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Integral Spinor Norm Groups Over Dyadic Local Fields

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Integral Spinor Norm Groups Over Dyadic Local Fields View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector ARTICLE IN PRESS Journal of Number Theory 102 (2003) 125–182 http://www.elsevier.com/locate/jnt Integral spinor norm groups over dyadic local fields Constantin N. Belià Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, RO-70700 Bucharest, Romania Received30 September 2002; revised3 December 2002 Communicatedby J.S. Hsia Abstract The spinor norms of integral rotations of an arbitrary quadratic lattice over an arbitrary dyadic local field are determined. The results are given in terms of BONGs, short for ‘‘bases of norm generators’’. This approach provides a new way to describe lattices over dyadic local fields. r 2003 Elsevier Science (USA). All rights reserved. MSC: 11E08 Keywords: Quadratic forms; Dyadic fields; Spinor norms; BONGs Introduction In the theory of quadratic lattices over global fields, where the local–global principle does not exist, the genus of a lattice usually contains many classes. The spinor genera are an intermediate step between classes and genera. (In the indefinite case when the rank is at least three the notions of spinor genera andclasses coincide.) In the theory of spinor genera in particular for the purpose of calculating the number of spinor genera in a genus, a key element is the knowledge of spinor norms of integral rotations associatedto the localizations of the given lattice at all primes. Until now these groups have been computedfor the case when the local fieldis non- dyadic by Kneser [K] or 2-adic by Earnest and Hsia [EH2]. More recently Xu [X3] ÃCorrespondence address: Institute of Mathematics ‘‘Simion Stoilow’’ of the Romanian Academy, 21, Calea Grivitei Street, RO-781011 Bucharest, Sector 1, Romania. Fax: +40-21-212-51-26. E-mail address: [email protected]. 0022-314X/03/$ - see front matter r 2003 Elsevier Science (USA). All rights reserved. doi:10.1016/S0022-314X(03)00057-X ARTICLE IN PRESS 126 C.N. Beli / Journal of Number Theory 102 (2003) 125–182 solved the case of dyadic fields where the ramification index e :¼ ord2 is 2. In this paper we completely solve this problem for arbitrary dyadic fields. See Theorems 1 (Section 5) and2 (Section 7) below. Throughout this paper F will be a dyadic local field, O ¼ OF the ring of integers, p the maximal ideal, O the group of units in O; e :¼ ord2 ; p a fixedprime element in F; Dð:Þ the quadratic defect function and D ¼ 1 À 4r a fixedunit with DðDÞ¼4O: Also we denote for any two lattices NDM by XðM=NÞ :¼fsAOþðFMÞjNDsMg the set of relative integral rotations. All unexplainednotations are from [OM]. In order to simplify the notation we put yðLÞ :¼ yðOþðLÞÞ and yðM=NÞ :¼ yðXðM=NÞÞ: If V is a quadratic space we denote LðVÞ the set of all lattices on V: We make the convention that if V is a quadratic space then det V is regarded as an element of 2 F’=F’2 while if L is a lattice then det L is regarded as an element of F’=O : (Hence if aAF’ we have det L ¼ a iff det FL ¼ a andvol ðLÞ¼aO:) One of the main difficulties in studying lattices over dyadic local fields is the fact that, unlike in the non-dyadic case, lattices no longer have orthogonal bases. Instead they split into unary or binary improper modular lattices. The 2-adic case is somewhat easier because the improper binary unimodular lattices are totally improper andthere are only two possibilities; namely, Að0; 0Þ and Að2; 2rÞ: When e41 things become more complicated. In order to overcome this we introduce in Section 2 the notion of a BONG (short for ‘‘basis of norm generators’’) of a lattice which is a generalization of the usual concept of an orthogonal basis. In many ways BONGs, especially the so-called‘‘goodBONGs’’ (introducedinSection 4) behave like orthogonal bases. This allows us to study a general lattice almost as if it is diagonalizable and thereby we simplify considerably both the reasonings and the statements which are often very complicated when dealing with lattices over dyadic fields. On the other hand, the use of BONGs is not intended to replace the usual Jordan splittings. In fact, our proofs incorporate both concepts and sometimes we switch back andforth between them. We shall use the notation L ¼ !x1; y; xng to denote the fact that x1; y; xn is a BONG for L: The approach we use for calculating yðLÞ is quite classical, namely finding generators for OþðLÞ: O’Meara andPollak [OP1,OP2] provedthat in the special cases when L is either modular or when F is 2-adic (i.e., e ¼ ord2 ¼ 1) the group OþðLÞ is the group XðLÞ generatedby SþðLÞ andthe Eichler transformations. Since the Eichler transformations always have trivial spinor norms they do not contribute to yðLÞ: Hence, we still have yðLÞ¼yðSþðLÞÞ: This fact allowedthe calculation of yðLÞ in the case when L is modular by Hsia [H] andin the general 2-adic case by Earnest andHsia [EH2]. Unfortunately, when e41 it is not known whether OþðLÞ¼XðLÞ andthe generation problem of OþðLÞ it is still unsolved. As already observed in [EH1] that since one is only interestedin yðLÞ one may assume that yðLÞaF’: This restriction on L simplifies the problem as it severely limits the possible structures of the Jordan splittings of L: Xu [X2] gave a set of complicatednecessary conditions for a lattice L to satisfy in order that yðLÞaF’: He also showedthat these conditionsimply that ARTICLE IN PRESS C.N. Beli / Journal of Number Theory 102 (2003) 125–182 127 OþðLÞ¼XðLÞ; thereby bringing us back to calculating yðSþðLÞÞ: Nevertheless, the explicit calculation of yðSþðLÞÞ is still a very technically formidable challenge. One of the benefits of using BONGs is that we can give necessary conditions such that yðLÞaF’: These conditions, called ‘‘property B’’ (see Section 4), are similar to those given in [X2] in terms of Jordan decompositions. But, they are expressible in a more compact form andit is not necessary to distinguish between the unary andthe binary components. Another advantage of using BONGs is that the formulas for yðLÞ given for the binary lattices in [H,X,X1] can be unifiedso that the modularand non-modular cases need not be separated. Our main result is the theorem given in Section 5. Its proof is split into two parts; namely, first we prove yðLÞ+G in Section 5 andthen yðLÞDG in Section 6. Here G is the group predicted for yðLÞ: The first part is easier andinvolves some techniques of multiplying the spinor norm groups of some binary lattices which are developed in Section 3. As for the secondpart we employ a technique from [X2] except that the Jordan decomposition used there is replaced by a BONG here. More specifically, given a sAOðLÞ; L ¼ L1>y>Lt; the necessary conditions in [X2] are usedto 0 0 express s as a product of symmetries of L anda s ; s AOðL2>y>LtÞ: This reduces the case to a lattice with fewer Jordan components. In our case, if we express L ¼ !x1; y; xng then property B allows us to have a similar expression for s with 0 þ s AO ð!x2; y; xngÞ: Next, we prove that the spinor norm of the above product of symmetries of L belongs to G: In fact, these symmetries belong to the orthogonal groups of some binary lattices of the form !x1; yg which have their spinor norms lying in G: Although our main result is given in terms of BONGs it can be readily translated to the traditional language of Jordan decompositions. However, the formulas will become somewhat less compact. It seems that the calculation of yðLÞ can also be done without the use of BONGs by using the methods of [X2] andthe formulas for the binary case obtainedin [H,X,X1]. However, this wouldrequire considerably more effort. Finally Section 7 is devoted to lattices not having ‘‘property A’’ (see Section 4). In this case yðLÞ can only be F’ or OÂF’2 andwe settle their distinctions. The use of BONGs has some further applications. In a future paper we will compute the relative spinor norm group yðM=NÞ in terms of goodBONGs of M and N: Also this methodprovidessome new ways to approach the problem of representation of lattices over dyadic local fields. 1. Duality lemmas ’ ’2 Definition 1. We define d : F=F -N,fNg to be the order of the ‘‘relative quadratic defect’’ by dðaÞ¼ord aÀ1DðaÞ: If a ¼ pRe with eAO then dðaÞ¼0ifR is odd and dðaÞ¼dðeÞ¼ord DðeÞ if R is even. Hence dðF’Þ¼f0; 1; 3; y; 2e À 1; 2e; Ng: ARTICLE IN PRESS 128 C.N. Beli / Journal of Number Theory 102 (2003) 125–182 Since ð1 þ pkÞF’2 ¼faAF’ j dðaÞXkg for any positive integer k we make the a ’2 ’ convention that ð1 þ p ÞF ¼faAFjdðaÞXag for any aAR,fNg: In particular N ’2 ’2 a ’2 ’ ð1 þ p ÞF ¼ F and ð1 þ p ÞF ¼ F for ap0: Lemma 1.1. d satisfies the domination principle: dðabÞXminfdðaÞ; dðbÞg: Proof. If one of a; b has odd order then the above inequality is trivial since the right side is 0. Otherwise, by multiplying with squares we may assume that a; bAO andmoreover a ¼ 1 þ a; b ¼ 1 þ b where DðaÞ¼aO and DðbÞ¼bO: We have ab ¼ 1 þ a þ b þ ab so dðabÞXordða þ b þ abÞXminford a; ord bg¼ minfdðaÞ; dðbÞg: & ’ ’2 ’ ’2- 7 Let ð ; Þp be the Hasse symbol.
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