| Let F be a number field,B be a quaternion F-algebra and K/F be a quadratic extension.Suppose there exists an F-embedding K→B.Let (?) and S be (?)_F-orders in B and K respectively.We define and study oriented optimal embeddings from S to (?).As a refinement of the classical trace formula,we give the oriented trace formula.Each optimal embedding from S to (?) induces a map between their ideal class of fractional ideals.We will characterize the image of the map,and provide a sufficient condition for this map to be injective. |
