Arithmetic properties of the representations of ternary quadratic forms

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dc.description학위논문 (박사)-- 서울대학교 대학원 : 수리과학부, 2016. 8. 오병권.-
dc.description.abstractIn this thesis, we discuss some arithmetic relations on the representations of (positive definite integral) ternary quadratic forms. Let r(n,f) be the number of representations of an integer n by a ternary quadratic form f and let p be a prime such that f is isotropic over Z_p. We show that under some restrictions, r(n,f) can be expressed as a summation of r(pn,g)'s and r(p^3n,g)'s with some extra term that can be explicitly computable, where each quadratic form g is contained in the same genus determined by f and p.

In the second part of the thesis, we discuss genus-correspondences between ternary quadratic forms respecting spinor genus. We modify the conjecture given by Jagy and prove this modified version. We also construct genus-correspondences satisfying some additional properties. In particular, we construct infinite family of genera of ternary quadratic forms that possess (absolutely) complete systems of spinor exceptional integers.
dc.description.tableofcontentsChapter 1 Introduction 1

Chapter 2 Preliminaries 6
2.1 Definitions 6
2.2 Spinor norms of local integral rotations 11
2.3 Spinor exceptional integers 16

Chapter 3 Watson transformations 21
3.1 H-type lattices 21
3.2 Generalization of Watson transformations 24

Chapter 4 Finite (Multi-) graphs of ternary forms 33
4.1 Definition of the graph G_{L,p}(m) 33
4.2 Connected components 36
4.3 Simplicity and Regularity 47

Chapter 5 Arithmetic relations of representations 54
5.1 The case when m = 0 55
5.2 The case when m = 1 60

Chapter 6 Genus-correspondences & Representations 68
6.1 Genus-correspondences 68
6.2 Reduced genera 73
6.3 Spinor character theory 76

Bibliography 82

국문 초록 84
dc.format.extent1394368 bytes-
dc.publisher서울대학교 대학원-
dc.subjectRepresentation of ternary quadratic forms-
dc.subjectWatson transformations-
dc.subjectGraph of ternary quadratic forms-
dc.subjectComplete system of spinor exceptional integers-
dc.titleArithmetic properties of the representations of ternary quadratic forms-
dc.contributor.affiliation자연과학대학 수리과학부-
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College of Natural Sciences (자연과학대학)Dept. of Mathematical Sciences (수리과학부)Theses (Ph.D. / Sc.D._수리과학부)
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