Arithmetic properties of the representations of ternary quadratic forms

Abstract

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.