Order Number 9807786
Effectiveness in representations of positive definite quadratic form s
Icaza Perez, Marfa In6s, Ph.D. The Ohio State University, 1992
UMI 300 N. ZeebRd. Ann Arbor, MI 48106 E ffectiveness in R epresentations o f P o s it iv e D e f i n it e Q u a d r a t ic F o r m s
dissertation
Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University
By
Maria Ines Icaza Perez,
The Ohio State University
1992
Dissertation Committee: Approved by
John S. Hsia Paul Ponomarev fJ Adviser Daniel B. Shapiro lepartment of Mathematics A cknowledgements
I begin these acknowledgements by first expressing my gratitude to Professor John
S. Hsia. Beyond accepting me as his student, he provided guidance and encourage ment which has proven invaluable throughout the course of this work.
I thank Professors Paul Ponomarev and Daniel B. Shapiro for reading my thesis and for their helpful suggestions. I also express my appreciation to Professor Shapiro for giving me the opportunity to come to The Ohio State University and for helping me during the first years of my studies.
My recognition goes to my undergraduate teacher Professor Ricardo Baeza for teaching me mathematics and the discipline of hard work.
I thank my classmates from the Quadratic Forms Seminar especially, Y. Y. Shao for all the cheerful mathematical conversations we had during these months.
These years were more enjoyable thanks to my friends from here and there. I thank them all.
Finnally I express my deepest gratitude to my family, especially to my parents
Sergio and Ines. Your support and encouragement was very important to me. V i t a
Sept. 21, 1959 ...... Born in Santiago, Chile.
1983 ...... B.Sc., Universidad de Chile.
1986 ...... M.Sc., Department of Mathematics, The Ohio-State University, Columbus, Ohio.
1984-1986 ...... Graduate Teaching Associate, Department of Mathematics, The Ohio State University, Columbus, Ohio.
1986-1987 ...... Instructor, Department of Mathematics, Universidad de Santiago, Santiago, Chile.
1987-1988 ...... Graduate Teaching Associate, Department of Mathematics, The Ohio-State University, Columbus, Ohio.
1988-1991 ...... Instructor, Department of Biology and Chemistry, Facultad de Ciencias, Universidad de Chile, Santiago, Chile.
iii 1992 ...... Graduate Teaching Associate, Department of Mathematics, The Ohio-State University, Columbus, Ohio.
F ie l d s o f S t u d y
Major Field: M athem atics.
Studies in Number Theory. Professor Robert Gold Professor Manohar Madan Professor Karl Rubin. Studies in Algebraic Geometry. Professor Warren Sinnott. Studies in Quadratic Forms Professor Ricardo Baeza Professor John S. Hsia Professor Daniel B. Shapiro. T a b l e o f C o n t e n t s
ACKNOWLEDGEMENTS ...... ii
VITA ...... iii
CHAPTER PAGE
I Preliminaries ...... 1
1.1 Introduction ...... 1 1.2 General Definitions ...... 3 1.3 Some Local R esults ...... 6 1.4 Some Global Results ...... 11
II An effective version of a Theorem of J. S. Hsia, Y. Kitaoka and M. Kneser 14
2.1 Estimations ...... 14
III ^-invariants of R ings ...... 41
3.1 The (/-invariant for local rin g s ...... 41 3.2 The (/-invariant of Global Rings ...... 48 3.2.1 The g-invariant of Z for n < 5 ...... 48 3.2.2 Study of gz(n) f°r n 6 ...... 51 3.3 The totally real c a s e...... 55 3.4 The imaginary case ...... 67
BIBLIOGRAPHY 70 C H A PT E R I
Preliminaries
1.1 Introduction
In 1978, J. S. Hsia, Y. Kitaoka and M. Kneser proved the following theorem about representations of positive definite integral quadratic forms. See Theorem 1 in [HKK].
T h e o rem 1 . 1 . 1 Let M be a positive Z-lattice of rank m > 2n + 3. There is a constant c — c(M) such that M represents any Z-lattice N of rank n, provided that
Mp represents Np for each prime p and fi(JV) = A/m{Q(a:)|0 ^ x € iV} > c.
The main purpose of this dissertation is to estimate a bound for the constant c by applying arithmetic methods rather than analytic ones.
In the case in whichn = 1 we obtain, up to an absolute constant, an explicit bound for c. This bound depends upon the determinant of M, its class number and some constants related to Minkowski-reduction theory (see Chapter 11,2.1.15). For n > 1, we obtain a bound for c that is recursively generated for each n. It depends as well on reduction-theory estimates and on the determinant and class number of A/
(see Chapter 11,2.1.16). The method we will apply in order to get our estimates will be to construct, at each stage, the proof of Theorem 1.1.1. Most of the constructions will be based on general results from lattices in quadratic spaces, in particular some
1 structure theorems for quadratic lattices over local fields. We will also make use of
known results concerning Minkowski- reduction theory (see [Ca], Chapter XII and
[VdW]). Finally we will use an estimate for the norm of a prime ideal in a given
ray class in a ray class group of a certain quadratic extension of the field of rational
numbers. The desired bound is then obtained by putting together all the particular
estimates.
Once we have obtained a bound for c(M), we give an application, both of 1 . 1 .1
and the methods used to obtain 2.1.15 and 2.1.16 to study the following invariant.
Let us consider for any integer n > 1 the set
Sn,z = l > • • ■) -^n ) 2 |Ll5..., Lrare linear forms overZ L i=i i.e. En,z consists of all sums of squares of linear forms in n- variables over Z. We
define the g — invariant of Z
any q £ En>z has a representation q = L\ + ----- 1- L* 1 gz(n) = min jt with integral linear forms L\,..., Lr J
The following facts concerning gz(n) were proved by L. Mordell (see [Ml], [M2])
and by Ch. Ko (see [Ko]). First of all gz{n) = n + 3 for n < 5. Also that n = 6 is the
first dimension in which there is a positive definite integral quadratic form which does
not belong to En,z- In chapter III we study gz(n) and prove that it is always finite;
moreover we give an explicit bound for gn>z for all n (see 3.2.4 and 3.2.7). We then generalize the definition of the g — invariant for the ring of algebraic integers of any number field and obtain similar results as in the case of Z (see section 3 in chapter
III). We devote the rest of this chapter to giving general definitions and results that will be needed in the next chapters. 3
1.2 General Definitions
In this chapter we introduce some of the basic definitions and results that will be needed troughout the dissertation. We will follow the notations and use most of the definitions as given in [O’Ml] and in [0 ’M2],
Let F be a field with characteristic of F not 2, O = O(S) be a Dedekind domain defined by a Dedekind set of spots S on F and / = I(S) be the group of fractional ideals. Let V denote an n-dimensional vector space over F which is assumed to be endowed with a symmetric bilinear form B : V xV— *F with associated quadratic form Q satisfying
2B{x,y) = Q(x + y) - Q(x) - Q(y) for x,yeV.
We define a quadratic lattice in V as follows
Definition 1.2.1 Let M be a subset ofV that is an O- submodule from V . We say that M is a lattice in V if there is a base x ^,..., xn for V such that
M C O x\ + • • * + O xn we say that M is a lattice on V if in addition F M — V
Given a € F and Ql € I we put
a M = {aarja: € A/}, 21M = : /3 €21 ,® € M}. Jin
The rank of a lattice L, rankL is given to be rankL = dimFL. A set of vectors
®i,... ,x n is called a base for L if and only if it is a base for FL with L = Ox\ + • • • + Oxr. A lattice which has a base is called free. Every lattice L is almost free in the sense that it can be expressed in the form L = Slxi + O x2 + h O xr with 31 a fractional ideal and x\,...,xr a base for FL.
Theorem 1.2.2 Given lattices L and K on the space V . Then there is a base
*ii • ■ • j for V in which
L = Sit®! + • • ■ + 3tXn, K = QliJHjXi + ' ‘
where 31; and fH, are fractional ideals with JH; D 9t2 2 • • • 2
The ideals are called the invariant factors of K in L. If K C L then one can prove that the index
(L -.K)= n < 0 : ffc) <=1 Suppose that a lattice L is the direct sum of the sublattices Li,...,Lr, with the bilinear form B(L>i,Lj) = 0 for 1 < i < j < r, then we say that L is the orthogonal sum of the sublattices L; and we write L = L\L ■ • ■ J-Lr. We call the Li the components of the splitting.
We will need the following
Definition 1.2.3 Consider a lattice L in the quadratic space V. By the scale of L, s(L) we mean the O- module generated by the subset B(L,L) of F. We define the
norm of L, fif(L) to be the O- module generated by the subset Q{L) of F.
Both the scale and the norm are fractional ideals. It follows easily that 2s(L) C
Af(L) C s(£). In the case in which the norm of a lattice coincides with its scale we say that the lattice isproper, otherwise we say it is improper. One also delines another fractional ideal associated to the lattice L as follows.
Let L = d i + ■•* + Qlxr, with the 2 h in I. Then the volume of L, v(L) —
9l| • • .. ,arr). Here d(x = det(B(xi,Xj)). In the case in whichL is free , we will directly refer to the volume of L as the determinant of L and we will denote it both by det(L) or by d(L).
Suppose that the scale of a lattice L is the fractional ideal 91. Then v(L) C 2 lr. In the case in whichL satisfies v{L) = 3lr we will say that the latticeL is Ql-modular.
We call L unimodular if it is 51-modular with 21 — O. In the event in whichL is free,
L is unimodular if and only if the associated bilinear matrix is unimodular.
Definition 1.2.4 Let L be a lattice on the regular space V , and let 21 be a fractional ideal. We say that L is 21-maximal on V if the norm of L, N(L) C 21 and if for every lattice K on V which contains L we have hf(K) C 21 implies that K — L.
We say that L is maximal if it is maximal with respect to some ideal 21.
Let 93 be any spot in the Dedekind set of spots S. Consider the quadratic space
V*p = F*pV. By the localization of L in V*p, Lv we mean the C?v-module generated by L in V*p. If L = 2 lia:i + • • • + %-xr, then L*p = 2 li*p;ci + • • • + 2lr Consider the lattice L in V. Let U be another quadratic space over F and let K be a lattice in U. Definition 1.2.5 We say that K is represented by L, and write K — * L if there is a representation a : FK — * F L such that aK C L . If a is an isometry and aK = L, then we say that K and L are isometric and write K = L. 6 1.3 Some Local Results In this section we use the same notations as before except that the field F is taken to be a local field, S the single spot ?p and O is the ring of integers at tp. Consider a non-zero lattice L in the regular quadratic space V. Then according to [0 ’M1;91 C], L can be decomposed into an orthogonal sum of one and two dimensional modular lattices. If we group the modular components of such splitting for L we find that it has a splitting L — L\ ± • • • ± L t in which each component is modular and s(Lj) D • • O s(Lt). We call such decomposition a Jordan splitting for the lattice L. If one assumes that the local field is non-dyadic, then it can be proved ([0’M1;92]) that every lattice has an orthogonal splitting into one dimensional lattices, i.e. every lattice L can be written as L = Ox i-L • • • ± O xT. This is no longer the case when the field F is a dyadic local field. Since we will be interested in representations of integral lattices, we set up some extra notation and definitions in order to be able to state some results concerning representations in the local case. If the field F is dyadic, we need to give the following Definition 1.3.1 For a lattice L, the set Q(L) + 2s(L) is an additive subgroup of F. We shall call this subgroup the norm group and we will denote it by g(L). It follows then that 2s(L) C g(L) C Af(L)> Definition 1.3,2 Let w be a uniformizing element in the non-dyadic field F, let L = L] ± • • • LLi be a Jordan decomposition for a lattice L in the quadratic space V 7 over F. Then we define Ci to be the orthogonal sum of the Lj for which s(Lj) 3 n'O. Definition 1.3.3 Let F be a dyadic field in which 2 is a prime element We define £(,) to be the orthogonal sum of those L j ’s for which the norm Af(Lj) D 2*0. Next we put iC[i] to be the orthogonal sum of the Lj for which s(Lj) D 2*0 and, in addition, that value j, if any, for which s(Lj) = 2'+lO with N’(Lj) C s(Lj) We now state a theorem for integral representations of quadratic forms over non- dyadic local fields (see Theorem 1 in [0 ’M2]). Theorem 1.3.4 Let F be a non-dyadic local field. Let L — L]L" ■ L L t and I = /i _L • • • -L/r be two lattices with their corresponding Jordan decomposition . Put Ci, defined as before for L, and correpondingly for the lattice I, Then I — ► L if and only if the quadratic spaces Fti and FC ,• satisfy FI, — ► FC,• for all i. The corresponding theorem for the case in which the field F is dyadic but 2 is a prime in F , requires some extra definitions. Namely Definition 1.3.5 Suppose F is a dyadic field in which 2 is a prime. For a lattice L we define A,- in the following way. If L has a proper 21+1 - modular component, put Ai — 2’+10 ; failing this, put A; = 2,+20 if L has a proper 2, + 2 -modular component; otherwise put Aj = 0. Define D, — det(Ci)0; if Ci = 0, put Di = 0 For any ideal 93 C Q and any quadratic space V over F, write 93 — ► V if there exists u> € V such thatQ(u>)G = 93. We shall write 93 —* if 93 — » Fx with Q{x) = fi. Finally we give the following definition (see [0 ’M2], pg. 858). 8 Definition 1.3.6 Given two lattices L and I, suppose that Ci (resp. U), Di (resp d,) and A< (resp Si) are the quantities defined above for L and I. Then we say that I is of lower type than L if the following hold for all i 1. rankti < rankCi 8. diDi —* 1 ifrankii — rankCi 3. Si C Ai + 2i+2G and Aj_i C + 2,+1 C> if rankt-i = rankCi 4. A,_i C 6 ,.! + 2,+a ifrankC — 1 = rankti > 0 and diDi — > 2,+ 1 5. Si C A,• + 2,+20 if rank Ci — 1 = rankti > 0 and diDi — ► 2* We are now ready to state the next theorem (see [0’M2; pg. 864]). Notation. Let p G F such that p is a unit and the polynomial x 2 + x + p is irreducible in F , hence 1 —4p ^ (F*)2. Write a(l + 4u?) — > U if either a — ► V or q ( 1 + 4p) — > U. Also if a 6 F, denote by < a > J_L the lattice OxLL with Q(x) = a. Theorem 1.3.7 Let I have a lower type than L. Then I — > L if and only if the following conditions hold for all i : (I) A,—(/^ , where the orthogonal complement is taken in (FC(i+2)), (H ) ft — . where the orthogonal complement is taken as in (I), (III) (^[i])'L — Fw implies A {Si C Sf. Here H denotes a hyperbolic plane, (IV) 2*(1 + 4oj) — ► (Fti)x, where the orthogonal complement is taken over ( F p l C ^ ) ) , 9 (V) 2*( 1 + Auj) — > (F£[t])'L> where again ike orthogonal complement is taken in (F(2«_LC(i+1))). In the particular case in which the latticeL is unimodular one has the following result (see [0’M2,pg. 862-863]). Theorem 1.3.8 Let L be unimodular and let I be an integral lattice with s(l) C O. If FI — ► FL and Af(l) Q Af{L). Then I — ► L if and only if (I) A_x — > (F^f-i])1, with the orthogonal complement taken in FL (II) ( F ^ - i ] ) 1 = (FH) implies A_i£_i C With the orthogonal complement taken as in (I). If we want to consider the case in which 2 is neither a unit nor a prime in F we have to quote the results due to C. Riehm (see [Ri]). Before that we must give a definition valid for any lattice. Definition 1.3.9 Let L be a lattice in any quadratic space V. We define the dual of L, L* = {x e FL\B(x,L) € O} We now state the following theorem which gives necessary and sufficient conditions for representations by unimodular lattices. Theorem 1.3.10 Let L be a unimodular lattice of rank > 3, and let I be another lattice with u — rankL — rankl > 0. Then necessary and sufficient conditions for I — ¥ L are : (i) I f v = 1 , l±(d(l),d(L))L. 10 (ii) If v = 2, /±{o:)±(ad(/)rf(L)) — > L, where a is a non-zero field element such that FLL(a) — ► FL and a € 4(s(/*))-1. (iii) If v > 3, «(/) C s(L) and g(l) C g{L). Necessary and sufficient conditions for (i) and (ii) to hold are given in [Ri], First Main Theorem. We will only be interested in the case of codimension being at least 3. Finally we give the following definition (see [Ki4]). Definition 1.3.11 Let L be a quadratic lattice over O. Define the level of L, £(L) to be the least integer such that ^ L^Q(L*) C (2). Define also j(L) to be the order of the scale of the last component of a Jordan decomposition for L, which according to [O'M 1,91:9] is independent of the particular Jordan splitting. Remark 1.3.12 From the definition, it follows at once that if *p is non- dyadic, £(L) — j(L). If is a dyadic spot, it then follows that £(L) < j(L) + e2 (^p), where e2 (^p) denotes the ramification index of 2 at Related to the last definition, we will make use of the next corollary (see Cor. 5.4.3 in [Ki4]) Corollary 1.3.13 Let L = OvH------1-Ovn and M = Ow H---- hOwn be quadratic lat tices over O and suppose that F L is not singular. IfB(v{, vj) = B(w,, Wj)mod^L^ x O and Q(vi) = Q(wi)mod2 7j(vi) = Wi‘modyit(L)+1 M * . 11 1.4 Some Global Results In this section we give some global results and definitions. We now assume that the field F is a global field and S is a Dedekind set of spots that consists of almost all spots in F. For any quadratic space V over F, we denote by 0(V) the orthogonal group of V and by 0'(V) the group of all rotations in 0(V) with trivial spinor norm. For K and L on V, we say that they belong to the same class if K = oL for some <7 6 O(V). This defines an equivalence relation on the set of all lattices on V. We denote the class of a lattice L by clsL. We now introduce the definitions of genus and spinor genus of a lattice Definition 1.4.1 One defines the genus,genL of the lattice L on V to be the set of all lattices K on V such that for each ?p E S there exists an isometry ay such that K y = ayLy. We immediately have genK =genL if and only if clsKy= clsLy for all E S. We say that K and L belong to the same spinor genus if there is an isometry a in 0{V) and a rotation £ clsL QspnL QgenL, The classes, genera, and spinor genera, partition the set of all lattices on V. Let L be a lattice on a quadratic space V over Q with dimension of V > 3. Let d'(Lp) denote the usual discriminant of the localized lattice Lp if n is even and \d[Lp) if n is odd. We say that L is good at p if the set of elements represented by Lp, Q(LP) C Zp and d'(Lp) E Z*. Let K be another Z- lattice on V that also satisfies the conditions Q(KP) Q Zp and d'(Kp) € Z*, i.e. K is also good at p. According 12 to [0 ’M1;82:19], Lp and K p are Zp-maximal and then by [0’M1;82:23], there exists a local basis for Lp, satisfying Q(e{) = Q(fi) = 0 , s (ei>/j) = > 0 = B(ei,ej) = ) for i ^ j , B{eu zk) = B(fi,zk) = 0, and such that the subspace spanned by Zat+i, • • •,z n is anisotropic with = (pai )et + (p”a>)/! + • • • + (p“‘ )et + (p-“' )ft + Zpz2 t+ 1 + • • • + Zp*„ where a, € N. Then [.LP : Lp n Kp] = [Kp : Lp n Kp] = p“i+-+“‘ Definition 1.4.2 According to [BH], one defines the global graph R(L : p) by taking for vertices those lattices in genL such that Kq = Lq for all q ^ p. The distance between the lattices L and K , d(K, L,p) = ?• where [Lp : Lp fl K p] = pr. We will say that L and K are neighbors when r = 1. Remark 1.4.3 Two vertices are connected by an edge only when they are neighbors. From [O’M: 104:5], we obtain Corollary 1.4.4 I f K belongs to the graph R (L : p), then R (L : p) contains a representative of every class in spnK. Remark 1.4.5 If F is a number field and O is its ring of integers, then all definitions above generalize when changing the prime p by a prime ideal ?p 6 O. It then follows that if two lattices L and K are good at a prime [£/qj : n '• n = iV o r m^ ) 7 13 where Norm'S denotes the norm of the ideal B The following corollary is a known result that arises from the fact that isometric lattices have isometric neighbors, Corollary 1.4.6 Let L be a Z-lattice, let {Li... L^i)} be representatives of the classes in the spinor genus of L. Let p be a prime such that L is good at p. Then jjML)-1Li C L for all i. Remark 1.4.7 This same result generalizes for number fields. We finish here this chapter; any further result or definition that we may use will be given when needed. CHAPTER II An effective version of a Theorem of J. S. Hsia, Y. Kitaoka and M. Kneser 2.1 Estimations Our goal here is to give au explicit estimate for c = c(M) in Theorem 1 . 1 .1 , by applying arithmetic techniques. In order to get this estimate we first need to prove several lemmas . The first of them gives an estimation for the constant c = c(K,qaL,') in lemma 1.3 of [HKK] . We will make use of some results about Minkowski reduction theory that were not included in Chapter 1. Most of them are stated in chapter XII from [Ca] and in [VdW]. For the sake of completeness we state lemma 1.3 here. Lemma 2.1.1 Let L be a positive Z - lattice of rank I > n + 3 . Let q be a prime such that Lq is isotropic . Assume that genL = spnL . Let s be an integer so that L represents every Z - lattice N satisfying qaLv represents Np at each p. Let V C L be a maximal lattice , K be a positive Z - lattice of rank k > n . There is c = c(K, qaU) such that K _L L represents every positive Z - lattice N = 5 ^ = 1 %vi °f rank n satisfying 14 15 1. Kp _L q‘L'p represents Np at each prime p 2. The matrix (B(vi, Vj) — cln) > 0 Notice that the integer s given in the statement of the lemma exists according to 1.2 in [HKK] , and if we assume that the prime q is good, then from chapter one, corollary 1.4.6 , it can be taken to be h(L) — 1 , where h(L) denotes the spinor class number of the lattice L . Before stating the next lemma we introduce the following notation (see [VdW]) Let pk = (2 / 7 r)fc[r ( 2 + k/2)]2. Define x _ f H 1 if * < 4 I A£, (5)*<*"3)(*"4) if* > 4 Finually put t* = A^1. Then with all notations as in the previous lemma . L em m a 2 . 1 . 2 Let S be a finite set of primes containing 2 and a prime q at which Lq is isotropic and such that Lp is Zp maximal for p & S. Then c = c{K,q3L') < ^nk2qi^ V^ \'\lp ford^ L‘\ kdetK p es Proof. For p € S we need to choose rp so large that pTp € N {qBL'p). Since dimL > n + 3, we may choose rp for each p as follows: 1. r, = 2s + ordqN (L ’q) = 2(h(L') — 1) + ordqN(L'q) for p = q 2 . t*2 = 1 + m 'd zN ^'f) for p = 2 3. rp = ordpJ\f(L'p) for p ^ q, 2 16 By approximation , there is a finite set of n-tuples (vf,..., (k = 1,..., t) of vectors v{* G K such that for any collection (x,>) (p G 5, i = 1,..., n, xl)P G K p) there is some h with vjl = XiiP (modprpKp) (2 . 1 ) at every p in S. Then c = c(K,qaL’) has to be taken so large that (cIn -B (v ’>,v$))>0 for h= 1 ,... ,t. Write vj‘ = J2j=i aijej where K = Zej + • • ■ + Ze*. and {ei...e*} is a Minkowski reduced basis for K. Let [K\ be the bilinear matrix for K with respect to {ei,..., e*}. Then (B(«,\vJ)) = («a)TW (a«) But if [K\ = (rj()ft= 1 and [Ao] = DiagK is the diagonal matrix with diagonal elements j’ii, ... ,i%kk , then from lemma 1.3.3 in [Kil], we get [K] < k[K0]. Therefore v'j) < (n«j)TA;[/fo](a,i). Since K was assumed to be Minkowski reduced we have fc[A'o] < kQ(ek)hi 17 thus B(vf,v}) < (ayffaM fa). Equation (2.1) together with the Chinese Remainder Theorem allow us to assume w.l.o.g. that 0 < ajj < 2q2WL')-1)( J ] pes and we then have B(vf, „f) < (^ O ^ ) < Tr[(aij)r(a^)}kQ(r.k)lu < nk22 2 p es From Minkowski reduction theory, we know that Q(ek) < T h jj~ ‘ Where Dk - 1 de notes the determinant of the sublattice generated by {ei,... e*_i}. (see [VdW]) . Therefore < ni24(TT Pes Thus if we choose c = c{K,qaU) = 4nk2(Y [p)2ord^ L\ kq4^ L'^ d e t K p es we get that (c/n — J5(v|l, v1})) > 0. And the proof of lemma 2.1.2 follows. Q.E.D. In order to get an estimate for c = c(M) we will follow the proof of Theorem 1 . 1 .1 step by step . We will give the proof here and simultaneously provide all the lemmas 18 that are necessary to complete our computation of c(M). We begin with the proof for Theorem 1.1.1. Let S be a finite set of primes such that Mp is unimodular for p & S and containing 2 and a prime q at which the Witt indexindqQM > 2 (this condition is needed only if n = 1 ) . For each p € S choose finitely many Np(j) of rank n so that any other sublattice Np of Mp of rank n is representable by some Np(j). This family exists according to lemma 1.5 in [HKK] . To each collection J = (jP)Pes construct by lemma 1.6 in [HKK] a sublattice K(J) = Zx,- of M satisfying K(J) = Np(jp) for p € S and d(K(J)) £ Z*U rZ* for p,r £ S. Notice that the primer depends on the collection J. We call the prime r the exceptional prime for K(J). We give in the next lemma an estimate for the determinant of the lattice K(J) Lemma 2.1.3 With all notations as before det{K(J)) < 2nr{detMf Proof. We work out separately the dyadic and non-dyadic case. Let p ^ 2,p G S. Then = (paiPi) -L • ■ • -L (p°"+Jp„+3) X • • • -L where ai < a2 < • • • < a„, and € Z*. Let NpU) = X • • • 1 (pinwn) with 61 < • • • < bn and u>i € Z*. Let us consider the following cases. 19 1. b{ < a „ + 3 for all 1 < i < n . Then ordp(d(Np(j))) = bi + -----[-bn < fln+3 + • • • + < ordp(d{Mp)). 2. bj > an+3 for some j , 1 < j < n. In this case , consider the lattice K U ) = (Pbl^ ) -1 * • 1 1 where b* = a n+ 3 + £*(mo«f2) , e = 0 or 1. Then NpU) — K ti) Also by theorem [0 ’M2] KU) — Mr. W.l.o.g. we may assume N^(j) C Mp . In order to finish case 2 we still have to consider two possible values for 0 ,1+3 . namely. (a) a n + 3 > 1 . Then. ordp(d(Np{j))) < n (a n + 3 + l) < 2 (a „ + 3 + a n + 4 + • • • + am) < 2ordp(d(Mp)). (b) a„ + 3 = 0. Then the latticeMp contains a unimodular component, there fore a maximal component of rank at least n+3. In this case then, Mp represents 20 all possible Zp- maximal lattices of dimension n. By [O’M l,82:18], any Np(j) is contained in a maximal Zp- maximal lattice Np(j) which then is represented by Mp. The results stated in 91:2 ,91:3 in [O’Ml] show that or'dp{detNp(j)) < 2. Therefore, w.l.o.g. we may assume that in this last case ordp(detNp(j)) < 2 Now we study the dyadic case. Since the lattice M2 could be unimodular we have to consider separately the cases d(M2) € ZJ or M2 is not unimodular over Z 2 . Suppose M2 is unimodular. Then let N2(j) = lj 1 • • ■ 1 la be a Jordan decom position for N2{j) with each /, = 62i-modular. Then each U has matrix [li\ = 2h /,• JJ. Now theorem 1.3.8 implies N'(j) = l'i± ---L K ~ ^M 2. Therefore in this case we may assume ord2(d(N2(j))) < n. On the other hand, if M2 is not unimodular, let M2 — Ij\ J. * • * -L Lt with eachLi is 2 °*-modular and dimLi = m,. By [O’Ml] we may assume that each Li is either a one or two dimensional lattice. Consider the lattice L\ J_ • * * ± Lk so 21 that dim(Lt _L • • • _L Ljt) = n + 3 or n + 4. Let N2(j) = h ± ---± la be a Jordan decomposition for N2(j) with matrix of U = [I,] = 2biAi with A< unimod ular and dimension of = n,-. Then as in the non-dyadic case, consider the following cases. 1. Suppose b{ < dk for all 1 < i < n. Then ord2(d(N2(j))) < na/t < nk+iOk + • • • + ntdt < on'd2(d{M2)). 2 . Suppose bj > a/, for some j, 1 < j < s. In this case for i > j , replace li by /J with matrix of /J , [IJ] = 2ak+eiAi with £,■ = 0 or 1 and &,• = a* + £j(mod2). Consider the lattice Then W) — > ^ 0 ')- Now we have to use theorem 1.3.7, in order to prove that 1*2(3) — ► m 2. First we want to prove that Np is of lower type than M2. From the definition of lower type, if dimCi — dim£i > 3, then conditions (1),...,(5) in 1.3.6 are all satisfied. On 22 the other hand, since N2(j) — > M2, then l\ -L • • • JL /;_i is of lower type than M 2 by proposition 19 in [0’M2]. Since «i < 61 < • ■ • 6 j_i < a* and all 6 ;, 1 < i < j — 1 , all a,-, 1 < * < fc — 1 lie in between a\ and a* we get that C, and I, satisfy conditions (1),...,(5) of the definition of lower type (notice that C, = 0 for i < 1 ). For i > j , li coincides either with £k or with 4+i and in both cases, dimC,—diml, > 3 therefore N'2{j) is of lower type than M2. In order to prove that N2{j) — ► M2 we must also check that conditions (I),...(V) in Theorem 1.3.7 are satisfied. Those conditions follow from the next two remarks: • ^ = 0 for i < 1 and since 11 J_ • • • JL lj~ 1 — ► M2, conditions (I),... (V) in 1.3.7 are satisfied for 1 < i < j — 1 . • For * > j, dim(C(i+ 2)/£[i]) > 3 and then by proposition 16 in [0’M2], conditions (I),...(III) hold. On the other hand for i > j also the codimension ofi[,-] in £(i+i) is at least 3 then (IV) and (V) hold. Then KU) — ► M2 thus if ak = 0 we may assume ord2(d(NM))) < n If ord,(d(JV£(j))) < n(a*-t-i) < H------H mtat + n < 2ord2(d(M2)) 23 All these cases together imply the assertion of lemma 2.1.3. Q.E.D. Remark 2.1.4 We point out here that for the special case dimN = 1 in Theorem 1.1.1, we get, for the non-dyadic as well as for the dyadic case, ordp(detNp ) < ordp(detM). Therefore for this particular case we have det(K(J)) < 2r(detM). Once we have an estimate for det(K(J)) we move one step further in the proof of theorem 1.1.1. The next step of the proof of the theorem is to construct on the orthogonal complement of K(J) a sublattice L(J) of M subject to the conditions L(J)P is maximal if p € S and L(J)P = K(J)p for all p & S. We first state the following result (see 1.2 in [HKK]) which we will need for our estimations Lemma 2.1.5 Let L be a positive Z- lattice of rank I > 3 , q a prime such that Lq is isotropic, and assume that the genus of L coincides with its spinor genus: genL = spnL. There is an integer s such that L represents every Z-lattice N satisfying qaLp —- Np at every prime p 24 R em ark 2 . 1 . 6 In [HKK, 1.2] the condition genL = spnL may be replaced by the following: For each lattice N represented by the genL there is a lattice V in the spnL such that N is represented by V . It is important to point out that according to [H;pg. 148] the condition in remark 2.1.4 always holds if dimL — dirnN > 3. In the next lemma we proved that if by choosing larger values for rp, p € .S' in lemma 2 .1 . 2 one does not need to take the lattice U in 2 . 1 .2 to be maximal contained in L. Namely: Lemma 2.1.7 For the proof of theorem 1.1.1 , it is enough to take W ) , = « (J )i for all p Proof. From the proof of Theorem 1.1.1, it follows that it suffices to take in lemma 2.1.1 the lattice V = L. From the proof of 2.1.1 we now need to show that the lattice L satisfies the following condition: For N = E ”=i Zuj, if H = £ ”= 1 Zpi is a positive lattice with matrix A = (B(vi,vJ)-B(v?,v})) where {u'1} satisfy (2 . 1 ) from 2 . 1 .2 , then Hp ^ q ' L ( J ) p for all p. 25 If p ^ S, by the local version of 2.26 in [Kitl] d(L(J)p) | d((M)p)d(K(J)p). Since d(Kp(J)) € Z*UrZ* for p £ S and also d(Mp) € Z* it follows that d(Lp{J)) e Z* UrZ*. Therefore by [82:19 in O’Ml], LP(J) is Zp-maximal and by [1.1 HKK] H„ — . q-l(J), = L{J)P for p £ S . If p € S, by construction of the lattices K(J), there is J such that Np — > K(J)P for p € S. Thus B(v{,Vj) = B{xi,pjxhp) xi,pjxj,p € K{J)P By (2 . 1 ) Xi’p = v-1 + prp-i,P with z<,p €K{J)P, therefore, ,4 = (£ (v a-,u,-) - B(v'i,Vj)) = vj) + PTpB(vi>zi,p) + PrpB(zw v'j) + PirpB(zi,P,z3,p) - B (v! \ Vj)) = PTp(B {vi, Zi,p) + B(zitp, v}) + prpB{zitP, zjiP)) hence jV(Hp) c f - Z , for all p 6 S. We now choose rp for each p according to 2.1.2 as follows. 26 1. rp = 2(h(L(J)) — 1) + 3ordq(det(Mq)) for p = q 2 . 7-2 = (n + 1 ) + Zordiidet^Mz)) for p = 2 3. rp = 3ordpdet(Mp) + 1 for p ^ p,q. Here h(L) denotes the spinor class number of L. With these choices for rp we get that for p ^ 2 , the value of rp is always bigger than or equal to the order at p of the determinant of the last Jordan component of L(J)P. Then by [0’M2] Hp ► <7 Lp. The choice of t‘2 makes / / 2 of lower type than L{J)i and also satisfying conditions (I),...,(V) of 1.3.7. We use here the fact that dimL(J) > dimH + 3 for all primes p. Now we use remark 2.1.6 to get the result from 2.1.7 as follows. Since, for all primes p, Hp — ► q*Lpi there is q*Li £ gen(qaL) such that H — > Since ( dim(qsL') — dim(H)) > 3, by [H] there is a lattice qaU £ spnq3L such that H — 7 qaV. By the choice of a qaV C L and lemma 2.1.7 is thus proved. 27 Q.E.D. Corollary 2.1.8 With all notations as in the previous lemma det(L(J )) < (detM)(detK(J )) < 2nr{detM f. Proof. The inequality follows from the fact that det(L(J)) | det(M)det(K(J)). See 2.26 in [Kit 1 ]. Q.E.D. We give, in the next corollary a restatement of lemma 2.1.2 for the particular cases in whichV = L(J) = K{J)X , K = K{J) and k = n. Corollary 2.1.9 For L(J) and K (J ) as before c(K(J),q'L(J)) < ^q^^-^detLiJYidetK iJ^Tnr-2 < Tn23n+2n3rq4{h(L(J])~l)(d etM )s. As a particular case we mention separately Corollary 2.1.10 For n = 1 = dimN c(K(J),q'L) < 2Bq4ihM J'»-V(det(M))*r. 28 In the proof of theorem 1.1.1, the construction of the lattice K{J) is based on lemma 1.6 of [HKK]. Since our last step in the construction of c(M) is to estimate a bound for the exceptional primes r corresponding to the lattices K(J), and the construction of this prime appears in lemma 1 .6 , we now give a full statement of that result Lemma 2.1.11 let M be a h - lattice of rank m, S a finiteset of primes containing 2, such that Mp is unimodular for p ft S. For each p £ S let x\iP, • • •, x1i 1. at p £ S, Xi approximates a :t)P 2. for p 0 S, d(xj} - • • ,xn) £ Z* with precisely one exception p — r, where , * ■ ■, £ rhT. We are using the abbreviation d(zilP, * • •, ®n,p) = d e t(B (x Xj)) , (i,j — 1, • • •, n). For estimating c(M), the casedimN = \,dimM > 5 is the key part for obtaining a general bound, and the exceptional prime arises from the proof of that case(see [Kit2]). We study separately the size of the exceptional primes for K(J) when dimK(J) = 1. In order to bound the prime r, we will use the following result which is a corollary of Theorem 1.1 in [LMO]. Corollary 2.1.12 Let K be a quadratic extension of Q. Let 9JI be a modulus of ideals in Q containing all primes that ramify in K and containing only finite primes. Then there is a computable absolute constant A\ such that in each ray class modulo £0 t of 29 K , there is a prime ideal of degree one with Norm( fp) < 2(D(K)l^K)NoTm^)N o r m m WK)Norm{m)-'l'>)Al Here D(K) and h(K) denote the discriminant and class number of the field K. With the previous theorem we may now give an estimate of the exceptional primes r in the following: Lemma 2.1.13 Suppose that the lattices K(J) are one-dimensional. Let Km := 16m2( J J p)2 ^ JWp>+,)(JJp ) 2 Tmde Cm := (4/3 ) ,,l_ 1 /2(detM)l/,n. P:=4(np)i w ( n p)- p£S P^StpKKmOm Then there is a computable constant C depending only on M such that for Ai given in 2.1.12, the exceptional primes corresponding to the lattices K(J) constructed in the proof of theorem 1.1.1 satisfy r < 2CAl for all K(J). Remark 2.1.14 From 2.1.12, it will follow that an explicit value for the constant C is C = (4(KmCm) ^ )KmCmP3 pi(VKrnCmp*-l)) 30 Proof. Let M — Zej + • • • + Zem where m > 5 and {ej}"_i is a Minkowski reduced basis for M. Let Np(j) = ZpXiiP for each p € S and Np(j) C Mp. Put x hp = 0liPeH \-P,n,Pem 6 Zp for 1 < i < m. Since dimNp(j) = 1 , from the construction of Np(j) we know from 1.3.12 that for the lattices Np(j) and Mp, one has the inequality j(Np(j)) Let = {p\det(M) £ Z*} U U = {p\p < (4/3Ym~ ^ 2(det(M))^m} D {p|d(ei) £ Z;}. Since MP is unimodular for p ^ 5, for each p G (U — S) there exists a vector yp G Mp with d(yp) G Z*. Suppose that for each p G S', X\tP is given. We choose a vector x'j such thatx\ approximates x1)P for p € S and such that x\ approximates yp for p G (U — S). We also want that Zx\ = Zx\tP. From 1.3.13, it is enough to take the degree of approximation to be j(M p) + 1 if p ^ 2 and to be j(M p) + 3 for p = 2. Then x\ satisfying the above conditions also satisfies the following one. Let t = {P i smw) $ z;} then for p G T, d(ei) G Z*. W.l.o.g. we may assume that x\ , e\ are linearly independent . Consider the lattice L = Zei + Zx\. For our following computations we will need an estimate for d(e\,x\) which we get 31 now. Put x\ = a\e\ + < • • + amem then, for p ^ 2 <*i=pi,p ( n py(Mp)+i( n ?) pg£,p/2 pgt/—S for p = 2 , a,- = A.2 (2j^>+3) We may assume then that 0 < a,' < 4( p)tJ(Mp)+1)( J J p). p €5 pgtZ-S Now if we put [ M] = bilinear matrix of M, = (c*i... a m)T[M](a! ... a m) < (aj ...a m)T(ai ... am)m<2(em) < m21 6 (n P )20(Afp)+1)( I I p)2rm(det(M)). ?es P e(v-s) Put Km = 16m2( n P ) 2° (Mp)+1)( I I p)2TmdeiM. pgS pg U-S By reduction theory <*(*;, ei) < Q(x\)Q(e,) < Km(4/3)m- ll2(detMY'm. Put Cm = (4/3) m —1 t \ d e t M y and define V = {p$S\p< I For the lattice L, there exists a basis {u>i,u>2 } such that W ^ ) ) = s ( ba/2 )■ 32 With (a,6 ,c) = 1 , and 8 E Q*. If 6 $ Z*, p ^ 2, then 5 6 pZ* since L has integral scale. Then Q[LP) fl Z* would be empty, but from the construction of x[, ei, Q(Lp) H Z* ^ (j> for all p £ S. Hence, 8 $ Z* implies p E S already. Choose a vector v E L such thatv approximates x\ for p E S U V and such that at p E S the degree of approximation is j{M p) + 1 if p ^ 2 and j(M p) + 3 for p = 2. For the primes in T let, v approximate e\. Via scaling by £ - 1 it is enough to consider a binary lattice N with symmetric matrix equivalent to f a 6 / 2 \ \ V 2 c / ' Moreover we may assume that a is a prime with (a, P) = 1 . Therefore we are now in the situation in which we are given a binary lattice with matrix as above, the set of primes S' = S U T U V and a vector v in the lattice with the approximation conditions given before for the primes in S’. Notice that S' contains the primes dividing D := b2 — 4«c. Let F = Q (/}’/2), A = Z[a, ( 6 + £)V3 )/2], and let A be the ideal generated by a and by ( 6 + D1 ^2 ) /2. Let q(x) = where N(x) denotes the norm of the element x E F. Thus we may consider A, as L, $(s) respectively. In this setting it is enough to prove that for v in A, there is an element x\ in A such thatX\ = v (modP) and Q(xi) € Z* U rZ* for p ^ S U T U V. Put v = BD , with (P,D) = 1. Consider the ray class DA ~ 1 in Ip/i(Fpj) (see [Ja], pg.109). Then by corollary 2 .1 . 1 2 , there exists a prime ideal of degree one in 33 the ray class such that r = Noi-m(* p) < 2{D{F)h^ orm^p'>Norm(P)WF)Norm(‘F)-'l))Ai. Since !p belongs to the ray class of DA~1, we have an expression for *p = u£M - 1 with u = 1 (modP ). Put x\ — uv ; then (xj) = VpAB Q A, = v (modP ). More over Q(xi) = = NormysNoi'm(B) = rNoi'm(B). By construction, Norm(B) € Z* for p S U T U V. Then Xi is the desired element. The remark follows from the fact that (D(F)h^ Norm^JVorm (P)W F)Narm(p)~1)) < (4K,nCm)*KmCmP3 P2^ KmCmP2~lK We have use the fact that the number of inequivaleut positive binary lattices of a given determinant D,is at most ^det(L) (see remark 2.1.18). The proof then follows. Q.E.D. Corollary 2.1.15 Prom, lemma 2.1.8, with the same notations as before we obtain that if dimK(J) = 1 , detK(J) <4{detM)CAl. We may now get an estimate for the exceptional prime r corresponding to the lattices K(J) with dimK(J) = n > 1 . We give a bound for r in the following 34 Corollary 2.1.16 With all notations as in the previous lemma. For m > n , n > 1 there exists a computable constant Cn(M, A \ ) such that r < Cn{M,Ai) The constant Cn(M, Aj) will be described recursively in the proof. Proof. We proceed by induction. First take a;ljP, p e S and construct x\ as in the previous proof. We know that Q(x\) < Km. Then construct x\ as before. In this case, in order to coutvinue by induction, we will need an estimate for the value of Q(xj). We may obtain such estimate by changing the modulus P so that at the primes p, p ^ ?*, with p/Q(xi) (i.e. at p € S — {r}), Zpx\ = Zpx\. From 1.3.13, it is enough to take at those primes the degree of approximation to be ordpQ{x\ ) + 1 for p ^ 2 and ord, 2 Q(x\) + 3 for p — 2. Therefore in this case our modulus, which we denote by P\, will be a = 4 (nf>)*m+t( n *>)• Then we estim ate that Q(xj) < rKm = Ci(M,4i). Here r is the exceptional prime when taking the ray classes with respect to the modulus Pi as given above. Suppose now that ati,.. .a;n_i have been constructed and let p be the corresponding exceptional prime. Write xn Vn,p € EE? Qp*i = Yp and *n,„ € ( E ^ i Qp* ,) 1 = Y x . Let Y = EE? Q and denote by M' the orthogonal projection of M into Y x , the lattice M' may have non-integral scale, but at p ^ S U {p}, A/' is unimodular contained in Mp. 35 At p one can find x„iP € M„ such that H— 1 ZpSj+ z px„tP i= 1 is unimodular, with y„tP 6 Yp and z„(P = pr{xn Since M PI Y x C M' C y x, if we put I = [M ': M H Vx], then the isomorphism M * M' (M n y )±(m n y x ) “ 1' ( A /n y x) gives det(MnYx)det(MnY) 1 /2 ” ( det(M) ] det(M ny-*-)^!,...,^-!).]^ - 1 det(M) } The last inequality follows from the fact that Zxi + • • • + Zx„_i is a sublattice of (M fl y ). But det(M H y x) I rffx),... yxn^i)det(M), and by induction hypothesis, , x„_i) < C7„_i(A/, Ai). Therefore one has I < Cn-\{My Ai). Also det(M') < det(M n y x) < Cn_i(M , A1)det(M). Let ^ = » < a l-i(M,A1 )} 2 H p |/} and let A/' = Ze'j ------|- ZeJn_^l_1j be a Minkowski-reduced basis for A/'. Then I2Q(e\) < pd^detiM ’)* ^ < := C'm. 36 Also I2Q{e\) € Z. If U = {p|p < C , then, U\ C U and the set {p|p | I2Q(e\)} C U. Since Mp is unimodular outside -?U{p}, for each p e U-(Sli{p})take zp € Af' withQ(z p) G Z*. Take z'n € M' , such that z'n = zn,p (modpttM^+1 Af') /or p ^ 2 p G .S'. < = zn,2 (mod 2j^ +3Mlt) . = zn,P (modpMp ). = Zp (modpMp) for p e U - (.S’ U {p}). Then like in the case n = 1 , Chinese Remainder Theorem together with Minkowski- reduction theory imply Q{z'n) < K,n where Km = 16(m — n + l)2(fj p)2(J(Mp)+1){ J J p)2Tm- n+idet(M') p es peuup-s < 16(m-n + l) 2 (n p ) 2(i(Mp)+iy ( n P?rm-n+xCn^{M,A,)det{M). Pe s peu-s Put Q\{x) — I2Q(x). The inequality above implies that «?,(<) < I2Km < A\))2Km := K'm Let T = {p|p g S U {p} U f/,p < K'm} 2 {p|p 0 *S U {p} U U,p \ C?i(.z,'J} and consider the lattice V = Zz'n + Ze\ with the quadratic form Qi(a:). Let d\(z'n, e^) denotes the determinant of the lattice V . Then it follows that d\{z'n,e\) < Q\{z'n)Q\(e\) < Let V = {p^5U{p}UTUf/|p< K'mC'm}. By the case n = 1, there exists 37 zn € L' such thatzn approximates z’n for p £ .S' U {p} U U U V , zn approximates e\ for p € T — (S U {p} U U U K), and Qi(zn) £ Z* U rZ* for p outside S U {p} U U U T U V. At p € .S' U {p}, approximate yHiP by yn £ f|(Y H Mp) where the first intersection is taken over p ^ S U {p}, and such that at p £ S' the degree of approximation is either j(M p)+l or j(A / 2 )+ 3 . Put xn = yn+zn- Then xn £ Af sinceforp ^ SL^p}, Af' C Mp and at p £ £U{p}, xn approximates x,liP. Also 1.3.13 implies that ZpxH |-Zpxn = Zp®i,p + • • ■ + Zpx„iP. It also holds that d(xi,..., xn) = d(x j,..., xn_j)d(zn). Then at p , by construction, d(xi,..., xn) approximates d(x i,... xn_i,x„tP) £ Z“ . At p £ t / —(5'U{p}), zn approximates zp thus Q(zn) £ Z*. A tp £ V, Q\(zn) approximates Q\{z'n) € Z* thus Q(zn) £ Z* and the same holds for p £ T. The rest follows from the fact thatd(x j,...,xn) = d(x i,..., xn-i)d(zn). In order to get an estimate for the prime r, it suffices to consider in this case the modulus n p) p€$ pgSu{p},p6V and apply corollary 2.1.12 as in the proof of 2.1.13 for the quadratic extension of Q associated to the lattice U. One gets then r < 2 (^ K ,mCln)^K^c^ P n ^ K'mC'mF^ Y l = Cn{M,Ax) Q.E.D. In the next proposition we summarize our results in the particular case in which dim N = n = 1 , and give a bound for the constant c(M) of theorem 1.1.1. We will make use of the following theorem due to Y. Kitaoka [Ki3] 38 Theorem 2.1.17 Let N be an n-dimensional positive Z-lattice , then then exists a constant c(n,e), for each e > 0 , such that h(N) < c{ny£)det(N)^-l^ 2+e Unfortunately there is no estimate for the constant c(n,e). On the other hand since for the case n — 1 the lattice L(J) is a four-dimensional lattice, we still may get an explicit estimate for h(L(J)) by using reduction theory. Namely Remark 2.1.18 The class number of L(J), when L(J) is four-dimensional, satisfies h W ) ) < (l)6r4(2r(det(M))2)3. In general, for any n-dimensional integral lattice L of determinant D h(L) < (i)»(»-i)/ar (*-a)/j»-i. O The proof of the above remark follows just by counting the number of possible Minkowski-reduced matrices of a given determinant D Once we have all previous estimates we may get a bound for c(M) in 1.1.1. The next lemma will be needed (see 1.7 in [HKK]). Lemma 2.1.19 There are constants bn > 0 such that for any Minkowski-reduced ba sis (t\) of a positive Z-lattice N, the matrix (B(vi,vj) — bnfi(N)In) is positive definite . R em ark 2 .1 . 2 0 The constants bn can be estimated by using reduction theory, lemma 1.8,2 and lemma 1.3.3 in [KilJ. One gets that bn can be taken as 39 We now finish with the proof of 1.1,1 and see how an estimate for c(JVf) can be obtained. By construction, for any N locally represented by M for all p, there is a J such that K{J)p represents Np for p € S. For p £ S, L(J)P represents Np by 2.1.7. Hence the conditions of lemma 2.1.1 are satisfied with K = K{J), L = L(J) , if for suitable basis (u,-) of N the matrix ( ) — c(K(J),qs^L(J)Jn) is positive definite. By the previous lemma, this is true if p(N) > jj-m axc(I<(J)yqa{J)L(J)) = c(Af). bn J All our estimations give then the following results. First for the one-dimensional case and then in general Proposition 2.1.21 Let M be a positive Z-lattice, let N be a one-dimensional Z- lattice, then M represents N provided that Mp represents Np for each prime p and p(N) > c(M) where c[M) satisfies c(M)l € eq2W{^ nin2rdelMi?~X){detMfr here the primesr satisfy the conditions from lemma 2.1.13. Finally we summarize our rjjjesults to get an estimate for c(Af) in the general case n > 1 T heorem 2 . 1 . 2 2 Let M be a positive Z-lattice of rank m > 2 n + 3. Let N be. an n-dimensional Z-lattice. Then M represents N provided that Mp represents Np at each p, and p(N) > c(M) where c(M) satisfies. 40 c(M) < i r „ 2 3-*+ 2 »3 r / («3 ("f^ '( M))3) Where 7 ’ satisfies the inequality in lemma 2.1.16. Remark 2.1.23 For an explicit bound for the class number, one may change the expression c(n, e)2ndet(M)3r by {i)n{n- 1)/2 r ”- 2 (2,1r g-invariants of Rings Let R be a ring. Define the ^-invariant of order n for R, as in chapter I, section 1.1. The purpose of this chapter is to study the (/-invariant for the case in which the ring R is a local ring and then, by applying the methods in chapter II, study the invariant for the ring Z and for the ring of integer of any number field. We begin by computing the (/-invariant of local rings. 3.1 The ^-invariant for local rings Throughout this section a local ring means the ring of integers of a local field K with charK = 0. We denote by O the ring of integers of K and by R the residue field which is assumed to be finite. We denote by re (= Q a uniformizing element. For any integral quadratic form (f> we denote by Lj, its associated O —lattice and we set V n ^(* 1 ,. . .Xn) = 5Z aijxixji aij — aH € O f OF 1 < i j < 71 «,j=l be an n-dimensional quadratic form over O. We will show now that if 2 is a unit or a prime in O then 41 42 Lemma 3.1.1 Assume that 2 is either a unit or a prime in O.Then there exist linear forms Li(x i,..., xn) = J ] kjXj bij € Ofor 1 < i< r l Q.E.D. From 1.3.4 and 1.3.8 it also follows that go(n) is finite when 2 is a unit or a prime in O. We will prove later that this also holds when 2 is neither a unit nor a prime in O, The next result, which is valid over any integral domain R with 2^0, enables us to interpret the g-invaTiant of a ring in terms of representations of forms by forms, or in a more geometric sense, in terms of representations of lattices by lattices {see [BLOP] for the case in which R is field). Proposition 3.1.2 Let ^(Xj, . . . , 33n) = fljjXjXj, ttij = Oij € R be an integral quadratic form over the domain R. Let Li{x t,...,arn), 1 < i < r be r linear forms over R in the variables X j , ..., xn. Then 43 1/ and on/y t/ the form Lgi, —» Ir. Where as usual Ir denotes the lattice associated to the form rx < 1 >. Proof. Let Lj = bj 1X1 H + bjnx n, bji £ R for 1 < i < r. Assume that ... ,3!n) — xj H- • • * H- bjnx n) j=i We compare coefficients to get a««= 1 < i< n j=i aud since 2 ^ 0 I . 1 ^ 2 ^ Tl au = 2^bjibjt I < £ < n. 3 = 1 — — We define a mapping as follows. Let = .Re* ® ••*© Re„ with <^(ej) = aif, ^(e,-,ej) = a,j. Consider in Ir vectors = (6ti,..., 6ri) and let cr(ei) = for 1 < i < n. Then the above equations exactly mean that a is an isometry. Conversely, any representation 44 Q.E.D. We now give some straightforward properties of the ^-invariant. Lemma 3.1.3 Let R be a domain with the property that any element in R which is a sum of squares in K = Quot(R), is a sum of squares in R (e.g. Z; Zp). Then 9K(n) < gii(n) for all n Proof. Let $n = H H a„x£ be an n-dimensional quadratic form over K which is a sum of squares of linear forms of dimension n in K. Then we may assume that oi,..., an G R, and since they are sums of squares in K, a i,... ,a„ must be sums of squares in R. Therefore the form a\Xi + • ■ • + anx% over R is a sum of gR(n) squares of linear forms over R. Hence 9x(n) < gri(n). Q.E.D. Corollary 3.1.4 If O is Z, the ring of algebraic integers of a non-totally real number field in which 2 does not ramify or a local ring, in which 2 is either a unit or a prime, then 9K(n) < 9o(n) for all n > 1. 45 Remark 3.1.5 There are few rings of algebraic integers which satisfy the condition of lemma S. 1.3. For example if K is a totally real number field, then its ring of integers O satisfies the mentioned condition if and only if O = Z or Z If K is not totally real, then the hypothesis of lemma 3.1.4■ always satisfied provided that 2 does not ramify in K. Since for any local and global field K it holds that gK (n) = n + 3 for all n > 3, we obtain the following. Corollary 3.1.6 With the same hypothesis as in 3.1.4, <7 c>(n) > n + 3 for all n > 3. In the next proposition we give a general bound for the ^-invariant of a local ring. Proposition 3.1.7 For any local ring it holds ffo(n) < n + 3 for all n > 1. The proof of proposition 3.1.7 for the non-dyadic case follows from theorems [63.21], [82.18] and [91.2] of [O’Ml] and the following lemma, (see lemma 1.1 in [HKK]). 46 Lemma 3.1.8 Let (/, V be two regular quadratic spaces over the local field K with dim(/+3 < dim V. Let L be an lb-maximal lattice on V, where 05 C K is a fractional ideal. Then L represents all lattices M on U for which the norm N(M) C Af(L). Proof. From theorem 63.21 in [O’Ml], it follows that V represents U over K, so that we may assume that M C V. Since X{M) C Jf{L) C 05, it follows from 82.18,[O’Ml] that M is contained in an 05—maximal lattice M 1 on V. But since any two 05—maximal lattices on V are isomorphic (see 91.2 in [O’Ml]), we get L ~ A/', and hence L represents M. Q.E.D. Now we give the proof for proposition 3.1.7 in the non-dyadic case. Proof. Let (j>n be an n-dimensional quadratic form over the non-dyadic local ring O with K = Quot(Q). Since O is non-dyadic, the 0 —lattice f(n+ 3 ) is O—maximal. But dim (/f *L^n)-|-3 = n + 3 and J\f(L^n) QAf{I(n+3) = O (since <£„ is an integral quadratic form). So that applying the previous lemma we conclude h $ n ^ f(n + 3) i.e. go[n) < n + 3 for all n > 1. Q.E.D. Let us now consider dyadic local rings. The proof of 3.1.7 in the case in which 2 is a prime is a straightforward conclusion of Theorem 1.3.8 , since we have that 47 in conditions (I), (II) in the theorem, the codimension is at least 3. Therefore both are satisfied. The case when 2 is not a prime is solved by using Theorem 1.3.10 as follows. Proof. Let Z^n and /(n+ 3) corresponds to (iii) from theorem 1.3.10 and we only need to see that s ( ^ „ ) Q s (I{n+3)) — O and g(L) C flr(/(n+3)) in order to conclude that L$n -+ /{„+3 )- The first condition is clearly satisfied and the second follows from the assumption that L^n is represented by /r for some positive integer r. If r < n + 2 we are done, if not then the codimension of This concludes the proof of proposition 3.1.7 for any arithmetic local ring. Q.E.D. Remark 3.1.9 From condition (iii) in the previous theorem we also see that ,as we mentioned in last section, there are integral forms that are not sums of squares of integral linear forms. Namely, let it be a uniformizer element in O, then ft is never represented by IT for any value of r since, by condition (iii) from the above theorem, it is never represented by Ir for r > 4 Remark 3.1.10 From corollary 3.1.5 we have proved go(n) = n + 3 for all n > 3 48 where O is any arithmetic local ring in which 2 is a unit or a prime. 3.2 T he g-invariant of Global Rings In this section we study g-invariants for global rings i.e. either the ring of rational integers Z or more general, the ring of algebraic integers of a number field F. For these cases all computations become much harder and we study the behavior of those g-invariants rather than trying to obtain explicit values for them. 3.2.1 The g-invariant of Z for n < 5 We give the first application of the local results obtained in section 3.1. This ap plication is related to the g-invariant of the ring Z,gz{n). We first mention some of the work already done in this direction particularly by Mordell [M 1,2,3,4] and by Chao Ko ([Ko]). In 1930, Mordell proved that ([M4]) proved that there exists a 6 -dimensional integral quadratic form over Z which is positive definite but it is not the sum of any number of squares of linear forms over Z. Namely: 6 6 d{x ) = + (52 x r)2 - ‘2x^X2 - 2x2X6. r = l r = l This form has determinant det g — 3. We give a proof of the fact that g{x) is not represented by Ir for any r by using only lattice theory. 49 Lemma 3.2.1 The form g(x) given above is never represented by IT for any positive- integer r Proof. Let us denote by Lg the Z-lattice associated to the form g{x). Let be a Z- basis for Lg. Since each Q(vi) = 2, if there were a representation cr of Lg by IT for some r, then B(v2, vf) = 1 imply that one needs at most one extra to express A_i = O (see 1.3.5), and the orthogonal complement of Q 2 l[-i] in Q 2 A is isomorphic to (3). According to (I) in 1.3.8, it must happen that O —* (3) (see pg. 7). This is a contradiction. Q.E.D. We now see how our local results from section 3.1 allow us to recover at once Ko’s results for n < 5. Namely. Proposition 3.2.2 gz(n) = n + 3 for n < 5. Proof. Let —► Zp/(n+3) for any prime p 50 and * ®^(n+3) since Therefore we obtain that there exists K £ Z^n —> K over Z. But it is known (See [O’Ml], Chapter X) that for n < 8 the genus of /(„+ 3 ) , is a genus of one class. Therefore L Sz(n) < w + 3 for n < 5. Once again from lemma 3.1.3 we have: n + 3 = #q(n) > £fz(n) f°r n and Proposition 3.2.2 follows at once. Q.E.D. Remark 3.2.3 Notice that last step could also have been proved by realizing that the integral form x^ + x] + -----1- x f^ + 7x^ is a sum of n -f 3 and not fewer squares of integral linear forms. 51 3.2.2 Study ofgz(n) for n > 6 In this section we study the behavior of g !• Is gz(n) < 0 0 f°r n 6? 2 . Can one give an explicit bound for gz(n ) in case we get an affirmative answer for question 1 ? We first give a positive answer for question 1. We strongly rely on Theorem 1.1.1 from the chapter I. Theorem 3.2.4 For Z, the ring of rational integers gz(n) < oo Proof. The proof is by induction on n. Let c = c(I‘2n+3 ) denote the constant from Theorem 1.1.1 corresponding to the integral lattice / 2„+3 . Let N be any n-dimensional positive definite integral lattice that is represented by l r for some r. Then we consider two different cases for the minimum of N, fh(N). Namely 1. n(N) > c. Theu by 1.1.1, the latticeN is represented by / n+ 3 . 2. n(N) < c. 52 In this case, let fc=l where Lk(x t " ,xn) = bikx !+••• + bnkxn. By comparing coefficients we get on = 5Zftij j=i and by assumption, an < c. Therefore at most c of the integral coefficients b\j are nonzero hence c r ^ = 2[i*(*i»” ‘i*n)]a+ ]£ [M*ar--I*n)]a. fc=l fc=c+l Since Lk(x2 , • • •, x„) are (n — 1)- dimensional linear forms we get by induction fifz(n) < c + gz(n - 1 ). Therefore flrz(ll) < 00. Q.E.D. Corollary 3.2.5 From the proof of 3.2.4 & follows that 9z{n) < ( $ 3 c(^3i+e)) + 8. t= 6 We now give an answer for question 2 by exhibiting an explicit bound for the g- invariant of Z. We first need to prove one lemma. 53 Lemma 3.2.6 Let c = c(l2n+s) be the constant given in 1.1.1 associated to the lattice hn+e■ Let bn be as in 2.1.20. Denote by h(L) the class number of a lattice L. Then c(/2n+6) < ( 1 / 6 ,l)n 2 2 4^/l(/2n+B)-1). Proof. Let K — L = / „ + 3 be two copies of the integral lattice /n+ 3 . Let N be any integral n-dimensional positive definite lattice.Write N = 52"= j Zut\ We now follow the notations from lemma 2.1.1. Take the set consisting of a single prime S = {2 } and choose s = {h(L) — 1) with h(L) being the spinor class number of L. We want to find conditions for N and for c(K,2*L) = d so that B(vi, Vj) - c 7 n > 0. We proceed as in chapter 2 . According to 2.1.1 we choose (v{*, • • •, v£) (h = 1 , • • • ,<) vjl G K so that for any collection (a:,, 2 ) € K 2, there is some h so that v f = Xipimodi2* K 2). (3-1) From the proof of 2.1.1 it follows that we must choose c1 so large that c'/n — B(v(l, vjf) is positive definite for h = 1 , • • •, f. Let K — Ze-i -p • • ■ + Zen^ . Then >1+3 j=i and = (« y )T[A.+a](«ij) = (<*v)T(<*v) < Tr[(ay)T(oy)]/». 54 Equation (3.1) implies that w.l.o.g. we may assume 0 < ay < 2 2'. Therefore Tr[(a1J)T(aij)]/tt < n 224aIn. Thus B{v£,v}) < n 22 4aIn and we may take c' = c(K,2 aL) = n 2 2 4(/l(L)- 1J. According to 2.1.19, there are constants bn > 0 so that is positive definite. Then if we choose c = c(/2n+6) = (l/^ )n a 2 4 W,»+3>-, > we have that for any N such thatfi(N) > c , N is represented by hn+s- Q.E.D. We now give an answer for question 2 in the following Corollary 3.2.7 9 z(n) < (l/6n)24VdM3)-i)(4!L+lEE!L±2l _ 55) + 8 b Proof. Let N be any n-dimensional integral quadratic lattice that is represented by /T for some r. As we did iu the proof of the finiteness of the (jr-invariant of Z, we first assume H(N) > c (/ 2n+3) then N is represented by / 2n+6 • On the other hand if fi{N) < c , then 3.2.5 implies 9z ^ c(hn+s) + • ■ ■ + cf/is) + 8 = E c(^«+e)] + ^ i=6 From the previous lemma we get I > ( / 2n+6) < E ( i 2 /^ ) 2 4{M/i+3)_,) i= 6 i=6 < ( l / 6 n )24(/l(/"+3)_1)(n (n + l)(2 n + l ) / 6 ) - 55) And we have then the desired result. Q.E.D. 3.3 The totally real case In this section we study the g-invariant for the ring of integers of a totally real number field F . We will not give full proofs of the results that easily generalize from the ones already given in the case of rational integers. We first give some notations. Notations: 56 F = totally real number field of degree g over Q O = Ring of algebraic integers of F N f/ q (oi) = Norm of an element a 6 F Tvp/q(<*) = Trace of an element a € F Since we are interested in generalizing the results obtained in the previous sec tion, we must have a generalization of Minkowski’s reduction theory for the case of positive definite quadratic forms over number fields. This theory, due to P. Humbert ([Hul],[Hu2]), will be needed throughout the next computations. We first recall the following. Definition 3.3.1 Let We now collect some results about Humbert’s reduction theory of integral quadratic forms over F. Theorem 3.3.2 Let 7- < Ci s'jj for any 1 < j, any er,r, embedding of F into R , S. |s?fc| < c2 a[t- for 1 < k < n, r as in (1), 8. * 1, * s t22 ■ ■ • sjm < c3(det 4>)T t as in (1), where all constant C\, c2 , c3 are positive and depend only Fon and n. We will also need the following result due to P. Humbert (see [Hu2,pg.296-299]). Proposition 3.3.3Any free O -lattice , N , has a basis u,- so that R(u,-, vj) = TlDT, with T unipotent triangular with bounded entries, D — (d\, - • •,dn) diagonal with p(N)/di and d»/rfi+i bounded (i.e. bounded with all conjugates bounded ). Here p(N) = A/in{JVjr/Q(Q(a:)jO / x 6 M} G M } denotes the minimum of the lattice N .The proposition above gives the generalization of lemma 1.7 from [HKK] that we state in the following Lemma 3.3.4 There are positive constants bn such that any Humbert-reduced basis (vi) ”= 1 of a positive O-lattice N, the matrix (B (vi, vj) — bnp(N ) /n) is positive definite. Since the proof of the finiteness for the ^-invariant in this cases uses the general version of Theorem 1.1.1 (see Theorem 3 in [HKK]) we give the statement of that Theorem here. Theorem 3.3.5 Let M be a positive O-lattice of rank m > 2n+3. There is a constant c = c(M) such that M represents any positive O- lattice N of rank n, provided that p(N) > c and Mp represents Nv at every prime p of F. 58 We are now ready to establish the following Proposition 3.3.6 170 (« ) < o° fo r all n . Proof. The proof is essentially the same as the one given for the finiteness for the fir- invariant of Z. It follows by induction on n, noticing that firo(l) < 5. In this case one has firo(n) < c(/2n+6) + flr0 (n - 1 ). Here c(M ) denotes the constant given in theorem 3.3.5 for an O - lattice M. Q.E.D. From corollary 3.2.5 we get as well Corollary 3.3.7 n 770 (n) < £ c(^2<'+e) + 5. i= 2 In order to give an explicit bound for the g invariant in the case we are studying in this section, it is enough then to obtain an estimate for the constant c(/ 2„+e). All the lemmas that we state now will give us such estimate. Lemma 3.3.8 Let L be a positive definite O-lattice of rank I > 3. Let q be a prime ideal in O such that Lq is isotropic. Then there exists an integer s € Z such that L represents any n-dimensional positive definite O - lattice N, provided that q 3LP represents Np 59 for every prime p of F Proof.See [HKK, pg. 139] Q.E.D. Remark 3.3.9 Since we are interested in obtaining an explicit bound, we get from remark 1.4-7 that we may take s = h(L) — 1. As before h{L) is the spinor class number of L We will later need the following Lemma 3.3.10 Let be all the embeddings of F into the real numbers. Let 8 = min (max I <7 i(a,) |) i Where the minimum is taken over all possible {a.}, integral basis for O. Suppose 0 = | &i(ol>3) | for some integral basis {u>i}f=l, some 1 < s < k. Then k I t = {a = ^ 2 a* I € Z, 0 < a,- < pr — 1} i=i is a set of representatives for O f n ? = i P i® f or Pi such that p, f*l Z = (p). Moreover, for all a € 72-, | o^a) |< (pr — 1)0. Proof. Since pr ei= | V l = i=1np? then o / f o — . o / n ^ o i=1 60 applies the right hand side onto the left hand side. But the set H is a system of representatives for 0 /p r0 , and the first assertion of the lemma follows. The second assertion is clear from the definition of /?. Q.E.D. Another result that generalizes for the case of free lattices over O is lemma 2.1.1. Instead of giving the full statement of the totally real number field version of 2.1.1, we only give a corollary of it which is what we need for our purposes. At this point we have to assume that 2 is not ramified in F. We will treat the case in which 2 ramifies in F separately. Corollary 3.3.11 With the same notations as the previous lemma .Let L = /„+3 , A/ = / tl+ 3 n > 2. Then there exists c = c(M, q*L) such that K LL represents any free lattice N = Ovi such that 1. Np — >■ K v±qaLv for any prime ideal p £ O. 2. The matrix (B{v{, vj) — cln) is positive definite. Proof. The proof follows just like the proof of 2.1.1. One only has to notice that if S is the set of prime ideals in O that contains all primes above 2 , then the integer r,, that we chose in 2.1.2, for each p, has to be so large that prp C A^(q*(/n+ 3 )p) . According to remark 3.3.9, it is enough to choose = r = 2(h(L) — 1 ) Q.E.D. Like in the previous section in order to obtain the desired bound, it is enough to estimate the constant c. We give such estimate in the next 61 Corollary 3.3.12 With all notations as before, c < n 22 4 /*('n+ 3 ) - 2 / ? 2 Proof. From the proof of 2.1.1 it is enough to choose c large enough so that it satisfies c/n — (B(vi,Vj)) is positive definite. That means it is positive definite in each embedding into the reals. By approximation, there is a finite set of n-tuples (v{1,... ,v£,), 1 < k < t of vectors vj1 € K such that for any collection (xiiP) , P € S, 1 < * < n,XiiP € Kp ,there is some h with v-1 = X{iP (modpTKp) for every p € S. Let v-1 = i aijej where ej, 1 < j < n + 3 is a basis for the lattice M. Then for each embedding a of F into the real numbers, a(B(v<,Vj)) = Lemma 3.3.10 together with the Chinese remainder theorem tells us that we may assume that 0 < | ty)) < Q.E.D. 62 Altogether we have proved Theorem 3.3.13 go(n) < (1 /bnWWPW'+tW'E**) + 5 i= 2 Where the b{’s are given in 3.3.4 • Proof. According to the proof of 3.3.5, (see also pg. 37 in chapter II), c(/2n+6 ) < (l/&n)c(/n+3,qVn+3)’ Q.E.D. We now assume that the prime 2 ramifies in F. The problem now is that one cannot choose the lattice L in 3.3.11 to be / „ + 3 since in this case the only representation theory due to C. Riehm (see 1.3.10) deals with unimodular lattices or particular cases in the non- unimodular situation and with that choice for L the proof of 3.3.11 will not follow. Instead, we have to choose L so that it is either maximal or containing a maximal lattice at the dyadic primes. We show how to define L in the following lemma Lemma 3.3.14 Let L — /n+ 3 . Let k = [F : Q], Then for all dyadic prime p £ O, Lv contains a 2 ‘2fc(n + 3 )2 -maximal lattice Np with volume v(Np) D 2 2fc^n+ 3 ^Op. . Proof. According to [0,Ml,82.18], there exists an Op-maximal lattice N \tP such that L C ATliP. Then by [O’M, 82.11], v(Lp) = V M Nhp) 63 where V is the product of the invariant factors 9ti,... , 8 ^ + 3 of in Ni)P (see 1.2.2) . Since v(L) = O p it follows that O p = V2v(Nx*) thus = (ai, - • a w s ) 3 now from [O’M l,81:12] W .p : L>\ = [Op : *1] ‘ •' [Op = Wn+s] Therefore L,\ < [O, = [O, : (#(JV, but Op C v(NjiP) C (l/2 "+3 Op) therefore 2n+3 O p C (v(Wi,p) ) - 1 and [Op : (v(Nhp))~ ']n+3 < [Op : 2n+3 O p] n + 3 = 2 fc(n+ 3 )2 thus [iV,,p: Lp] < 2 fcb‘+ 3) 5 and Np = 2 febl+3 )3 yy1^ C Lp. The lattice Nf is 2 a*tn+ 3 )2 -maximal. The condition about the volume of Np follows by construction. Q.E.D. We will also make use of the following 64 Lemma 3.3.15 Let F be a totally real number field, met m € N.Then every lattice of rank m contains a free sublattice M' such that [M : M'\ is bounded by a constant depending only on the field F. Proof. From [0 ’M1,81:5] we can express M = Oe^ -\------1- Oem_i + T>em where O is the ring of integers of F and V is some fractional ideal. By replacing em by remt for r € Z we may assume thatD € O. Take now whose norm I tfjr/qfo) |< ( - r 4 ( l \y /2Norm(V) < ( - ) Ta(l DF \y>Norm(V). IT Tl T Where r2 are the complex embeddings of F into C which by assumption equals 0. Put M ' = Oei + • • • + OeTO_i + r}Oemi then Q.E.D. We are now ready to give a bound for the ^-invariant of the ring of integers of a totally real number field in which 2 ramifies. We give that bound in the following Theorem 3.3.16 Let F be a totally real number field with discriminant divisible by 2, let O be its ring of integers. Then 9o{n) < + 5 i= 2 Where t = [DiscfF)^/2)] with [] being the least-integer function, and CF is a constant depending only on the field F which is explicitly estimated in 3.3.17. » 65 Proof. Let ^n* Let K = / n + 3 and define an O- lattice L\ , according to [O’M l; 81:14], such that locally it satisfies (r \ _ / (At+3 )P if P non-dyadic Op \ Np if p dyadic where N p is as given in 3.3.14. By lemma 3.3.14, v{L\) D (2fc2 ("+3 )3 0). According to 3.3.15, take L to be a free lattice in L\ with index bounded by t < [(£V)1^2]- We will use, instead of the lattice Li, the free lattice L to be the one correspoding to / „ + 3 in the statement of 3.3.11. With the appropriate choices for rp we get then a similar result as that of 3.3.11 for the ramified case. Since in our previous computations, the estimation of the size of the constant in 2.1.1 and 3.3.11 was the key step towards the desired result, we compute a bound for it in the next result. Let S be the set of primes containing all dyadic primes in O and all primes dividing (<), let q be one dyadic prime in S. As before, we want to choose for each p 6 S the integer rp large enough so that prp C C A/”(qaL). Let ep be the ramification index of the prime p at p ,with p U Z = p. Then 1 . r p = 2(h(L) — 1) + 2ep(ord 2 (i) + 2 A:(n + 3 ) 3 + 1 ) for p = q. 2. rp — 2ep(ordp(t) + 2k(n + 3) 3 + 1) for p ^ q . By choosing rp in the above way we assure that all steps in 3.1.11 follow. What is left now is to estimate a bound for the constant c = c(K, q*L). Once more let 66 (vf) € K such that for any collection €/fp,p€5,l v1- = x,-iP mod(prKp) for 1 < j < n , 1 < h < r and for each p 6 S. Set where {ey} , 1 < j < (n + 3) is a basis for / n+ 3 . Then for each embedding a of K into R we have From 3.3.10 and the Chinese Remainder Theorem we may assume 0 < | <7(oy) |< 22(/'W-l)(JJp)2fc(or v/ 1 where we have used the bound ep < k for all p . Thus At this point, corollary 3.3.7 tells us that 5o(») < (l/6„)24( We will finish our result by giving an estimate for h(L) in the following lemma Lemma 3.3.17 h(L) < | NF/clDet(L) |"<»+i)/a Proof. The proof follows by using Humbert reduction. The value given in the lemma correspond to the number of reduced matrices with fixed value for the norm of its discriminant. 67 Q.E.D. We now finish the proof of 3.3.16. det(L) = I ^F/Q(Det(L)) |< t2fc(H+3)22^ We define then CF = (24c 3fc)"("-1)/2 (^2 fc(fl+3 )22A3 This concludes the proof of 3.3.16 Q.E.D. 3.4 The imaginary case We conclude this work considering the case in which the field F is non totally- real number field. This case is much easier than the totally- real one and with the aid of the local results from section 3.1, it will follow 9 QK(n) = n + 3 for n > 1 The proof is based on two results. The first one which we state in the next lemma (see {H]). 68 Lemma 3.4.1 Let V, W be two regular quadratic spaces over F, where F is a non totally real number field. Suppose dim V + 3 < dim W and lei L (respectively M) be an O —lattice on V (respectively on W ). Assume that Lp is represented by Mv for all discrete spots p on F K, is represented by Wp for all infinite spots q on F Then there exists M € Spn(M) such that L is represented by M. The other result needed is related to indefinite set of spots. Namely Definition 3.4.2 Let S be a Dedekind set of spots consisting of almost all spots on the global field F. Let V be a regular quadratic space over F. We say that S is indefinite for V if there is at least one spot tp outside S at which V The next needed result was already used in chapter I (see remark 1.4.3) and it is given in [O’Ml]; 104:5. For the sake of clarity we state that result next. Theorem 3.4.3 Let V be a regular quadratic space over a global field F with dim V > 3, S an indefinite set of spots for V, and L be a lattice on V with respect to S. Then clsL = spnL. Since the set S of all discrete spots in F is an indefinite set of spots for the quadratic space F /(u+3) = V}(n+3), 3.3.20 implies clsI(n+3) = spnl(n+3). Let K be any n-dimensional positive Z-lattice. From the local results obtained 69 before, we know that the hypotheses of Lemma 3.3.18 are satisfied, so that there exists L € spn(/(n+3) such that K — y L and the proof follows. B ibliography [BLOP] : R. Baeza, D. 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