Traces in Arithmetic Fuchsian Groups, Original Research Article

Let image be a quaternion algebra over an algebraic number fieldFof degreenover image such that imagecircle times operatorimageimageall equal toM2(image)×imagen−1, where image are the Hamiltonian quaternions. If image is an order in image, then image1={xset membership, variantimage: N(x)=1} can be embedded inSL2(image) as a Fuchsian group. Let imageimage={t: t=Tr(γ), γset membership, variantimage1} be the set of traces of the elements in image1and imageimage(r)={tset membership, variantimageimage: 0less-than-or-equals, slanttless-than-or-equals, slant2r}. The first purpose of this paper is to show thatimagewhereDFis the discriminant ofF,d(image) is the (reduced) discriminant of image, andmimage(image) only depends on the completion imagepfr;of image for nonzero prime ideals image dividingd(image). We also calculate the invariantsmimage(image) for important families of orders.