Action of Frobenius

Frobenius(P, q): PtEll[FldFunRat], RngIntElt -> PtEll

The \(q\)-th power Frobenius map on the point \(P\) of an elliptic curve that can be defined over a function field with constant field \({\mathbb{F}}_q\)).

FrobeniusActionOnPoints(S, q : parameters): [ PtEll ], RngIntElt -> AlgMatElt

gram: AlgMatElt                    Default: 

A matrix representing the \(q\)-power Frobenius map on the subgroup of the geometric Mordell–Weil group (modulo torsion) with the given basis \(S\). (This subgroup is assumed to be invariant under the \(q\)-power Frobenius.)

The optional parameter gram should be the Gram matrix with respect to the height pairing of the points in \(S\).

FrobeniusActionOnReducibleFiber(L): < Tup > -> AlgMatElt

Given reduction data \(L\) for an elliptic curve \(E\), such as given by the command LocalInformation, this function returns a matrix representing the Frobenius action on the non-identity components of corresponding fibres.

FrobeniusActionOnTrivialLattice(E): CrvEll -> AlgMatElt

Given an elliptic curve \(E\) defined over a rational function field over a finite field, returns a matrix representing the Frobenius action on fibre components and the zero section of the corresponding elliptic surface.