Quaternion orders and quadratic forms -- code and computational data

This page contain source code and data related to my diploma thesis Quaternion orders and quadratic forms and an article of the same name based on it, alongside some explanatory comments.

The version of the thesis as defended is available here. The preprint of the article version is available here.

The code and data are available here.

The zip archive contains four Magma files. Once unpacked, each may be loaded in Magma with load "<filename>";:

• io.m specifies a format for describing quaternion orders and provides a function readOrder for reading an order from such a description. This description writes an order O as a tuple < K, D, a, b, G >, where:
  • K is a list of coefficients of a monic polynomial f, ordered from the absolute to the leading coefficients, specifying the number field K = Q[x]/(f).
  • Following this, if α is a root of f and f has degree d, elements of K are always given as d-tuples of rational numbers describing coordinates with respect to the Q-basis 1, α, ..., α^d.
  • Ideals of O_K are then given using a list of generators written as above. (Though only principal ideals are ever written in the files provided.)
  • D is the (reduced) discriminant of O, written as a list of generators as above.
  • a and b are elements of K describing the quaternion algebra A containing O.
  • G is a list of generators of O, each of these quaternions is written as a quadruplet of elements of K, describing the quaternion's coordinates with respect to the basis 1, i, j, k of A.
• maximals.m creates a list maximals containing descriptions of all definite maximal orders with class number 1 up to isomorphism. These may then be read by readOrder. This is taken from the data of M. Kirschmer, D. Lorch: Algorithmic enumeration of ideal classes for quaternion orders, SIAM J. Comput. 39 (2010), 1714-1747, available from here.
• search.m contains an implementation of the algorithmic search for perceptive suborders of a given order as a function search. On loading the file, the algorithm is run on all maximal orders from maximals, with the results (as representatives up to isomorphism) gathered in a list perceptives (as Magma objects).
• perceptives.m contains a list perceptives containing descriptions of the perceptive suborders found by search.m. These may then be read by readOrder.