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algorithms

this page contains a few files with algorithms either written in pari-gp or in magma. most of these are obsolete now.

my new and favourite package for doing computational number theory is sage. everything I have and will contribute to it is included in the latest version.

selfpoints

here you can find more details on the computations of self-points.

shark : computing tate-shafarevich groups via iwasawa theory

shark was originally a package of algorithms written in magma that computes upper bounds on the rank and on the p-primary part of the Tate-Shafarevich group of an elliptic curve over Q. it has now been transferred to sage and is part of the latest version.
on request i can provide you with the obsolete magma package.

elliptic curves over function fields

the file ecff.m contains certain functions to work with elliptic curves over function fields in magma. note that the current version of magma has all of this function implemented. this was written for an earlier version.
the following functions, already defined in magma for number fields, are exended to curves E over FldFunG; here v stands for a place PlcFunElt :
  • LocalInformation(E, v)
  • LocalInformation(E)
  • Conductor(E)
  • TamagawaNumber(E,v)
  • TamagawaNumbers(E)
  • BadPlaces(E)
  • Reduction(E,v)
  • EllNp(E,v), added, the number of elements in the reduction
  • TorsionSubgroup(E) see the file readme.ecff.m for more information are some examples how to use the functions. see also on the help page of magma on the corresponding functions for number fields.

    heights on the fine selmer group

    the file hell.gp, containing functions defined in pari-gp can be used to compute the canonical p-adic height on an elliptic curve over Q. note that mazur, tate, and stein have found a much faster algorithm for computing these heights. it has been improved even further by david harvey. His algorithm is implemented in sage.

    then the file, fine.gp contains most of the algorithms used in my thesis to do computations on the fine selmer group, such as the p-adic regulator and the upper bound on the euler-characteristic.

    the fine tate-shafarevich group

    for the computation of the fine tate-shafarevich group, see here.