Quaternion rings, ternary quadratic forms, and Fourier coefficients of modular forms on PGSp(6)

The recent work of Gan, Gross, and Savin to develop a theory of Fourier coefficients of modular forms for the split algebraic group G 2 makes use of a correspondence between an orbit problem and certain algebraic structures; specifically, a bijection is established between orbits of integral binary cubic forms for a certain twisted action of GL2( Z ), and isomorphism classes of cubic rings over Z . In this thesis, we use this work as a model and establish how Fourier coefficients of holomorphic modular forms for PGSp6 are parameterized by orders in definite quaternion algebras, making use of a correspondence between orbits of integral ternary quadratic forms for a certain twisted action of GL3( Z ) and isomorphism classes of quaternion rings over Z . Under this correspondence, the primitive quadratic forms correspond to Gorenstein quaternion rings.;As an application, we use the language of quaternion orders to describe Fourier coefficients of degree-3 Siegel Eisenstein series; we then combine this description with an exceptional theta correspondence to obtain a formula for the number of embeddings of an arbitrary definite quaternion order into Coxeter's ring of integral octonions. This result generalizes a previously known formula, proved using different techniques, that counts embeddings of a maximal definite quaternion order into Coxeter's octonions.