MAGMA Computational Algebra System

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 GF(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 fiber components and the zero section of the corresponding elliptic surface.