Constructor quaternion algebra over number fields from ramification #37189

Eloitor · 2024-01-29T13:26:17Z

This PR adds a new input format for the constructor of quaternion algebras.

It uses PARI alginit function to construct one form the ramification information.

Fixes #16948

The title is concise, informative, and self-explanatory.
The description explains in detail what this PR is about.
I have linked a relevant issue or discussion.
I have created tests covering the changes.
I have updated the documentation accordingly.

Eloitor · 2024-01-29T13:34:00Z

I've seen this other PR #37173 that implements .ramified_places().

S17A05 · 2024-01-29T14:58:13Z

I'll review this once the necessary checks have completed.

src/sage/algebras/quatalg/quaternion_algebra.py

S17A05 · 2024-01-30T18:28:08Z

Let me know (e.g. by commenting here) once you think this is ready for another full review; in the meantime, Lint recognized some whitespaces in the blank line 302.

Eloitor · 2024-01-31T12:25:10Z

ready for review

S17A05 · 2024-01-31T13:42:38Z

ready for review

Great, I'll review it again when I find the time for it; this will probably not be before Friday though.

src/sage/algebras/quatalg/quaternion_algebra.py

AurelPage · 2024-01-31T16:03:57Z

Did you check that the real embeddings are treated in the same order as in .real_embeddings()? This does not seem automatic as the real embeddings are computed from the roots of the defining polynomial, not from PARI, but you are simply forwarding the second argument to PARI's alginit. Maybe this should be checked, documented, and tested (by checking the signs of the embeddings of a,b).

github-actions · 2024-01-31T18:11:33Z

Documentation preview for this PR (built with commit 6808417; changes) is ready! 🎉
This preview will update shortly after each push to this PR.

S17A05

Sorry for the delay. For the future, please don't force push big changes and instead put them into a new commit, otherwise it makes the review process a lot harder because everything has to be re-checked again.

S17A05 · 2024-02-11T19:49:49Z

src/sage/algebras/quatalg/quaternion_algebra.py

+            if category(K) is not NumberFields():
+                raise ValueError("quaternion algbera must be defined over a number field")


The check category(QQ) is not NumberFields() returns True; you should instead place this check after handling the case of the rational field.

sage: category(QQ) Join of Category of number fields and Category of quotient fields and Category of metric spaces

Is it possible to use this to check if K is QQ or a number field?

It might be possible, but even then I would advise against it: You need to handle the rational field separately anyways (I ran into the same issue in #37173), and up to the check is_RationalField(K) I don't see any code that doesn't work for either setting; hence it is probably easier to place the above category check in the else block after the treatment of the rational field case.

src/sage/algebras/quatalg/quaternion_algebra.py

S17A05 · 2024-02-11T20:16:25Z

src/sage/algebras/quatalg/quaternion_algebra.py

+                    A = pari(K).alginit([2, [fin_places_pari, [QQ(1/2)] * len(fin_places_pari)], arg2], maxord=0)
+                    a = K(A.algsplittingfield().disc()[1])
+                    b = K(A.algb())


Unfortunately there is a major issue here that @AurelPage correctly detected: The order in which the real embeddings are computed in PARI is indeed not the same as the order of embeddings given by .real_embeddings().

Here is an example:
sage: K.<v> = NumberField(-3x^5 - 11x^4 - 4x^3 + 1)
sage: A = QuaternionAlgebra(K, [], [1/2, 0, 1/2])
Quaternion Algebra (-3v^4 - 14v^3 - 15v^2 - 4v, -8) with base ring Number Field in v with defining polynomial -3x^5 - 11x^4 - 4x^3 + 1

Using the ramified_places code from #37173, one can check that this algebra ramifies at the two embeddings sending $v$ to $-0.738229069310693$ and to $0.467355922387407$. In K.real_embeddings() these are the second and the third embeddings, respectively, but the desired output should be ramified at the third embedding as well as the first embedding sending $v$ to $-3.25393276038648$.

This could be fixed by first precomputing the order in which PARI lists the embeddings, and then permuting the list of local Archimedean invariants accordingly. I am not sure if this is feasible, and maybe @AurelPage has a better idea on how to work around this discrepancy between Sage and PARI.

@S17A05 It is not difficult to find the bijection. Consider a generator of the field $K$, and take an $\varepsilon$ that separates its real embeddings. Consider the sequences of real embeddings computed by sage and pari respectively, at precision smaller than $\varepsilon$. Sort them both using indirect sort (with the permutations) and compose the permutations in the right way.

However, this procedure should probably not be hard-coded in the middle of the quaternion algebras code, but should be put in a place that ensures it can be reused. Maybe it is already somewhere? I don't know where that should be.

@S17A05 It is not difficult to find the bijection. Consider a generator of the field K, and take an ε that separates its real embeddings. Consider the sequences of real embeddings computed by sage and pari respectively, at precision smaller than ε. Sort them both using indirect sort (with the permutations) and compose the permutations in the right way.

However, this procedure should probably not be hard-coded in the middle of the quaternion algebras code, but should be put in a place that ensures it can be reused. Maybe it is already somewhere? I don't know where that should be.

I don't think this is implemented yet. I'm not sure if this is the most elegant solution, but I would suggest adding an optional boolean parameter (e.g. called pari_perm, set to False by default) to the .real_embeddings()-method of a number field such that, if the parameter is set to True, the method returns both the real embeddings and a permutation of the indices that correctly re-indexes the embeddings to be in the order they are computed in in PARI.

S17A05 · 2024-02-11T20:28:03Z

src/sage/algebras/quatalg/quaternion_algebra.py

+        sage: QuaternionAlgebra(QQ, [(2), (3)], [0])
+        Quaternion Algebra (-1, 3) with base ring Rational Field
+        sage: QuaternionAlgebra(QQ, [(2), (3)], [1/2])
+        Traceback (most recent call last):
+        ...
+        ValueError: Quaternion algebra over the rationals must have an even number of ramified places
+        sage: K.<w> = NumberField(x^2-x-1)
+        sage: P = K.prime_above(2)
+        sage: Q = K.prime_above(3)
+        sage: A = QuaternionAlgebra(K, [P,Q], [0,0])
+        sage: A.discriminant()
+        Fractional ideal (6)
+        sage: A = QuaternionAlgebra(K, [P,Q], [1/2,0])
+        Traceback (most recent call last):
+        ...
+        ValueError: Quaternion algebra over the rationals must have an even number of ramified places


The error warning in line 211 needs to be adapted, see the comment on line 314. Also, once #37173 has been reviewed and merged it would be nice to adapt the examples given here to show that the constructor gives an algebra with the desired ramification.

S17A05 · 2024-02-13T19:19:47Z

The modifications of the constructor also seem to mess up some examples according to the details of the Build & Test / build failure, and on certain inputs there seems to be a SignalError: Segmentation fault thrown by A.algsplittingfield().disc(); we can look at these things once the code gives the correct mathematical result, though.

S17A05 · 2024-04-14T01:15:14Z

@Eloitor I have found a fix for the communication between Sage and PARI - the main problem is that PARI optimizes the defining polynomial, which mixes the embeddings up a bit. By mimicking this optimization in Sage, one can easily compute the permutation there with a simple sorting (since the real embeddings in Sage are sorted already), though. If you want to, I can implement the solution if you give me the necessary access to your branch (I could also fix the merge conflicts and the issue raised in the other review comment).

Eloitor · 2024-04-14T08:53:45Z

This is great news!! Please go ahead. You should have access now :)

S17A05 · 2024-04-14T21:26:42Z

Should be fixed; the new test examples aren't deterministic though (it can return different, but necessarily isomorphic algebras), so I've made sure that they are not actually tested right now - I'll make them into proper tests once #37173 has been merged. Also quickly mentioning myself here (@S17A05) so that the PR is a bit easier to find for me :)

Amend: Round 2

S17A05 · 2024-10-15T14:36:25Z

The errors in the tests seemed to be outdated, so I updated the branch to re-trigger the tests

S17A05 · 2024-10-16T10:59:57Z

The errors in the tests seemed to be outdated, so I updated the branch to re-trigger the tests

Whoops, turns out I was looking at the wrong thing - the new commit should fix the errors.

S17A05 · 2024-10-16T12:11:15Z

The timeout in src/sage/rings/tests.py seems unrelated, but should probably be investigated either way.

Eloitor force-pushed the quatalg_from_ramification branch 5 times, most recently from 8b4220c to 44fc263 Compare January 29, 2024 14:13

Eloitor changed the title ~~Constructor quaternion algebra over number fields form ramification~~ Constructor quaternion algebra over number fields from ramification Jan 29, 2024

S17A05 reviewed Jan 29, 2024

View reviewed changes

Eloitor force-pushed the quatalg_from_ramification branch 5 times, most recently from 5be2ea6 to d96f42e Compare January 30, 2024 11:52

S17A05 added t: enhancement p: minor / 4 s: needs review c: algebra labels Jan 30, 2024

S17A05 self-requested a review January 30, 2024 14:47

Eloitor force-pushed the quatalg_from_ramification branch 3 times, most recently from 882ba53 to 55809f3 Compare January 31, 2024 12:23

Eloitor force-pushed the quatalg_from_ramification branch 3 times, most recently from 3be5ad7 to 0844c28 Compare January 31, 2024 12:33

AurelPage reviewed Jan 31, 2024

View reviewed changes

src/sage/algebras/quatalg/quaternion_algebra.py Outdated Show resolved Hide resolved

Eloitor force-pushed the quatalg_from_ramification branch from 0844c28 to 2657bae Compare January 31, 2024 15:40

S17A05 suggested changes Feb 11, 2024

View reviewed changes

S17A05 added s: needs work p: major / 3 and removed s: needs review p: minor / 4 labels Feb 11, 2024

S17A05 force-pushed the quatalg_from_ramification branch from 118178d to d4bc980 Compare April 14, 2024 21:22

S17A05 force-pushed the quatalg_from_ramification branch from e56386d to 2ae8288 Compare April 14, 2024 21:34

S17A05 added s: needs review and removed s: needs work labels Apr 14, 2024

vbraun force-pushed the develop branch from eba5e19 to e5f42fa Compare June 3, 2024 22:15

Eloitor force-pushed the quatalg_from_ramification branch from d96b447 to b025292 Compare October 3, 2024 08:14

Eloitor and others added 6 commits October 3, 2024 13:18

Resolve conflict

97a0a4b

Fix error messages in quaternion_algebra.pỳ

bdb792c

Resolve conflict

fea271f

Lint fixes

94e112d

Amend: Round 2

Small docstring fixes

b025292

Merge branch 'develop' into quatalg_from_ramification

efbf567

Replace is_RationalField by isinstance(...)

0c80936

Merge branch 'develop' into quatalg_from_ramification

6808417

Provide feedback

Saved searches

Use saved searches to filter your results more quickly

Constructor quaternion algebra over number fields from ramification #37189

Constructor quaternion algebra over number fields from ramification #37189

Eloitor commented Jan 29, 2024

Eloitor commented Jan 29, 2024

S17A05 commented Jan 29, 2024

S17A05 commented Jan 30, 2024 •

edited

Loading

Eloitor commented Jan 31, 2024

S17A05 commented Jan 31, 2024

AurelPage commented Jan 31, 2024

github-actions bot commented Jan 31, 2024 •

edited

Loading

S17A05 left a comment

S17A05 Feb 11, 2024

Eloitor Feb 12, 2024

S17A05 Feb 13, 2024 •

edited

Loading

S17A05 Feb 11, 2024 •

edited

Loading

AurelPage Feb 12, 2024

S17A05 Feb 13, 2024

S17A05 Feb 11, 2024

S17A05 commented Feb 13, 2024

S17A05 commented Apr 14, 2024 •

edited

Loading

Eloitor commented Apr 14, 2024

S17A05 commented Apr 14, 2024 •

edited

Loading

S17A05 commented Oct 15, 2024

S17A05 commented Oct 16, 2024

S17A05 commented Oct 16, 2024

		if category(K) is not NumberFields():
		raise ValueError("quaternion algbera must be defined over a number field")

Constructor quaternion algebra over number fields from ramification #37189

Are you sure you want to change the base?

Constructor quaternion algebra over number fields from ramification #37189

Conversation

Eloitor commented Jan 29, 2024

Eloitor commented Jan 29, 2024

S17A05 commented Jan 29, 2024

S17A05 commented Jan 30, 2024 • edited Loading

Eloitor commented Jan 31, 2024

S17A05 commented Jan 31, 2024

AurelPage commented Jan 31, 2024

github-actions bot commented Jan 31, 2024 • edited Loading

S17A05 left a comment

Choose a reason for hiding this comment

S17A05 Feb 11, 2024

Choose a reason for hiding this comment

Eloitor Feb 12, 2024

Choose a reason for hiding this comment

S17A05 Feb 13, 2024 • edited Loading

Choose a reason for hiding this comment

S17A05 Feb 11, 2024 • edited Loading

Choose a reason for hiding this comment

AurelPage Feb 12, 2024

Choose a reason for hiding this comment

S17A05 Feb 13, 2024

Choose a reason for hiding this comment

S17A05 Feb 11, 2024

Choose a reason for hiding this comment

S17A05 commented Feb 13, 2024

S17A05 commented Apr 14, 2024 • edited Loading

Eloitor commented Apr 14, 2024

S17A05 commented Apr 14, 2024 • edited Loading

S17A05 commented Oct 15, 2024

S17A05 commented Oct 16, 2024

S17A05 commented Oct 16, 2024

S17A05 commented Jan 30, 2024 •

edited

Loading

github-actions bot commented Jan 31, 2024 •

edited

Loading

S17A05 Feb 13, 2024 •

edited

Loading

S17A05 Feb 11, 2024 •

edited

Loading

S17A05 commented Apr 14, 2024 •

edited

Loading

S17A05 commented Apr 14, 2024 •

edited

Loading