Construct a quaternion algebra over a number field by specifying its ramification

[mmasdeu]

There is a need for the constructor QuaternionAlgebra to take as input a number field F and a ramification (specified by an ideal D in F and a set of real places S of F) and which constructs a quaternion algebra over F with ramified precisely at the primes dividing D and at the places in S.

We have code to add here, which is now being tested and documented.

CC: @tornaria

Component: number theory

Keywords: quaternion algebra, ramification

Author: Marc Masdeu, Xavier Guitart

Branch/Commit: u/mmasdeu/16948-review @ fcf1a69

Issue created by migration from https://trac.sagemath.org/ticket/16948

[mmasdeu]

Commit: 2d2cfa9

[mmasdeu]

comment:1

The code passes all doctests and runs in reasonable time. I think it is ready for review.

[mmasdeu]

Branch: u/mmasdeu/16948-quatalg-from-ramification

[jdemeyer]

comment:2

Just a remark: this functionality is in PARI now, so you could just try to wrap PARI.

[fchapoton]

comment:3

Please use python3 raise syntax: raise Error('msg')

see patchbot report in plugin raise_statement for the lines with a bad raise syntax

[fchapoton]

New commits:

de9cb12	Merge branch 'u/mmasdeu/16948-quatalg-from-ramification' into 6.8
2c21a0b	correct py3 syntax for raise

[fchapoton]

Changed branch from u/mmasdeu/16948-quatalg-from-ramification to public/ticket/16948

[fchapoton]

Changed commit from 2d2cfa9 to 2c21a0b

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from 2c21a0b to b29e4f8

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Branch pushed to git repo; I updated commit sha1. New commits:

b29e4f8

trac #16948 some details in code and documentation

[mmasdeu]

Changed commit from b29e4f8 to e5c0916

[mmasdeu]

New commits:

e5c0916

Added missing docstrings in number_field.py so that patchbot is happy.

[mmasdeu]

Changed branch from public/ticket/16948 to u/mmasdeu/16948

[roed314]

comment:9

This no longer merges with sage 7.2 (according to trac).

[fchapoton]

comment:10

merge done, was easy

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New commits:

fe1008a

Merge branch 'u/mmasdeu/16948' in 7.3.b7

[fchapoton]

Changed commit from e5c0916 to fe1008a

[fchapoton]

Changed branch from u/mmasdeu/16948 to public/16948

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from fe1008a to cdae032

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Branch pushed to git repo; I updated commit sha1. New commits:

cdae032

Merge branch 'public/16948' in 7.4.b5

[nsirolli]

comment:14

A small remark regarding the ValueError in line 636: the exponents in the discriminant factorization should be equal to 1, and not just odd.

By the way, is there a reason why the input is an ideal of F instead of a list of prime ideals of F? If I want to construct a quaternion algebra with a prescribed ramification set of finite primes, I'd have to use their product as input, and then the constructor would have to factor it back (see line 634).

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from cdae032 to 5284a37

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Branch pushed to git repo; I updated commit sha1. New commits:

5284a37

Merge branch 'public/16948' in 7.6.b4

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Branch pushed to git repo; I updated commit sha1. New commits:

73272e3

trac 16948 fixing bad merge

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from 5284a37 to 73272e3

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from 73272e3 to 355c58d

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Branch pushed to git repo; I updated commit sha1. New commits:

355c58d

trac 16498 some details in weak_approximation

[mmasdeu]

Changed commit from 355c58d to fcf1a69

[mmasdeu]

comment:18

Thanks for the comments! Please take a look at the changes I have introduced.

Replying to @nsirolli:

A small remark regarding the ValueError in line 636: the exponents in the discriminant factorization should be equal to 1, and not just odd.

By the way, is there a reason why the input is an ideal of F instead of a list of prime ideals of F? If I want to construct a quaternion algebra with a prescribed ramification set of finite primes, I'd have to use their product as input, and then the constructor would have to factor it back (see line 634).

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New commits:

fcf1a69

Trac 16948: adopted changes proposed by nsirolli.

[mmasdeu]

Changed branch from public/16948 to u/mmasdeu/16948-review

[nsirolli]

comment:19

Thanks for accepting my proposal.

Now the new input format, QuaternionAlgebra(K, "list of primes", S) is quite different from the one you are extending to number fields other than QQ, QuaternionAlgebra(D). I feel this controversy should be avoided. The problem could be addressed by letting this new constructor accept both a discriminant and a list of primes as input. A more radical solution would be to change the classical QuaternionAlgebra(D) to take a list of primes as input, but I guess that it could cause trouble.

Besides this, here are some remarks about the changes introduced in this commit:

• Line 245 of quaternion_algebra.py should be fixed, since it does not reflect the new input format.

• The documentation needs to take account of these changes; see lines 92-94, 605-607, 625 of quaternion_algebra.py and line 1835 of number_fields.py

• Line 589 of quaternion_algebra.py and lines 1804, 1815, 1817 of number_fields.py: shouldn't the name of the list of prime ideales be L or P instead of I?

[fchapoton]

comment:20

needs rebase, does not apply