A package for Magma to make computation with étale algebras. An étale algebra is described by a direct product of number fields This guarantees that every operation on elements is very fast.
For the theory on which this code is based, see the References section at the bottom.
Please send comments and bug reports to stefano.marseglia89@gmail.com.
We introduce new type AlgEtQ, AlgEtQElt, AlgEtQOrd and AlgEtQIdl which correspond to étale algebras over the rationals, their elements, orders and ideals, respectively.
We have functionalities to compute isomorphism classes of invertible and non-invertible ideals for orders.
For complete descriptions and more details we refer to the List of commands for AlgEtQ.
To use them, use the magma command AttachSpec("spec"), after opening magma in the folder where you have downloaded the repo.
We also introduce the type AlgEtQMod, for modules over AlgEtQ. This code is functional but less mature.
For complete descriptions and more details we refer to the List of commands for AlgEtQMod
To use them, use the magma command AttachSpec("specMod"), after opening magma in the folder where you have downloaded the repo.
Functionalities for p-adic etale algebras, including how to build completion at rational primes, are developed by Casper Putz. Available here.
In the folder examples, you will find files containing the code to reproduce the examples from the papers in the references below, which should be of help to get a quick start on the functionalities.
Stefano Marseglia,
Computing the ideal class monoid of an order,
J. Lond. Math. Soc. 101 (2020), no. 3, 984-1007, DOI
Stefano Marseglia,
Cohen-Macaulay type of orders, generators and ideal classes, arXiv
Stefano Marseglia,
Modules over orders, conjugacy classes of integral matrices, and abelian varieties over finite fields, arXiv
Stefano Marseglia,
Local isomorphism classes of fractional ideals of orders in étale algebras, arXiv