Ternary quadratic forms over number fields with small class number

A canonical form for positive definite matrices

“…Several efforts have sought to enumerate lattice genera of either bounded discriminant or satisfying some arithmetic conditions such as small (spinor) class number. For example, the Brandt-Intrau tables [9] of reduced ternary forms with discriminant up to 1000, Nipp's tables [33] of positive definite primitive quaternary quadratic forms with discriminant up to 1732, and more recently the complete table of lattices with class number one due to Kirschmer and Lorch [22], to name a few.…”

Definite orders with locally free cancellation

“…Up to ‐algebra isomorphism there are 168 such orders, of which 149 have locally free cancellation. Our list is consistent with the (corrected) old classification. Kirschmer–Lorch computed all definite quaternion orders with type number at most 2, and made them available in electronic form . This includes all orders with , and such an order trivially has locally free cancellation.…”

“…Every quaternion order with trivially has locally free cancellation, and all orders with were enumerated by Kirschmer–Lorch .…”

Unit groups of maximal orders in totally definite quaternion algebras over real quadratic fields

“…[40]). This has been used to study various arithmetic problems concerning totally definite quaternion algebras, including the analogous Gauss problem and the cancellation property by several people [8,17,21,22,35,39]. Brzezinski [8] obtains a complete list of all orders (including non-Gorenstein orders) in definite quaternion Q-algebras with class number one.…”

“…Kirschmer and Voight [22] determine all Eichler O F -orders with class number ≤ 2. Kirschmer and Lorch [21] determine all Eichler O F -orders with type number ≤ 2.…”