Binary Hermitian Forms and Optimal Embeddings

honors thesisCollege of ScienceMathematicsGordan SavinLet L=K be a quadratic extension of global fields, and OL the ring of integers of L. We prove two correspondences between (i) binary L-hermitian forms which represent 1 and optimal embeddings of L into a quaternion algebra, (ii) integral binary OL-hermitian forms which represent 1 and embeddings of OL into a quaternion order. We then provide necessary and sufficient conditions for a binary hermitian form to represent 1. In the integral case, we assume that L is tamely ramified, a condition that we hope to remove in future work. Finally, in the correspondence (ii), we prove a relation between the discriminants of the order, the hermitian form, and OL, which is a first step in understanding how the structure of a quaternion algebra/order depends on the norm