Release Notes V2.8 (July 31, 2001)

This screen provides a terse summary of the new features installed in Magma for release version V2.8 (July 31, 2001). For more details, see the full release notes in the files relv28.dvi or relv28.ps, or see the full release notes at the top page of the HTML version of the Help system.

Documentation

There are 18 new chapters in the Handbook.

The Handbook now has bibliographies for each chapter, and much more information concerning the algorithms used has been included in several places.

Algebraic Geometry

All of the algebraic geometry types together with their basic support have been moved into the kernel. This includes the integration of many of the specialised packages like elliptic and hyperelliptic curves. This significantly increases the power of the geometry types, allowing one to apply the functionality of both a specialised curve type and the general scheme type to a single object. The change also makes available many generic constructors and standard set and sequence operations, which were previously not available for the geometry types.

Maps between schemes have been enhanced with new constructors and computations of images and pullbacks available in more circumstances. Maps can also be defined between schemes now rather than just between affine or projective spaces.

Plane curves have been tied very closely to the function field machinery in Magma. The result is that computations with divisors and their Riemann-Roch spaces can be made entirely in the geometric context. The applications include: gap numbers, canonical embeddings of curves, class group computation over finite fields, constructions of geometric codes from curves and many other standard methods in the geometry of curves.

A new package for computing with plane conic curves has been written. Its major feature is an implementation of a new algorithm by John Cremona (Nottingham) to find points with reduced coordinates on conics defined over the rational numbers. This implementation is very fast, improving on Cremona's initial test timings.

New functionality for computing automorphisms has been added to the hyperelliptic curve package by M. Stoll. P. Gaudry has implemented a number of methods for counting points on the Jacobian of a hyperelliptic curve. These include the Schoof algorithm for a genus 2 curve. Code for determining Igusa invariants has been provided by E. Howe while P. Gaudry has implemented an algorithm for constructing a genus 2 curve from invariants.

A package for modular curves has been developed by David Kohel of the Magma group. A modular curve is defined in terms of standard affine modular equations which are stored in precomputed databases. The possible model types are ``Atkin'', ``Canonical'', and ``Classical''.

A package for Brandt modules developed by David Kohel is now included.

A package for modular forms developed by William Stein is now included.

A database of K3 surfaces has been added. It contains characteristic data of K3 surfaces embedded in weighted projective spaces in codimension up to 4. The functions used to create the database are also included so that users can extend the database or to create similar databases.

All geometric chapters of the Handbook have undergone major revision. Many more examples have been included, some of which are extended calculations which illustrate different parts of the new geometry packages working together.

Coding Theory

A database of best known linear codes over GF(2) has been implemented (a joint project with M.~Grassl).

Magma now has facilities for linear codes defined over Z_4.

Commutative Algebra

Magma now provides facilities for computing with Groebner bases of ideals of polynomial rings over Euclidean rings (including the important case of the integer ring Z). Such Groebner bases are computed in Magma by an extension, due to Allan Steel, of Jean-Charles Faugere's F_4 algorithm which uses sparse linear algebra. Many of the standard functions based on Groebner bases over fields also carry over to ideals defined over Euclidean rings.

Groups

An algorithm for determining the maximal subgroups of a permutation group is provided. This is applicable to any group for which the non-abelian composition factors either have order less than 2.6 times 10^7 or have a permutation representation of degree less than 32. The machinery for finding all conjugacy classes of subgroups has been extended to a much larger class of groups. In particular, the former limitation that the trivial Fitting quotient have order less than 216,000 has been removed.

An algorithm for finding the automorphism group of a permutation group is also included. A variation tests isomorphism of two permutation groups.

The functionality of the new category of polycyclic groups introduced in V2.7 has been greatly increased.

A new category has been created for generic (finite) abelian groups. The creation of a generic abelian group is a new feature which allows the user to create an abelian group over any domain given that an identity and a group operation have been defined. Features include finding the order of an element, determining a presentation from generators, and computing the discrete logarithm of an element.

Several new databases of groups have been created, e.g. for perfect groups, almost simple groups, rational maximal finite matrix groups and finite quaternionic matrix groups.

New algorithms for computing the conjugacy classes of elements and the maximal normal p-subgroup in a finite matrix group have been installed.

Several functions for analysing the action of a matrix group over a finite field on subspaces of the underlying natural vector space have been introduced.

Some basic support machinery for finitely presented groups has been redesigned, yielding improved performance and increased applicability of key functions for fp-groups.

New versions of the algorithms for coset enumeration, simplification of presentations by Tietze transformations, Reidemeister-Schreier rewriting and computation of p-quotients have been installed. An interactive coset enumeration facility is now available.

The functions for the representation theory of finitely presented groups have been revised completely.

The function Order for finitely presented groups has been revised and now uses a more sophisticated strategy for computing the order of a group or proving its infiniteness.

A suite of functions for analysing central extensions of finite soluble groups has been added.

Incidence Structures

The nauty program due to B. McKay for finding automorphisms of graphs has been updated to the latest version (2.0) and its user interface within Magma has been enhanced. This new version of nauty is often much faster than the previous version installed within Magma.

A catalogue due to B. McKay et al of strongly regular graphs is now available.

B. McKay's program for the orderly generation of graphs is now accessible from within Magma.

Linear Algebra and Module Theory

Several new advanced algorithms for matrices over Z or Q have been implemented (and more will follow in the future).

Modules over orders are now supported.

Optimization

Basic facilities are now provided for linear programming.

Rings, Fields and Orders

An experimental implementation of the number field sieve (NFS) is now available.

Basic facilities for computing with Galois rings are now included.

Polynomials over the integers or rationals are now factored by the exciting new algorithm of Mark van Hoeij. The search for the correct combination of modular factors (which has exponential worst-case complexity in the standard Berlekamp-Zassenhaus algorithm) is now performed by van Hoeij's algorithm, which efficiently finds the correct combinations by solving a Knapsack problem via the LLL lattice-basis reduction algorithm.

Algebraic number fields and their orders have been modified in several ways since Magma V2.7. A new field type has been created to act as the field of fractions of orders of number fields. More functionality has been provided for relative fields and orders and their ideals and quotients by ideals can now be created.

The quadratic fields and rings have been rewritten to become a part of the algebraic fields and their orders.

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