lemkeoliver

Code related to the paper “Inductive methods for counting number fields”, by Brandon Alberts, Robert J. Lemke Oliver, Jiuya Wang, and Melanie Matchett Wood

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All references below are to the arxiv version of this paper, available here.

Files

• The file SavedDatabase_23_3.mag is a Magma file containing the output of the computation described in Section 7.1, which was recorded in Corollary 1.8. It was created by running the file CreateDatabase.mag described below.
  • More specifically, this database contains an array called Database of 4953 GroupCountingInfo records, corresponding to the 4953 transitive groups of degree up to 23.
  • The GroupCountingInfo record is a bespoke record type, whose objects have the following properties:
    • degree: The degree of the permutation group
    • label: The label of the permutaiton group within its degree, i.e. the group is TransitiveGroup(degree, label) in Magma
    • malleindex: The Malle index of the group
    • bound: The best known exponent in an upper bound for G-extensions that follows from our methods
    • conjecture: A boolean indicating whether the number field counting conjecture is known for G-extensions over Q.
    • isnilpotent: A boolean indicating whether G is nilpotent
    • issolvable: A boolean indicating whether G is solvable
    • isconcentrated: A boolean indicating whether G is concentrated.
• The file output.txt contains the text output of running the file CreateDatabase.mag.
• The file CreateDatabase.mag is the main Magma file used to generate the database.
  • Inside it, there are two important variables: MaxDegree (set to 23) and number_of_repeats (set to 3), which is the number of times to iterate through the list.
  • Otherwise, the mathematical content is compartmentalized into a number of separate Magma files in the utils/ directory:
    • utils/CountingInvariants.mag: Computes the index of a permutation group and the minimal index subgroup
    • utils/NumberFieldBounds.mag: Computes various upper bounds on the number of G-extensions
    • utils/InitializeDatabase.mag: Initializes the database, and contains the definition of the GroupCountingInfo record.
    • utils/GroupConstructions.mag: Creates certain different types of groups
    • utils/Comparing_Orderings.mag: Has code used to compare the different pushforward discriminants
    • utils/Pontryagin_Dual_Class_Group_Bounds.mag: Implements an upper bound on H1_ur using the Pontryagin dual, i.e. Lemma 4.1
    • utils/ConjectureIsKnown.mag: Runs through the database, and records some groups for which the field counting conjecture is known
    • utils/InductiveCounting.mag: Does the hard work, and actually iterates through the database, updating the best known upper bound and detecting whether we may infer the field counting conjecture.
• There is also a tarball inductivemethods.tar.gz containing all the contents of this folder.