Summary

Algebras

• A module for quaternion algebras has been implemented. Special constructions are implemented for orders over $\mbox{\bf Z}$, with given ramification and index in a maximal order. The left and right ideal arithmetic of orders in definite quaternion algebras over $\mbox{\bf Q}$ is treated, permitting enumeration of all one-sided ideal classes.

Algebraic Geometry

• A new package for computing with hyperelliptic curves and their Jacobians has been implemented. (Developed by Michael Stoll [Düsseldorf], with revisions and kernel support by members of the Magma group.)

• A major new initiative in modular forms and modular curves has been undertaken. As the first step, a Magma version of William Stein's Hecke package for computing with modular symbols has been implemented. (Developed by William Stein [Berkeley].) An alternative approach via Brandt modules associated to the ideal theory of quaternion orders has been implemented by David Kohel of the Magma group.

• Elliptic curves can now be created from a plane curve of genus one with given rational point. Functions for transformations of genus one hyperelliptic curves and elliptic curves allow for easy conversion between categories.

• The functionality for computing with elliptic curves has been expanded, including new functions for computing S-integral points, Weil pairings, and group structure over finite fields, and improved algorithms for height and rank computations over $\mbox{\bf Q}$.

• Resolution graphs and splice diagrams have been introduced as a means of encoding data generated by the resolution machinery. The package includes a resolution function for curves adapted to present its output in resolution graph format.

Coding Theory

• The advanced Zimmermann algorithm for minimum weight computation has been implemented. Several other new coding theory functions have been installed.

Commutative Algebra

• A major revision of the multivariate polynomials module has been achieved. This includes a new internal random-access array representation, a fraction-free coefficient representation and new variable-size monomial representation, yielding improvements in time and space usage. Several other improvements have been done and many bugs and leaks have also been fixed.
• Affine algebras have been expanded in their functionality considerably. Further support has been added for the case for when affine algebras are fields.

Groups

• A new category has been created for the family of groups defined by a polycyclic presentation. Note that such a group may be infinite. Algorithms for element arithmetic and (some) subgroup computations analogous to those for finite soluble groups have been implemented.
• The Smith algorithm for computing the automorphism group of a finite soluble group has been implemented (with improvements) by Mike Slattery.

Incidence Structures

• Categories for Incidence Geometries (in the sense of F. Buekenhout) and Coset Geometries have been implemented by Dimitri Leemans.

Linear Algebra and Module Theory

• A new feature that allows the creation of a matrix without having to predefine its parent structure has been introduced. A matrix may be expressed in a broad range of different formats. This is expected to make working with matrices a great deal easier. In addition, an extensive range of functions for creating matrices having a special structure (block, sparse, scalar, diagonal) has been provided. A new chapter in the Handbook brings together all the operations available for matrices.
• The existing facilities for K[G]-modules have been generalised so that the great majority of them apply to modules defined over a (matrix) algebra. As part of the restructuring, this class of modules is described in a new chapter that forms the first chapter in a new series (Handbook Part) relating to Representation Theory.

Rings, Fields and Orders

• A new system has been developed for computing with algebraically closed fields, which have the property that they always contain all the roots of any polynomial defined over them. One can now compute the variety of any zero-dimensional multivariate polynomial ideal over the algebraic closure of its base field.

• The new MONSTER random number generator of G. Marsaglia has been implemented, replacing a weak linear congruential generator.

• Index calculus methods for discrete logarithm computations in prime finite fields have been implemented.

• The algebraic number fields module has been upgraded to correspond to KANT/Kash V2.2. In particular, this provides the user with much improved algorithms for class group and unit group computation.
• The algorithms for computing the Galois group of a number field (more precisely, its splitting field) have been extended by Katharina Geissler so that it is now possible to compute Galois groups for fields having degree up to 20 (previously 15).
• The original KANT/Kash module for function fields has been completely revised and its facilities have been vastly expanded by Florian Hess. In particular, machinery has been introduced for working with places and divisors.
• Local rings and fields have been improved and expanded. A wider range of local rings and fields may be created since inertia rings can now be specified by an inertia polynomial as well as by degree.