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Order Number 9807786 Effectiveness in Representations of Positive Definite Quadratic Forms Icaza Perez, Marfa In6s, Ph.D

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Order Number 9807786 Effectiveness in Representations of Positive Definite Quadratic Forms Icaza Perez, Marfa In6s, Ph.D 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.
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