. r (P , ) is just x for any P,, as the P,-adic valuation is trivial on 1. It follows f r m t h s t t~at the map cav above is an I-linear map. (In fact, each x E 1' conjugates each Av tu itself, so by 16, Prop. 2.6 and Lemma 2.21. ulQ(2) = 0, and wQ(xf) = wQ(x) + wq(f) : u ~ ~ ( f ) . )

The image of ev is thus an I-subspace of 1"; this will be our noncor~l~r~uttrtive Coppa code associated to D I F and the divisors P and G. We denote it CD(P.C;).

We point out that while different choices in our sheaf const,ruction n~ay lead t,o different (e.g., non-isomorphic) sheaves. the ring of constants nQcx A4 IS uniquely determined up to isomorphism by t.hc division dlgebra D since k is finite, by a tlleorcl~~ of Schofield 110. Tileorem 1 . 1 1 . We are wurkit~g with a specific choice of sheaf to st^ ~llat 1 IS it sulm11g of t.he ring of constants. In fact, i t is not hard to see that. 1 - n,,,, ,Aq. so 1 is LIIC: full r111g 111' conslants of the slreaf A .

We determine in the next proposition the minimum dist.a~cc: ,u~tl tlw tlillw~slon of ~ I K ~ code C.'o(P, G) . Thc ~nethotls are analogous to those used fur wlriiliulatlve Coppa c d e x (scr 131). Note that si~~cc. - 1 for each i, we have deg(Ap,) [ I : I : ] . ,Therefore. tleg(P) = 7.71.

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3218 MORANDl A N D SETHURAMAN

Proposit ion 6. t lm(CLj(P (:)) ~ I I I I ( L ( ( ; ) ) - (hn(C((; - P)) u d c~(( '~)(P. (:)) 2 71 -

T - I tlcg(G) Moreover, ~Jdeg((:) < I 71, then dlm(Co(P, C)) - d m L(L')) 2 dcg((:) -I I - 0. where y zs the genus o j C)

Proof: The kernel of ev is { J E L(G) I vp,(J) 2 1, I = I . . . . .T I ) . which is pnwsc,ly C(G - P ) . The result 0 1 1 tht, c l~~~ lcws io~~ in~nnediately follows. As for. t,he t l ~ s t a ~ ~ c e , write d for d(C:o('P,G)) and let J E C(C), / f 0, be such t,hat w(cv(J)) - - (1. T11c.n / i s I I I

t.he maximal ideal of exactly 11 - d of the Ap, , say A q l , . . . , A p I t follows that J E

C(G - (A,;, t . . . t A,:., , ) ) . tlc~lcr. C(C; - (Apnl + - - - + A p , ~ , ) ) # 10). I t Collows that deg(G - (AP,~ t . - - t Ap,,, ,)) > 0 , since C ( H ) = (0) for a divisor 11 of ~~egat ive degree. Thus, deg(G) - r ( n - d ) > 0, or tl 2 n - r . - I deg(G).

For l l ~ e second statement, il tlrg(C:) < r.rr - deg(P), then deg(G - P) < 0, so L(G - P) {O). It follows from the peragr-npl~ abovc that, ev is then an injechve wap. Moreover, l),y the Kiemann-Roch theorem, wc have dinl(Cn(P. G')) = dim(L(G)) 2 deg(C) I 1 - g.

4. Relation between C U ( P , G) and CL(+(P) , @(G))

We continue to use the notation of t.he previous sections: P I , . . . , P, are rational points on X , and D is a division algebra with center F , the function field of Xlk, such that the local index of D a t each P, is equal to the index r of D, and the local index of D a t any point P # Pi is 1. Recall that we have a sheaf A of maximal orders on D, that A is a subsheaf of the constant sheaf U c D, and t.hat each stalk A p contains 1. Note that since Ilk is a cyclic Galois extension and 1, F are linearly disjoint over k, the extension L / F is also cyclic Galois. l i ~ t u he a generator of Cal(L/F). Since L is a maximal subfield of D, wr may write D as a cyclic crossed product D - (I , / F, o, a ) = 9)::; Lx', where x' - a 6 F and zbz-I - u(b) for all b E L. We will make use of this description of D in a number of places below.

If f{ is a field extension of t.', recall the canonical map 3~ : div(F) - div(K) that satisfies l / 'K (P) = C eqlpQ% where the sum is over all IC-points Q lying over P , and where e ~ , p is t,he ramification index of Q over P . In terms of valuations, eQ/p is t.he ramification index of f</F relat,ive to t l ~ c vaiuatio~~ ring of K corresponding to Q. We will refer to this valuatior~ ring by C3,i,Q. I.hr our tnituin~al subfield 1F = L of D, \ye writ.e for d:L. We also view the map li, as a map fro111 div(D) to div(L) by sending Ap to ri,(P). Since / , IF ' is everywhere unrarnified, eq/p 1 for every P and every Q lying over P, and. for each of the points P,, there is a unlquc point Q, lying over P,. This follows from the fr~ndan~el~tal - - equality [L : FI := Cu,peu i r f a ip~ where fytp = (OL,Q : OP1, and from the f ~ c t that thc residue field of any point ex(cwlnig f: is 1, an extension of k = of degrec IL : FJ . As above, let P - C, AI: and let C t ) c b a divisor with supp(G)n{Ap,, . . . , Apq} -= 0 . In the uext lenima we compare our code w ~ t ll $1 Coppa code constructed from the algebraic functioit field L/1. To refer to such a Coplm code we denote it by CL(E', G') if E', C' are appropriat.ely chosen divisors on L.

Lemma 7. The space L(qi(C:)) 85 cnnonzcully rsomorphzc lo C(C) f' I,, and, wzlh the zdcn- tzjtcalton oJ C(+(G')) us n ruh\pcwe oJ C(C), the zmage ev(C($(G))) oJ C(@(G)) under thc ct~nluatron map ev L(C) - I" zs /he Coppa code CL($J(P). @(G'))

Proof: I.rt f E L s;itlsfy ( J ) I r/,(G) 2 0 in div(L). Givwl 1' E S - { I J , . . . . . I J , , ) . 11.1 QI . . . . .Q, he the e x t r ~ ~ s i o ~ ~ s r ~ f I' to I,. Let. r ~ p be the coefficient of i l l (.: ' l ' l~us, I S

the coeflirie~~t, of each (2, i l l < I : ( ( : ) , C'onscqucntly, vy,(f) t- 71,. 2 0 for each 1 . lret R I)(: CL

uniforrnizcr for Op. T I I ~ I I K I S < L ~ ~ l ~ i i o r ~ n ~ z e r for each Q, since L I P is everywhere unramifird. So. V ~ , ( J T " ~ J ) 1 0 for all I . so x"" J lit% in all of the valuation rings that extend Op. Hut.

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DIVISORS ON DIVISION ALGEBRAS 3219

this means t,hat nnP J is i n the 1nt.egra1 c4os11rt. of Of in L, rv11wt1 IS l o p . (To see this, if B is the integral closure of Up i l l L , take n basis 1 1 , . . . ,I, of Ilk, and let l',, . . . .1: he the dual basis relative to Trl/k. Then l;, . . . .1: is also an F-bass of L. Suppose that b E B, and write b = C, lia, with a, E F. 'Then 61, = C, l,lJn,, so 'IYLIF.(W,) = C, 'Iklp.(ltl:)u3 = a,. But, TrLIF(B) O P SO each ( 1 , E O p . 'I'hus. U - C / ; U p = /Up.) tlrncr, snP f E Ap. Since T

is also a unifor~nizer for A p i m l is I I I its ctlltrr, wc8 have ci:p(s""J) - wp(f) t up 2 0, SO

J E L(G). For P E { P I , . . . , f:,), lOp, is ( l ~ e u~~iquc. valuat,lon ring ext~entlir~g O r z , and I T sa,y v: denotes this valuation on L . t,lien (J) 1- @(G) > 0 implies lhat v:(J) 2 0 for each i, which implies that / E Aps for each 1 'I'hus, ulpL( f ) 2 0 for each 1. 'I'llerefore, L(dr(G)) C L(G)n L.

For the reverse inclusion, t.ake f E L(G) n L. Then, for P E S - { P I , . . . , P,), wp(f) t np 2 0. If rr E Op is a uniformizer for Op and A r , then J - T - " ~ U for some u E Ap. But this puts u in Ap n L, wl~ich is integral over O f . Thus, 11 is in every valuation ring of L extending Op; in other words, tru(J) > -np for every valt~ation ring of L extending Op. For P E { P I , . . . , P,), we have urq( J) 2 0, so J E Ap, n L. As before, Apa is a valuation ring, so Ap, n L is the (unique) valuation ring of L that extends Opt. Thus, up,(/) 2 0 for each i. Thus, we have proven C($:(G)) = C(G) n L.

Consider ev : L(G) -+ l", where ev( f) = (f (PI) , . . . , f (P,)). I f / E L($(G)), we claim that f (QJ = f (PJ for each 1 , and so ev(L(d4G))) = {( / (&I) , . . . , j(Qn)) I j E L($(G))}. This is CL($(P),$(G)), finishing the proof of the lemma once we prove the c l a i m A d o this, note that if r n ~ , ~ , is the maximal ideal of OL,Q,, then j(Q,) = J + r n ~ , ~ , E O L . ~ . = - 1 = Ap<. Also, viewing J E D, we have J(P,) = f t J(Ap,). But, Ap, contains O L , ~ , and J(A.1 ~ O L . Q , = r n ~ . ~ , . so J t ~ L , Q . = J + J(Ap,). In other words, J(Q,) = f(P,). t

We say that an L-basis d l , . . . . d, of D is an orlhogonal baszs with respect to a maximal or- der Ap E X with associated value functjon wp, provided t,hat wp(C,l,d,) = min,{wp(l,d,)} for all 1, E L. If , in addition. urp(C,l,d,) = min, {wp(l,) + wp(dt)}, we will call the L-basis d l , . . . , d, a strongly orthogonal basis with respect to Ap.

Lemma 8. With D = @:,: LJ:' us tn the beginning of thts section,

I . I] D is splzt a1 P . lel d l . . . . .d, he unils in A p such that &, . . . ,h is a basis for over Ap fl L. Then d l . . . ,d, as u strongly orthogonul basts for D with respect to Ap.

2. Ij P = P,, lhen 1 ,s.. . . . Y"' is u stronyly orlhogonal basis for D with respect to A r

Proof: Suppose t l~at d l , . . . , tl, are u~~ i t , s in Ar such that &. . . . ,dT is a basis for over Ap C1 L. Note that u:p(d,) : 0 for eac11 1. and t.hat since 11, E A;, we have ,wp(ed,) = urp(e) for all e E D by the dcfi~~ition of the value ful~ction ,u~p (or by 16, Lemma 2.21). Suppose that min. {wp(l,d,)} = u!p(lk). Then wp(lt) = t for some t E Z, so if a is a uniformizer for Op, then wp(a-'1,) 2 0 for all i , and wp(s-'lt) = 0. These facts hold since a E F = Z(D). Thus, we may as sum that wp(1,) 2 0 for all i and wp(1k) == 0. Then C,l ,d, = C,cz f 0 since 5; # 0, so wp ( C , l,d,) - 0, as desired.

For the second part, suppose that P 7 P, is one of the rational points at which the local index of D is equal to r. - ~ n d ( D ) . Let Inv(Dp) r= k/r t Z with gcd(k,r) = 1. Then Dp =

(lFp/Fp,o, x k ) for some unik)rniizer s of Op. Since Y = s k u for some unit 11 E O;,. and since wplF = rvp if imp is the ~lorlnitlized valuat.ioli o n F wit11 vAuation ring Op, we see that wp(xl) = ik. Mor~owr. L/k' is ~~~lrdniifit 'd at P, so t.hc: valur group of L is TZ. ' ~ h u s , since gcd(r, k) = 1, t,he va111c.s c~.p(.r') iwc distinct ~notlulo the vitlue group of L. If x,1,:1:' E 11. each term 1,x' has dist111c.t v,tlur~, so / u p (C, I,.?) -~ lnin, /I)~(/ , .E') = n~in, {,wp(l,) t ,up(x')]: the latter equality I~ol(ls si~lc.v WJ, 1s ;I valuntion OII 11. 'I'hls shows that l , z , . . . ,zr-I is a strongly ort,hogonal basis of U over L w~t,li respect to Ap.

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3220 MORANDI A N D SETHURAMAN

Proposit ion 9. 7'he set { 1 , ~ . . . . , xV-I} zs an ortlro!loncd hnsi .~ u~zlh rr..?pccl 1 0 vm:h A r f w PEA'.

Proo[, I,et, 1' c ,Y. I S tJ I>,, t h t , h ~ J:' f 0 1 I I I :L SI I O I I ~ I ~ or1 I I O ~ O I I ~ I)P& with r ~ p ~ t , t.0 A p by Lellllna 8. Next., suppoac. t,hN I ) is splil il l P Iirbc.l~ll fro^^^ t l ~ v ~)~.uof or 1x111n1a 7 tirat 10,. is t.lle irltc.&r.al closllrc. ol 0,. i l l 1, I"I.OIII 1111s. wtx S(Y ' 1 1 1 d 1 11 /C)/$ is ~ ~ ~ n 1 ; ~ i n e d in the stalk Ap. By 14, Prop. 1.31, t.liere are 2, E A p such r h k t At, - r i i ; ' , j 112, with : , /I - u'(6)q for b E L, and z,z, = f (o',uJ),-,+, for some normalized rocycle f w~th vducs in H . Since 1) is split at P, the maxi~nal order A r 1s Azuntaya over 01,. 'l'l~us, J(A18) - r n ~ d ~ b - -ii,::Jrnp~;,, wher-e mp is the n~aximal ideal of Op. HOWCW. h j 14. t'rop. X I / . ./(A,') (I)::,: 1,2,, where 1, IS the product of the ~naxinial ideals iZ/I of A for wh~ch f ( u ' . a ') 81 hl . Since mpU is the product of all thr niaxin~d ideals or H , U I I I ~ I I V fiwtori~.llio~~ of i( l(ds i l l Ll force^ f (u', o-') @ M for every n~axirrral ideal M of H , so f(u' , o- ' ) E 11' Sor ( W I I r . 'L'her~, front the cocycle condition, we see that o'(J(o-', oJ) ) f ((n', n-" ' ) j ( u E 3 u-') j ( I , uJ) € B* , so all cocycle values are in fact in Be. Consequently, the residue r i ~ ~ g 2 is equal to lhe crossed -- - product (B/Op, o, f ) = @::;%, so Ihe f are linearly i~~tlepentlent over z. Therefore, the 2, form a strongly orthogonal basis with respect to Ap by Lemn~a 8. Note also that ~ ~ ( 2 , ) = O for each i from this description of the residue ring 7, To see that the powers of x: form an orthogonal basis with respect to A P , write x' = a,;, with a, E 1,. Wt. h,lvt.

since 2 0 , . . . ,z,-, is an orthogonal basis of D w~t h respect to A p Tl~us, l l ~ c I' form an orthogonal basis. I

T h e o r e m 10. IJ G is a divzsor on D with deg(G) < rn and supp(C) n {A p,, . . . . AP- } = 0, then C(G) C L.

ProoL. Let f = x , l i x ' E L(G), so wQ(f) t wy(C;) 2 O for all Q. However. .uiy(f) = ~nin,(w~(l,x ')} 5 1 u ~ ( 1 , ) by Proposition 9, so lo E L(G) . I,(*( 7 -- C::: I,.? - j-10 E L(C;). Then wp, (f') 2 0 for all j . Since uip, (f') = rni~~,,, (w i ; (1 , ) I- i lc4p, ( . r ) } . again by Proposition 9, each wp,(l,) + iwp,(z) >_ 0. But iwp,(x) f O in l'A,5/~~p,n~. for cac:li 1 # 0 , so in fact lurJ(l,) + iwp,(x) > 0 for each 1. Consequently ubp,(f') > O for cach J . T l ~ r coefficie~~t of AP, in (f') i G is then at least 1. The degrw or A,; is I . s i ~ w t l ~ t s residue ri t~p of Ap, is 1. This, together with (f') t G 2 0, forces dep((jl) I G) 2 r.71, while cieg((J') t C) - deg((f')) + deg(G) < deg(G) < rn , a contradictio~~ unless f ' - 0. 'I'lwreforc.. j T. 1" E 1,. I

Corollary 11. Let G be a divisor on D with supp(C;) n { A p , , . . , A/:, } -- 0 , and suppose that deg(G) < m = deg(P). I j $ is the naturnl 7nq1 11, : tiiv(1,') tliv(1,). then the code C o ( P , G) is qua1 to the code CL(ll.(P),*(C:)).

References

[2] E. I~ormanek, The Polynoir~ial Identitzes nnd liivcmc~rrls o/ r t x 1 1 Alakriccs. CBMS Notes, Vol. 78, Amer. Math. Soc. Providence. 1991.

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DIVISORS ON DIVISION ALGEBRAS 322 1

131 V. D. Coppa, Codes 011 i~lgelnaic. curves, Sornel h!ull~ Dokl . 24 (1981), 170- 172

141 D. Haile, Crossed-product orders over dlscrew va lua tm~ r~ngs. J Al!leb~u 105 (1987), 116-148.

151 1. R e i n ~ r , Mcrimul Orders. Academic Press, [ondon. 1975

161 P. M o r a ~ d i , Vdut' fu~~ct ions on cent,ral simple algebr<~s, 7hii.s A7n~7. Mnth Soc 315. 1989, 605-622.

[7] P. Mora~idi. Nonco~~~mritiitive Priifer rings. J. Algebru 161. (I!N:I). 324 311

181 M. Orzeclt and C. SIIIHII, 7he Rtauer Group of Coin~tnulutzoc I<zngs. Marcel Llckker. Inc., New York, 1975.

191 M. Van den Berg11 and J. Van Geel, A duality theorem Sor orders in central s i ~ n p l t algebras over function fields. J. Pure Appl. Algebm 31 (1984), 227239.

1101 M. Van den Bergh and J . Van Geel, Algebraic elements in divis~or~ algebras over function fields of curves, Israel J. hlath. 52 (1985), 33-45.

1111 J . Van Geel, Places ulld Vuluulions in Noncommututzve Hzrrg 'I'twr:y, Marcel Dckker. Inc., New York, 1981.

1121 E. Witt , Rien1a1111-Rocl~ster sat.z und 2-funcktion in1 hyprrko~nplexen. Math. Ann. 110 (1934), 12-28.

Received: June 1997

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