S-Space College of Natural Sciences (자연과학대학) Dept. of Mathematical Sciences (수리과학부) Theses (Ph.D. / Sc.D._수리과학부)
Arithmetic properties of the representations of ternary quadratic forms
- 자연과학대학 수리과학부
- Issue Date
- 서울대학교 대학원
- Representation of ternary quadratic forms; Watson transformations; Graph of ternary quadratic forms; Genus-correspondences; Complete system of spinor exceptional integers
- 학위논문 (박사)-- 서울대학교 대학원 : 수리과학부, 2016. 8. 오병권.
- In 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.