The Torsion Subgroup

TorsionSubgroup(E): CrvEll[FldFunG] -> GrpAb, Map

Given an elliptic curve \(E\) defined over a function field \(F\), this function returns an abelian group \(A\) isomorphic to the torsion subgroup of \(E(F)\), together with a map from \(A\) to \(E(F)\).

TorsionBound(E, n): CrvEll[FldFunG], RngIntElt -> RngIntElt

TorsionBound(E, n, B): CrvEll[FldFunG], RngIntElt, RngIntElt -> RngIntElt

Given an elliptic curve over a function field \(F\) and an integer \(n\), this function computes a bound on the size of the torsion subgroup of \(E(F)\) by considering the torsion subgroups of the fibres of \(E\) at \(n\) different places of \(F\).

When an integer \(B\) is given as a third argument then the subgroup of elements of order dividing \(B\) is bounded, rather than the whole torsion subgroup.

GeometricTorsionBound(E): CrvEll[FldFunG] -> RngIntElt

Given an elliptic curve \(E\) defined over a function field \(F\), this function computes a bound for the geometric torsion subgroup of \(E\). That is, the torsion group of \(E(K)\) where \(K/F\) is the smallest extension with algebraically closed constant field. In cases where a bound cannot be computed then \(0\) is returned.