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Ann Arbor, MI 48106 REPRESENTATIONS OF POSITIVE DEFINITE HERMITIAN FORMS DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy it the Graduate School of The Ohio State University By Juan Pablo Prieto-Cox, M.S. * * * * * The Ohio State University 1990 Dissertation Committee: Approved by Professor John S. Hsia Professor Paul Ponomarev Advise£r Professor Daniel B. Shapiro epartment of mathematics Dedicad© a la Memoria de mi padre Octavio Prietc Cmwtey-Bcevey (1927-197G) ACKNOWLEDGMENTS I would like to express my sincere gratitude to Professor John S. Hsia, he not only taught me some good mathematics, but also set forth an example of excellence as a teacher and as a mathematician. I -want to thank him also for the help he provided me in the preparation of this work. In retrospect, I know that I owe him a great debt for bringing me ashore when I was drifting. My trip to Ohio State (as well as that of many others) would not have been possible without the invaluable help of Professor Daniel Shapiro. I would also like to thank Professor Paul Ponomarev for helping me understand some geometry. I want to thank both of them for their capful reading of this thesis. I want to thank Professor Alice Silverberg who introduced me to the fascinating theory of Abelian Varieties. I would also like to thanlc Manuel OTlyan, a long time road companion, who shared his time with me, gave me good ideas and was willing to listen to my elusive arguments and proofs. I would like to thank my mother for her continuous understanding and encouragement Finally and foremost, I want to thank my wife (and companion in a long journey since my undergraduate days) Kenna, and my two amazing daughters, Maria Jesus and Carnila, for accepting the countless hours I could not share with them, for making my life richer and for giving me hope in those, not so rare, moments of anxiety and desperation. Hove you. iii VITA April 2, 1959 Born - Santiago, Chile 1983 .LicenciaturaenMatemdrlcas, TJniversidad de Chile, Santiago, Chile 1986 Master of Science, The Ohio Sate University, Columbus, Ohio 1984-Present Teaching Assistant, Dept. of Mathematics, The Ohio State University, Columbus, Ohio Summer 1986, 1988 and 1990 Research Associate, Supported by NSF grants DMS 850 3326, DMS 880 3805 of Professor J.S. Hsia PUBLICATIONS (with R. Baeza, D. Leep and M. O'Ryan), Sums of Squares of Linear Forms, Math. Zeit 193, 297-306 (1986) FIELDS OF STUDY Major Field: Mathematics Studies in Quadratic Forms; with Professors J.S. Hsia and D. Shapiro Studies in Number Theory with Professors R. Gold, J.S. Hsia, M. Madan, P. Ponomarev, K. Rubin and W. Sinnott,, Studies in Algebraic Geometry with Professors K. Rubin and A. Silverberg iv TABLE OF CONTENTS ACKNOWLEDGMENTS iii VITA iv INTRODUCTION 1 CHAPTER PAGE I HERMrriAN SPACES 9 § 1 Basic results and definitions 11 §2 The split case: E = FxF 15 § 3 The induced quadratic form 1 19 §4 Local and Global Hermitian spaces 23 H HERMITIAN LATTICES 28 §5 Generalities 28 §6 Local theory of Hermitian lattices 32 §7 Hermitian lattices over number fields 38 IE REPRESENTATION OF HERMITIAN LATTICES 43 §8 The induced quadratic form II 43 §9 The */-class and the S£/-class 49 § 10 Some local results 54 § 11 Global representations 58 v APPENDIX 73 §A1 The induced quadratic form HI: Going down to (Q 73 § A2 The induced quadratic form IV: The genus and the spinor genus ... .78 §A3 Further research 84 REFERENCES 87 vi INTRODUCTION Historical background In studying the representation of a number as a sum of four squares1 Hermite introduces, in 1853, the notion of a (binary) Hermitian form over IQ(^T). He notes that binary Hermitian forms have a remarkable number of common properties with binary quadratic forms and in a short number of pages he proves some important results (fractional equivalence, finiteness of the class number, reduction, etc.). Here is what he had to say about these new forms: Les considerations suivantes, que nous prisentons comme une premiere esquisse d'une thiorie vaste etfeconde sur laquelle nous reviendrons d I'avenir, offriront plusieurs exetnples de cette itroite analogie avec les formes binaires; mais on y verra en mime temps le germe de notions arithmitiques toutes nouvelles, qui miritent peut- etre de fixer 1'attention des Giometres. It is because of its own genesis and the similarities mentioned above that the theory of Hermitian forms has inherited many of the techniques of the quadratic theory. Surprisingly, the two theories have not developed at the same pace, most of the classical problems of representation have not been studied in the Hermitian theory, so in a sense this theory is underdeveloped. We have to admit that some of the classical problems, as seejHel] 1 2 sums of squares for example, are not as appealing in the Hermitian setting (sums of norms). There is a point worth mentioning. Hermitian forms arose, in part, as a tool to understand a problem in quadratic forms. This has been partially, if not totally, reversed in history. In most cases the theory of Hermitian forms has either been reduced to or followed the theory of quadratic forms. We think that this rich theory can have a useful role in understanding quadratic forms, especially as a testing ground for some open problems, although regrettably, our work is not an example in this direction. The problems of classification and representation are the two most classical and fundamental problems in the theory of quadratic as well as Hermitian forms. The desire to understand these problems has motivated the larger part of the work on Hennitian forms. The algebraic theory evolved faster than its integral counterpart, in 1936 Landherr proved his classical local-global principle, not only for Hermitian forms over fields but also over quaternion algebras. Soon after that, in 1939, Jacobson introduced the so called trace form of a Hermitian form and proved that we only need to understand quadratic forms, since the trace forms classify Hermitian forms (see theorem 3.1). There is an explicit characterization of all quadratic forms which are trace forms in [Lew] (see also proposition 3.3). The integral theory is not as well understood, certainly the questions of representation and classification are wide open in the global case, although the local case has been completely solved by Jacobowitz and Johnson (unlike its quadratic counterpart). Except for the work of Raghavan, that we comment upon below, the important results for the theory are somewhat new. In 1962 Jacobowitz gave an answer for classification over local fields using techniques similar to those employed by 3 CMeara for quadratic forms. In 1964, Shimura proved the Strong Approximation for SU(V) when V is indefinite, and as a consequence proved that SU-genus = SU~class. This has an advantage over the corresponding Approximation Theorem for 0+(V) that gives instead spinor genus = class. He also computed the class number of U-genus in the indefinite case. A complete solution to the local representation problem was given by Johnson in 1966. He made extensive use of the invariants provided by a Jordan splitting, very much like the quadratic case. Gerstein studied the decomposition of Hermitian forms in a paper in 1970 and proved (using the result of Shimura previously mentioned) that every global indefinite form is decomposable into components of rank at most 4 (see [Gel]). After the result of Shimura, much of the global work has concentrated on positive definite forms. The computation of the class number was initiated by Iyanaga in [Iy2], he uses the method of neighbor-lattices of Kneser to compute the class number of the standard unimodular lattice /„ over S3(i)t for small values of n. Later, in 1978 Gerstein studied lower bounds for the class number of positive definite forms (see [Ge2]), with special emphasis on unimodular forms. There are other works of Braun (1941) and Otremba (1971) that we have not read (for a comment on them see [Ge2]).