Overview

This repository contains Magma code to

• Enumerate suborders/superorders of a given quaternion order (originally based on code by M. Kirschmer and D. Lorch).
• Classify definite orders that are Hermite rings, respectively, have locally free cancellation.

Moreover it contains a computer readable list of all definite Hermite quaternion orders over a ring of integers in a number field.

The code accompanies the paper

D. Smertnig, J. Voight, Definite orders with locally free cancellation, Trans. London Math. Soc. 6 (2019), no. 1, 53–86.

Tested with Magma V2.24-5.

Usage

To use the package, it must first be imported.

AttachSpec("Hermite.spec");

Load the definite Hermite quaternion orders, respectively the ones with locally free cancellation.

orders := LoadOrders("hermite.dat");
orders_c := LoadOrders("cancellation.dat");

The ones with cancellations can also easily obtained via

orders_c := [ O : O in orders | HasCancellation(O) ];

but that takes a few minutes, as the class sets and stable class groups have to be computed.

The variable orders is a list of quaternion orders. We a priori fix representatives for the base fields, and orders contains the 375 orders up to isomorphisms R-algebra automorphism, where R is the center. To produce the 303 orders up to ring automorphisms from this list, use DedupOrders.

To rerun the classification yourself use

SetVerbose("Quaternion", 1);
res := FindHermiteOrders();
orders := OrdersAsList(res);
orders_c := [ O : O in orders | HasCancellation(O) ];

It takes less than an hour on my Laptop (i7-7600U).

Example: Enumerating suborders

This example demonstrates how to enumerate all suborders of a quaternion order up to local isomorphism and a bounded index.

QQ := RationalsAsNumberField();
ZZ := Integers(QQ);
B := QuaternionAlgebra< QQ | -1, -1>;
O := MaximalOrder(B);
p := 3*ZZ;

G := EnumerateSubordersAtP(O, p, 6);

Here 6 means to enumerate orders up to index p^6.

The output is organized as a graph by radical idealizers. The orders themselves are attached to the vertices as labels. The structure can be visualized:

SaveOrdersGraphToDot(G, "suborders.gv" : ColorFunc := func< v | ColorClassesAtP(3*ZZ, v) >);

Then call Graphviz from the command line

dot -Gsize=10 -Tsvg suborders.gv -o suborders.svg

This produces

To access the order O16 (it's Gorenstein, but not Bass, because it's pink!),

V := VertexSet(G);
O16 := Label(V.16);
print Generators(O16);

produces

[ 1, 27*i, -6*i + 3*j, 1/2 - 9/2*i + 3/2*j + 3/2*k ]

There is also EnumerateSuborders, which recurses over multiple primes, and EnumerateSuperordersAtP for computing superorders. To obtain representatives for the global isomorphism classes of orders, pass the optional parameter UpTo := "Isomorphism".

File layout

• EnumOrders/ Code for enumerating orders (independent of Hermite)

  • enum_suborders.m: Functions for enumerating suborders of a given quaternion order, possibly picking out a given class of orders using a selection function.

  • enum_superorders.m: Functions for enumerating superorders of a given quaternion order, possibly picking out a given class of orders using a selection function.

  • orders_aux.m: Auxiliary (mathematical) functions for quaternion orders: residue rings, radical idealizer, subalgebras of finite algebras.

  • list_helpers.m: Functions to help dealing with graphs or lists of quaternion orders.

  • tree_helpers.m: Helper to handle constructing the graph data structrue in enum_suborders.m.

• Hermite/ Code for classifying Hermite orders

  • classify_hermite.m: Contains the actual logic for running over the fields, algebras, and maximal orders that need to be checked. Calls EnumerateOrders.

  • bounds.m: Some helper functions related to the computation on bounds of indices and discriminants needed in the classification.

  • stable_class_group.m: Computation of the stable class group, right class set, and checking the Hermite property, respectively, cancellation for an order.

  • helpers.m: A few small helper functions that are used in saving, loading, and printing orders.

  • loadsave.m: Functions for saving and loading lists of orders (plus extra data) in computer-readable format.

  • print.m: Functions for creating a textual representation of the list of orders.

• run_classification.m: Runs the classification of Hermite orders and saves the output to a computer-readable file.

• load_classification.m: Loads the classification of Hermite orders from a previously saved file.

Data format

The output of the classification (hermite.dat and cancellation.dat) is formatted as follows. Each file contains a Magma list of tuples [ <O₁,e₁>, <O₂,e₂>, ... ] where Oᵢ represents the ith order, and eᵢ contains additional metadata about the order.

• Oᵢ = [ F, a, b, [γ₁, ..., γₘ] ] where
  • F = [f₀, f₁, ..., fₙ] represents a number field given by F≃Q[x]/(f) with defining polynomial f=f₀ + f₁ x + ... + fₙxⁿ. Denoting by α the image of x under this isomorphism, an element of F is encoded as [c₀, ..., cₙ₋₁], representing c = c₀ + c₁ α + ... + cₙ₋₁ αⁿ⁻¹.
  • a, b are elements of F, corresponding to parameters of the quaternion algebra B with i²=a, j²=b, k=ij=-ji over F. Elements γ of B are then represented by [g₁,g₂,g₃,g₄] with γ = g₁ + g₂i + g₃j + g₄k.
  • γ₁, ..., γₘ are elements of B that are generators for the order.
• eᵢ = <H, G, c, P, t, h> where
  • H is the cardinality of the (right) class set.
  • G is the cardinality of the stable class group.
  • c is 1 if the order has locally free cancellation, and 0 if it is Hermite but does not have locally free cancellation.
  • P is 4, 3, 2, 1, 0, -1 if the order is maximal, hereditary, Eichler, Bass, Gorenstein, or non-Gorenstein (that is P represents the most restrictive of these properties applying to the order)
  • t is the type number, that is, the number of isomorphism classes of orders locally isomorphic to Oᵢ
  • h is the class number of the underlying ring of integers.