same-class-number

This paper includes code related to the papers "The relative class number one problem for function fields, I, II, III" by Kiran S. Kedlaya. Most of the code is packaged in Jupyter notebooks; these are intended to be run using SageMath (tested using version 9.7). When indicated, there is also an external dependency on Magma (tested using version 2.27-1).

In the subdirectory "Shared":

• cyclic_covers.sage: Sage subroutines for finding cyclic covers of function fields. Requires Magma.
• weil_poly_utils.sage: Sage subroutines for managing Weil polynomials. This includes converting back and forth between Weil polynomials, point counts, Frobenius traces, and LMFDB labels for isogeny classes of abelian varieties; it also includes calling Sage's Weil polynomials iterator under specific conditions (e.g., with specified Frobenius traces).
• polys.xlsx: Excel spreadsheet listing candidate Weil polynomials for purely geometric extensions of function fields over F_2, and indicating which of these correspond to cyclic covers (when known). This is generated and modified by various notebooks.
• Curves by genus.ipynb: Process data files from Xarles (see below) and Dragutinović (see below) to build a table of curves of genus up to 5 over F_2 sorted by their Weil polynomials. Requires Magma. Creates a temporary file curves.txt.
• function_fields.sage: Loads curves.txt and provides an access function to create Magma function fields with a given zeta function.
• ListCurvesGenus4p2.txt: data file from Xarles.
• HyperellipticCurvesData.txt, TrigonalCurvesWithAutomorphisms.txt, Complete_Intersections.txt, Pts_Count_Complete_Intersections.txt: data files from Dragutinović.

In the subdirectory "Part I":

• Linear programming.ipynb: Compute bounds on numbers of rational points on curves over finite fields using the linear programming method.
• Relative class number 1 for q>2.ipynb: Compute candidate Weil polynomials for purely geometric extensions of function fields over F_q for q>2, and search for cyclic covers among these. Requires Magma.
• Weil polynomial bound for q=2.ipynb: Compute candidate Weil polynomials for purely geometric extensions of function fields over F_2. Requires curves.txt. Updates polys.xlsx.
• Cyclic covers for q=2.ipynb: Search for cyclic covers among candidate Weil polynomials listed in polys.xlsx. Requires Magma and curves.txt. Requires and updates polys.xlsx.

In the subdirectory "Part II":

• splitting_sequences.sage: Sage subroutines for computing splitting sequences for finite extensions of function fields.
• Non-Galois covers for q=2.ipynb: Perform various computations needed to rule out the existence of noncyclic covers over F_2. Requires Magma, curves.txt, and polys.xlsx.

In the subdirectory "Part III":

• orbits.sage: Sage subroutines for computing orbit representatives for the action of a finite group on subsets of a finite set. This code now has its own repository.
• linalg.sage: Sage subroutines for linear algebra.
• preamble.sage: Common declarations for the notebooks in this folder. This includes some initialization code plus a function closeout run at the end to report results.
• A series of Jupyter notebooks covering the Brill-Noether strata in genus 6. These all require Magma and polys.xlsx.
  • There is no notebook for hyperelliptic curves because these are handled in the text with no computation at all. Similarly, there is only a short computation needed for bielliptic curves; see below.
  • Trigonal curves are separated by their Maroni invariant: Genus 6 trigonal Maroni 0.ipynb, Genus 6 trigonal Maroni 2.ipynb.
  • Plane quintics are handled in Genus 6 plane quintic.ipynb.
  • Generic curves are handled in a series of three notebooks:
    • Genus 6 generic, part 1.ipynb performs the orbit lookup tree calculation. Creates a temporary file genus6-tuples.txt used by the other notebooks.
    • Genus 6 generic, part 2.ipynb find curves with 5 F_2-points.
    • Genus 6 generic, part 3.ipynb find curves with 4 or 6 F_2-points.
• A series of Jupyter notebooks covering the Brill-Noether strata in genus 7. These all require Magma and polys.xlsx.
  • There is no notebook for hyperelliptic curves because these are handled in the text with no computation at all. Similarly, there is no notebook for trigonal curves of Maroni invariant 3.
  • Trigonal curves of Maroni invariant 1 are handled in Genus 7 trigonal.ipynb.
  • Bielliptic curves are handled in Genus 6 and 7 bielliptic.ipynb. This notebook also includes a short computation needed to handle bielliptic curves in genus 6.
  • Curves with a g^2_6 are separated based on whether the g^2_6 is self-adjoint, rational over F_2, or irrational over F_2: Genus 7 self-adjoint g26.ipynb, Genus 6 rational g26.ipynb, Genus 6 irrational g26.ipynb.
  • Tetragonal curves (with no g^2_6) are handled in Genus 7 tetragonal, no g26.ipynb.
  • Generic curves are handled in a series of three notebooks:
    • Genus 7 generic, part 1.ipynb performs the orbit lookup tree calculation. Creates a temporary file 6-tuples.txt used by the other notebooks.
    • Genus 7 generic, part 2.ipynb find curves with 6 F_2-points.
    • Genus 7 generic, part 3.ipynb find curves with 7 F_2-points.

All files in this repository not attributed to another source are consigned to the public domain.