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Principal Ideal Problem

Code accompaining the paper "The principal ideal problem for endomorphism rings of superspecial abelian varieties". This code contains some function for solving the principal ideal problem and the kernel to matrix problem in dimension 2. We refer to the paper for a more detailed explanation of the functions.

Project Structure

  • pip.py: main algorithms
  • ideals.py: ideals helpers
  • misc.py: generic helpers
  • bench.py: test functions
  • qmat_iso.py and qmat_iso_q.py: isomorphisms between quaternion orders modulo n and 2x2 matrices

Testing the code

The following files can be used to test and benchmark the code, and as an example for its usage:

  • test_square.py: tests the function fill_square on random O0-ideals, checking that the matrix output has the correct norm and that the first column generates the input ideal; it also computes average timings and output sizes veryfing the claims of Sec. 4.4
  • test_kernel.py: generates a pair of points (P, Q) of l-torsion and checks that the function kernel_to_matrix returns a matrix of norm l with (P, Q) as a kernel

To run the tests simply run sage test_xxx.py or sage --python -O test_xxx.py to skip assertions.