# 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`

```magma
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.