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Description

This repository is a database of Shimura curves and their Atkin-Lehner quotients which is an accompaniment to arXiv.

The database contains defining equations for all Atkin-Lehner quotients of Shimura curves X_0(D,N) which are geometrically hyperelliptic. Furthermore, the set of rational points on the quotient curves is included when finite and a description is given when infinite.

A file for each quotient curve can be found in the folder ShimDB-v2 The name of each file will be of the form Shim-X(D,N)-g?-W.m where

  • D - is the discriminant of the quaternion algebra;
  • N - is the level;
  • g? - is the genus of the curve;
  • W - is a list of numbers representing the group of Atkin-Lehner involutions.

To interact with the database you will need to have Magma installed. Then you can clone the repository.

Usage

Once the repository has been cloned, to use the database first do

AttachSpec("spec");

To load the data associated to the curve X_0(D,N)/W;

s:=ShimDBRecord(D,N,W);

Each curve/file has a record which holds information about the curve, the attribute names of the record are

ShimLabel
ShimDiscriminant
ShimLevel
ShimAtkinLehner
ShimGenus
ShimModel
ShimTopCurve
ShimProjectionEquations
ShimRationalPoints
ShimPointsProvedCorrect
ShimPointsNotes
ShimPointsEverywhereLocally

One can also load the record associated to a curve from the filename using

s:=ShimDBRead(filename);

To load all filenames or records in the database use ShimDBFilenames() or ShimDBRecords() respectively.

As an example, to see all Hasse principle violations we can do

filenames:=ShimDBFilenames();
for file in filenames do
  s:=ShimDBRead(file);
  if s`ShimAtkinLehner ne [1] and Type(s`ShimRationalPoints) eq SetEnum then
    if s`ShimRationalPoints eq {} and s`ShimPointsEverywhereLocally eq true then
      file;
    end if;
  end if;
end for;

and the output is Shim-X(87,1)-g2-[1,3].m, Shim-X(93,1)-g3-[1,3].m, Shim-X(119,1)-g4-[1,7].m, Shim-X(6,29)-g2-[1,6].m, Shim-X(6,37)-g2-[1,3].m, Shim-X(39,2)-g3-[1,78].m.

Authors

  • Oana Padurariu
  • Ciaran Schembri