EOType:

Magma code for computing Ekedahl-Oort types of curves (function fields) over finite fields

Importing:

To use the code in Magma simply attach the EOType file

Attach("EOType.mag");

Examples:

The two files

• ExampleHyperellipticEO.mag
• ExamplePrymEO.mag

contain examples to compute lists of EOtypes of Jacobians of curves and Prym varieties respectively

---

Documentation:

The documentation is a bit light (apologies). Included in the folder is what is below and some slides from a (funny at the time) talk I gave on this codebase at the Canadian Mathematics Society Winter Meeting in December 2018.

Main functions:

The main function of this code base is EOType(Kn:FldFun) which returns the sequence corresponding to the Ekedahl-Oort type of the p-torsion of the Jacobian of the curve corresponding to the input function field.

EOType(Kn::FldFun) -> SeqEnum
  {Return the lenght 2g sequence of the Ekedahl-Oort type of Kn}

Basic functions on EO Types:

ExtendEO(EO::SeqEnum) -> SeqEnum
  {Given a length g EO type return the length 2g type}

IsValidEO(EO::SeqEnum) -> BoolElt
  {Checks if the length 2g sequence is a valid symmetric BT1 EO-type}

DecomposeEO(EO::SeqEnum) -> SetMulti[SeqEnum]
  {Return the EO type of each irreducible submodule}

ComposeEO(EOset::SetMulti) -> SeqEnum
  {Turns a multiset of EOs into a single EO} 

EO types of (mod p) Dieudonne modules (which we call (F,V)-modules)

H1dR(K::FldFun) -> ModRng
  {Return the (F,V)-module of dimension 2g}

EOType(D::ModRng) -> SeqEnum
  {Computes the EO type of the (F,V)-module}

FVModule(EO::SeqEnum , p::RngIntElt) ->  ModRng
  {Return the canonical (F,V)-module (over GF(p)) for a given type}

Conversion functions between presentations:

There are many equivalent ways to descibe the isomorphism type of a Dieudonne module. Below are a series of functions that convert between several of these different representations.

EOToPermutation(EO::SeqEnum)-> GrpPermElt
  {EO to Permutation}

PermutationToEO(c::GrpPermElt) -> SeqEnum
  {Permutation to EO}

PermutationToEO(c::SeqEnum) -> SeqEnum
  {Permutation to EO}

EOToFVRelations(EO::SeqEnum : P:=FreeGroup(2)) -> SetMulti[GrpPerm]
  {Return the (F,V)-module relations of the components}
  The group P is an optional parameter so the user can provide an <F,V> group globally. 

FVRelationToEO(FVelt::GrpFPElt) -> SeqEnum
  {Return the (F,V)-module relations to an EO type}

FVRelationsToEO(FVelts::SetMulti) -> SeqEnum
  {From a collection of (F,V)-module relations to an EO type}