MAGMA Computational Algebra System

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Summary

Algebraic Geometry

• Coherent Sheaves
  • A new package is included providing functionality for working with coherent sheaves on ordinary projective schemes. These are naturally represented by graded modules over the polynomial ring, that is, the coordinate ring of the ambient of the base scheme.

  • There are a number of basic constructors of sheaves, including one for the canonical (dualising) sheaf of an equidimensional, locally Cohen-Macaulay scheme. Further construction operations include tensors, direct sums, tensor powers, Homs and duals.

  • The initial focus, in terms of functionality, apart from the computation of important cohomological invariants of varieties, has been on invertible sheaves (or divisors) and the explicit computation of their associated rational maps into projective space. The map is computed from the ``global section submodule" of the sheaf, which in turn comes from the maximal module.

  • For a base scheme X and an effective (Cartier) divisor D on X defined as a closed subscheme by an ideal I, there is code to compute an invertible sheaf corresponding to the class of D. Here, computing the explicit divisor map is essentially the same as computing the Riemann-Roch space if X is a variety. In fact, the Riemann-Roch space can be recovered during the computation of the associated sheaf.

  • A test for isomorphism of sheaves enables linear equivalence of effective Cartier divisors to be tested. Other properties which can be determined include local freeness, and whether the maximal module of the sheaf is arithmetically Cohen-Macaulay.

• Algebraic Surfaces
  • For a curve defined over a global field, one may calculate a regular model of the associated arithmetic surface, locally at a given prime. From this, one may obtain information such as the component group of the Jacobian at that prime.

  • Parametrization of degree 5 Del Pezzo surfaces has been added. The routines for parametrization of degree 7 and 8 surfaces and for rational scrolls have been updated. The code was provided by Josef Schicho (RISC, Linz).

• Toric Varieties
  • Magma V2.16 contains the first stage of a large new package for toric geometry being developed by Gavin Brown, Jaroslaw Buczynski and Alexander Kasprzyk. It incorporates both the combinatorial and Cox ring approaches.

  • The package includes code for cones, fans and polytopes in a rational vector space. Standard operations and constructions are provided including the definition of structures, duality, and lattice point counting within finite polytopes.

  • Toric varieties are defined via fans. Combinatorial tests are provided for the usual geometric properties of the toric variety: singularity, completeness, projectivity. Standard fan-based constructions such as weighted blow-ups of toric subsets are included.

  • Support is provided for working with torus-invariant divisors and divisor-class groups. This includes arithmetic of divisors, equivalence tests, computation of the canonical divisor and the construction of graded cones of the union of Riemann-Roch spaces for all multiples of a divisor.

  • The Cox ring of a toric variety T may be computed. This allows T to be used as a very general single or multi-graded ambient space. Definition of arbitrary closed subschemes of T via homogeneous ideals in the Cox ring is supported. Conversely, Cox rings can be made as abstract objects and the corresponding toric variety and its combinatorics deduced.

  • The basic components of the minimal model program for toric varieties are incorporated, including extremal contractions, generalised flips and an explicit tour of the chambers of the mobile cone (in the sense of Mori dream spaces).

  • The package is integrated with the existing Magma scheme structures, using the Cox ring as the ambient coordinate ring. Many of the basic scheme operations work for subschemes defined via the Cox ring.

Arithmetic Geometry

• Elliptic Curves
  • The routine for determining the set of S-integral points on the curve has been replaced with a new implementation. In addition to being reliable, there are a number of new ideas which lead to improved efficiency.

  • The Cassels-Tate pairing between elements of the 2-Selmer group has now implemented for elliptic curves defined over number fields.

  • A new implementation of ``elliptic curve Chabauty'' based on the Mordell-Weil sieve has been included.

  • An algorithm based on the Mazur-Tate algorithm has been implemented to compute the p-adic height of a point on an elliptic curve over the rationals.

  • A routine is included to compute the automorphism group of an elliptic curve over an arbitrary field. It is returned as an abstract group together with a map to the group of actual automorphisms.

  • Some speedups have been introduced for point counting over small prime fields, in particular, for fields of cardinality less than 2³⁰.

• Hyperelliptic Curves
  • Two-cover-descent has been implemented by Nils Bruin for hyperelliptic curves. This is two-descent on a hyperelliptic curve, over the rationals or a number field. This is not the same as two-descent on the Jacobian, and it yields more precise information about rational points on the curve. One obtains the ``2-Selmer set" of the curve (which is defined to be the 2-Selmer group of an abelian variety).

• L-Functions
  • Many new types of L-functions have been added, together with utility functions for working with them. The most prominent new L-functions are for Hecke characters, and Hecke Grössencharacters, which are important in the development of Tate's thesis. This appears to be the first general implementation of L-functions for Hecke Grössencharacters.

  • The TensorProduct intrinsic has been widened in its application. The new intrinsic is quite powerful, and has been used to identify (numerically, via the functional equation) the spinor L-function for various Siegel modular forms.

  • A related addition is the construction of symmetric powers. These have been implemented in the simplest cases (for degree 1 L-functions), and for elliptic curves over the rationals. The tensor power construction is of interest in studies of the Sato-Tate conjecture, as it can compute the exact Euler factor at bad primes for any symmetric power of an elliptic curve over the rationals.

Arithmetic Geometry (Modular Forms)

• Arithmetic Fuchsian Groups
  • The fundamental domain routine has been further improved, and a function has been added to solve the word problem.

• Modular Symbols
  • A routine has been added to decide whether two given newforms are twists of each other.

• Hilbert Modular Forms
  • The existing package has been extensively reworked resulting in better performance and greater reliability. The restriction to squarefree level has been lifted, one can now obtain a NewSubspace relative to any level. A general procedure has been implemented for obtaining new (and old) spaces of a given space from knowledge of dimensions and the Hecke action on spaces of lower level. The field over which Hecke operators are expressed has been changed to the natural one, determined by the Galois structure of the weight (it is Q in parallel weight).

• Modular Forms Over Imaginary Quadratic Fields
  • A new package is included that computes modular forms over arbitrary imaginary quadratic fields. The method involves the `Sharbly' complex and Voronoi polyhedra, and was developed in practice by Paul Gunnels and Dan Yasaki. These techniques made it possible to automate calculations which in previous implementations had to be done separately for individual fields. The current version computes Hecke operators (at ideals that have odd order in the class group) on spaces of cusp forms of weight 2 with trivial character.

• Admissible Representations of GL₂(Q_p)
  • A new package, developed by Jared Weinstein, treats local Langlands theory for GL₂ over Q. Starting with a newform in a space of classical cusp forms, and a prime p, one can construct the local component at p of the associated automorphic representation. This is an admissible representation of GL₂(Q_p). One can compute key features of this, such as principal series parameters or a cuspidal inducing datum. Furthermore, the local Langlands correspondence associates to this a Galois representation on the absolute Galois group of Q_p. One can compute (the restriction to inertia of) that Galois representation.

Commutative Algebra

• Polynomial Rings
  • Multivariate polynomial multiplication and division has been made much faster, using a simple implementation of the heap-based algorithms of Monagan and Pearce.

  • The factorization of multivariate polynomials has been further optimised for some classes of input (in particular for polynomials over the integers with more than 2 variables).

  • Bivariate factorization has been improved via better use of deflation techniques.

• Gröbner Bases
  • An interface between Magma and the SAT solver Minisat has been developed, allowing one to apply SAT methods when solving polynomial systems over GF(2).

  • When applying the F₄ algorithm to systems of polynomial equations defined over GF(2), an early termination criterion is used based on the occurrence of linear polynomials. This often yields a non-trivial speedup in the hardest step.

• Ideals and Modules
  • Improvements to the algorithm for constructing the minimization of a free resolution has led to significant speed-ups.

  • The computation of the primary decomposition of an ideal has been improved leading to significant speed-ups in some cases.

  • The calculation of the colon ideal of an ideal is now significantly faster for some inputs. Being able to compute the colon ideal rapidly is critical to many key calculations in algebraic geometry.

  • The computation of a minimal basis for a non-homogeneous ideal or module has been improved through use of automatic homogenization.

  • The algorithm of Eisenbud and Sturmfels has been implemented for computing a maximal regular sequence of elements inside an ideal I of a polynomial ring over a field. It is designed to produce a sequence of reasonably sparse polynomials.

  • For an ideal I of an affine algebra R, given by an ideal in the polynomial ring of which R is a quotient, code is provided to construct the polynomial ideal J whose quotient algebra is isomorphic to the Rees algebra R(I) of I over R.

• Differential Rings
  • Routines for the factorisation of linear differential operators over differential Laurent series rings have been implemented by Alexa van der Waall. Both coprime index 1 factorisation and LCLM factorisation are supported.

Global Arithmetic Fields

• Number Fields
  • Dirichlet and Hecke characters, including Hecke Grössencharacters in some cases, have been implemented as the duals of RayResidueRing and RayClassGroup, and should allow group operations. This appears to be the first known general implementation for Hecke Grössencharacters and their L-functions; the only previous code was in PARI/GP and was only for finite order Hecke characters in special cases (largely Hilbert characters).

  • A new algorithm for the computation of the subgroup of K^* generated by a set of elements has been implemented. This enables one to conveniently work in subgroups of the multiplicative group.

• Algebraic Function Fields
  • A new algorithm has been implemented for the computation of p-maximal and maximal orders in Artin-Schreier extensions. The same techniques have also been applied to compute the prime splitting in those cases. In particular, when constructing arithmetic-geometric codes, those two new algorithms improve performance by several orders of magnitude.

• Galois Theory
  • Following the model of the highly successful implementation of the Galois groups over Q which allow the computation of Galois groups of (reducible) polynomials of arbitrary degree, a similar algorithm for Galois groups of function fields in positive characteristic has been implemented.

  • Support for the use of complex approximations in the computation of Galois groups over Q has been added.

  • A new algorithm for computing invariants for intransitive groups yields a reduction of computation time of several orders of magnitude for reducible polynomials.

  • A generic, field independent algorithm for the computation of subfields following new ideas of Klüners and van Hoeij has been implemented. While the new algorithm does not result in improved performance over Q, it is generic and thus, for the first time, allows computation of subfields of global function fields.

Group Theory

• Finite Groups
  • New algorithms for constructing all subgroups from maximal subgroups have been implemented, bringing speed improvements, and the ability to handle groups with larger abelian chief factors than the previous method allowed. With this machinery it is straightforward to construct the 111,004 conjugacy classes of subgroups of the simple group, Fischer Fi₂₂, which has order 6,456,175,165,400.

  • Code to produce standard generators for all quasi-simple groups having simple quotient with order up to 2 x 10⁸ has been provided by Derek Holt. Standard generators are used to create an isomorphism between a quasi-simple group in some arbitrary representation and a ``standard" version of it.

• Matrix Groups
  • A package developed by Alla Detinko, Dane Flannery and Eamonn O'Brien allows the user to determine whether or not a matrix group defined over a rational function field is finite.

  • A database of the maximal finite irreducible subgroups of Sp_2n(Q) for 1$\le$i$\le$11 (constructed by Markus Kirschmer) is included.

• Finitely Presented Groups
  • As a result of experiments which involved trying to establish whether certain fp-groups are finite of infinite, revisions were made to key tools such as the p-quotient and Todd-Coxeter functions. These changes have resulted in Magma being able to resolve a significantly higher proportion of examples. It is now possible to settle (without any human intervention) the question for all but 11 instances of the 13,646 distinct one-relator quotients of the modular group where the additional relator has length 36.

  • The newly-published Plesken-Fabianska algorithm for finding infinite PSL(2, K)- quotients of a finitely presented group has been implemented as part of V2.16.

  • The Homomorphisms function has been extended by Derek Holt so that it is now possible to search for homomorphisms from an fp-group into a (small) soluble group given by a power-commutator presentation.

  • A simple function has been provided which tries to construct the regular representation of a finite fp-group and then search for a permutation representation having much smaller degree. This function has been successfully applied to groups of order up to 600,000,000.

Lattices

• Lattice Reduction
  • Functions have been developed by Damien Stehlé for computing Hermite-Korkine-Zolotarev reduced bases of lattices. HKZ-reduction is an alternative to LLL-reduction. It is significantly more expensive to obtain, but it provides lattice bases of much better quality (i.e., shorter and basis vectors that are closer to being mutually orthogonal).

• Lattice Enumeration
  • It is now possible to prune the tree that is explored during the enumeration of short vectors. Although it may result in some vectors being missed, it can make the computations faster by factors higher than 100 if a small probability of an incorrect output is acceptable.

• Automorphism Group
  • An improved algorithm for computing the automorphism group of an integral lattice has been developed. The algorithm can handle lattices having a much larger number of vectors of minimal norm than its predecessor. The result is that it is much faster than the old algorithm and can handle significantly larger lattices. For instance, it is able to compute the automorphism group of some of the easier lattices of dimension 48 in the Sloane-Nebe database. A similar algorithm for determining isometry of a pair of lattices is also provided.

• Database
  • A new version of the Sloane-Nebe database has been constructed. This version contains the $\Theta$-series series and automorphism groups for most of the lattices. A number of errors in the original have been corrected.

Representation Theory

• Splitting G-modules and A-modules
  • A new Meataxe algorithm has been developed for splitting general A-modules, where A is a finite dimensional matrix algebra defined over the rational field. This yields an effective algorithm for decomposing a module into indecomposable summands. If the module is a G-module for some group G, extensive use is also made of character theory. Representations associated with characters having non-trivial Schur indices are properly handled. The difficult problem of splitting homogeneous modules (direct sums of the same indecomposable) is handled by decomposing the endomorphism ring of the module via a maximal order. Schur indices are properly handled. Modules having dimensions in the several hundreds are routinely split into indecomposable modules.

  • A method for extracting a particular G-module from a large degree permutation module defined over a number field has been implemented. The algorithm is based on Nickerson's ``Split-P" condensation method. The new feature is the use of the Michler-Weller algorithm for character values of constituents of a permutation module to identify an appropriate condensed vector to spin in the uncondense stage.

  • Tools for constructing the condensation of permutation modules, tensor products and induced modules over fields of either characteristic zero or characteristic p are included in the release.

• Irreducible Rational Representations
  • An effective algorithm has been developed for computing irreducible Q[G]-modules for a finite group G. Character theory is used to identify a (reducible) module M that contains the desired module. The Meataxe described above is then used to split the module M thereby yielding the required irreducible module. Condensation is applied to reduce the dimensions of the modules that have to be split.

  • Lattice-based techniques have been developed that control the growth of coefficients at every stage.

  • A variant of the algorithm can determine all irreducible Q[G]-modules. The machinery has been used to construct irreducible Q[G]-modules having dimensions well over a thousand.

• Integral Representations
  • A constructive version of the Jordan-Zassenhaus theorem for determining all classes of non-equivalent integral representations over a number field has been implemented (in the case where the representations are absolutely irreducible over the field).