Implemented `.ramified_places` and modified further methods to extend quaternion algebra functionality to number fields #37173

S17A05 · 2024-01-27T02:38:31Z

Implemented method .ramified_places for quaternion algebras over number fields. Integrated .ramified_primes() into it in the process.
Modified .is_division_algebra(), .is_matrix_ring() and .is_isomorphic to use .ramified_places instead of .discriminant(), thus extending them to base number fields.
Rerouted .discriminant() through .ramified_places since the original call to .hilbert_conductor also computed all finite ramified places.
Added .is_totally_definite() and moved is_definite().

Some more detail:

The new method .ramified_places returns all places at which the quaternion algebra self ramifies; this includes the infinite places by default, but can be reduced to only the finite places with the optional parameter inf. The old version of .ramified_primes() from Fixes and simplifications for .ramified_primes(), .discriminant() and .is_isomorphic of quaternion algebras #37164 has been integrated into .ramified_places, thus setting the former up for possible future deprecation; currently it calls self.ramified_places(inf=False) for backwards compatibility.
.is_division_algebra() and .is_matrix_ring() now instead check whether the list of ramified places (finite and infinite) is trivial. .is_isomorphic now compares the set of finite ramified places and, unless working over $\mathbb{Q}$, the list of infinite ramified places of both algebras. The latter can be compared as lists since the real embeddings of the number field are sorted independently of each algebras' invariants, but the former (probably) need to be compared as sets since the order of the list depends on the primes above the respective invariants. The docstring of .is_isomorphic (as well as some of the other docstrings) now includes an example of a non-split quaternion algebra with trivial discriminant, namely the algebra with invariants $(-1,-1)$ over the quadratic field $\mathbb{Q}(\sqrt{3})$.

Possible future work:

Extend functionality to all global fields (of characteristic not equal to $2$) [UPDATE: Will be done once both this PR and Implemented the (local) Hilbert symbol for global function fields #37554 have been merged]

…lity to number fields - Implemented method `.ramified_places` for quaternion algebras over number fields. Integrated `.ramified_primes()` into it in the process - Rerouted `.discriminant()` through `.ramified_places` - Modified `.is_division_algebra()`, `.is_matrix_ring()` and `.is_isomorphic` to use `.ramified_places` instead of `.discriminant()`, extending them to base number fields - Added `.is_definite()` and `.is_totally_definite()` methods - Added Voight's book "Quaternion Algebras" to the list of references

S17A05 · 2024-01-27T02:59:02Z

Fixed some whitespaces and blank lines discovered by lint. Also corrected the formatting of references to Voight's book - thanks to @grhkm21 for pointing this out to me!

S17A05 · 2024-01-28T13:03:59Z

After fixing some typos and some errors in the docstrings, I believe this should be ready for review now.

@yyyyx4

… `.discriminant()` and `.is_isomorphic` of quaternion algebras 1. Removed unnecessary product and factorization for `.ramified_primes()` 2. Adapted `is_isomorphic` to reduce unnecessary calculations 3. Fixed a bug in `.discriminant()` and `.ramified_primes()` where rational invariants caused errors 4. Removed `.hilbert_conductor` from `sage.arith.misc` import list and added `.hilbert_symbol` 5. Reduced restriction of `.ramified_primes()` to rational quaternion algebras in docstring in preparation for planned `.ramified_places()` function over number fields (currently being worked on) In more detail: 1. The original workflow for `.ramified_primes()` went as follows: - Call `.discriminant()`, which calls `.hilbert_conductor` - Inside `.hilbert_conductor`, the ramified primes are computed and their product (the discriminant of the quaternion algebra) is returned - Finally, factor the discriminant back into its prime factors Hence we have a redundant product and, more crucially, a redundant prime factorization. This fix modifies `.ramified_primes()` to instead directly build the list computed in `.hilbert_conductor()` (up to a bug fix described in 3.) and return it; the list might not always be sorted by magnitude of primes, so an optional argument `sorted` (set to `False` by default) allows to enforce this (small to large) sorting. Furthermore, `.discriminant()` has been adapted to directly take the product of the list returned by `.ramified_primes()` (only in the rational case, for now - see 5.) 2. Since the `.discriminant()`-function needs to compute all (finite) ramified primes (this was also true before this PR, it was just hidden inside `.hilbert_conductor()` instead), the function `.is_isomorphic()` now compares the unsorted lists of finite ramified primes to decide whether two rational quaternion algebras are isomorphic. 3. The function `sage.arith.misc.hilbert_conductor` requires its arguments to be integers (to create certain lists of prime divisors); since it was originally used to determine the discriminant (and, as explained in 1., the ramified primes), it raises an error when the invariants are proper rational numbers. To get around the analogous error for the method `.hilbert_symbol`, we instead look at the numerators and denominators of both invariants separately, using the fact that we can (purely on a mathematical level) rescale both invariants by the squares of their respective denominators without leaving the isomorphism class of the algebra. 4. The only call to `sage.arith.misc.hilbert_conductor` in quaternion_algebra.py was given in the old computation of the discriminant (the other `.hilbert_conductor` in the code, also in `.discriminant()`, refers to the one in `sage.rings.number_field`), so it was removed from the import list. The new approach to `.ramified_primes()` requires `sage.arith.misc.hilbert_symbol`, which was added to the import list. 5. As of now the `.ramified_primes()`-method is only supported for rational quaternion algebras. I'm currently working on a version over number fields, but once it works correctly this will be implemented as a new function `.ramified_places` (Update: see sagemath#37173) ~~to distinguish between different formats (prime numbers vs ideals) over $\mathbb{Q}$~~ (Update: this wasn't really feasible, see the issues discussed in sagemath#7596; thanks to @yyyyx4 for pointing me towards this discussion) ~~and, furthermore,~~ to not cause confusion using the term "primes" for the Archimedean real places where a quaternion algebra might ramify. Hence the implementation restriction in the docstring of `.ramified_primes()` was removed, but the method still throws a ValueError if not called with a quaternion algebra defined over the rational numbers. #sd123 URL: sagemath#37164 Reported by: Sebastian Spindler Reviewer(s): grhkm21, Sebastian Spindler

- Removed greek letter alpha in docstrings in hopes of this fixing infinite build loops - Updated `.is_definite()` to fit PR sagemath#37173 (up to the reference to Voight's book) - Other small modifications of docstrings, comments and error warnings - Slightly cleaned up code with respect to intermediately defined variables

@yyyyx4

… `.discriminant()` and `.is_isomorphic` of quaternion algebras 1. Removed unnecessary product and factorization for `.ramified_primes()` 2. Adapted `is_isomorphic` to reduce unnecessary calculations 3. Fixed a bug in `.discriminant()` and `.ramified_primes()` where rational invariants caused errors 4. Removed `.hilbert_conductor` from `sage.arith.misc` import list and added `.hilbert_symbol` 5. Reduced restriction of `.ramified_primes()` to rational quaternion algebras in docstring in preparation for planned `.ramified_places()` function over number fields (currently being worked on) In more detail: 1. The original workflow for `.ramified_primes()` went as follows: - Call `.discriminant()`, which calls `.hilbert_conductor` - Inside `.hilbert_conductor`, the ramified primes are computed and their product (the discriminant of the quaternion algebra) is returned - Finally, factor the discriminant back into its prime factors Hence we have a redundant product and, more crucially, a redundant prime factorization. This fix modifies `.ramified_primes()` to instead directly build the list computed in `.hilbert_conductor()` (up to a bug fix described in 3.) and return it; the list might not always be sorted by magnitude of primes, so an optional argument `sorted` (set to `False` by default) allows to enforce this (small to large) sorting. Furthermore, `.discriminant()` has been adapted to directly take the product of the list returned by `.ramified_primes()` (only in the rational case, for now - see 5.) 2. Since the `.discriminant()`-function needs to compute all (finite) ramified primes (this was also true before this PR, it was just hidden inside `.hilbert_conductor()` instead), the function `.is_isomorphic()` now compares the unsorted lists of finite ramified primes to decide whether two rational quaternion algebras are isomorphic. 3. The function `sage.arith.misc.hilbert_conductor` requires its arguments to be integers (to create certain lists of prime divisors); since it was originally used to determine the discriminant (and, as explained in 1., the ramified primes), it raises an error when the invariants are proper rational numbers. To get around the analogous error for the method `.hilbert_symbol`, we instead look at the numerators and denominators of both invariants separately, using the fact that we can (purely on a mathematical level) rescale both invariants by the squares of their respective denominators without leaving the isomorphism class of the algebra. 4. The only call to `sage.arith.misc.hilbert_conductor` in quaternion_algebra.py was given in the old computation of the discriminant (the other `.hilbert_conductor` in the code, also in `.discriminant()`, refers to the one in `sage.rings.number_field`), so it was removed from the import list. The new approach to `.ramified_primes()` requires `sage.arith.misc.hilbert_symbol`, which was added to the import list. 5. As of now the `.ramified_primes()`-method is only supported for rational quaternion algebras. I'm currently working on a version over number fields, but once it works correctly this will be implemented as a new function `.ramified_places` (Update: see sagemath#37173) ~~to distinguish between different formats (prime numbers vs ideals) over $\mathbb{Q}$~~ (Update: this wasn't really feasible, see the issues discussed in sagemath#7596; thanks to @yyyyx4 for pointing me towards this discussion) ~~and, furthermore,~~ to not cause confusion using the term "primes" for the Archimedean real places where a quaternion algebra might ramify. Hence the implementation restriction in the docstring of `.ramified_primes()` was removed, but the method still throws a ValueError if not called with a quaternion algebra defined over the rational numbers. #sd123 URL: sagemath#37164 Reported by: Sebastian Spindler Reviewer(s): grhkm21, Sebastian Spindler

@yyyyx4

… `.discriminant()` and `.is_isomorphic` of quaternion algebras 1. Removed unnecessary product and factorization for `.ramified_primes()` 2. Adapted `is_isomorphic` to reduce unnecessary calculations 3. Fixed a bug in `.discriminant()` and `.ramified_primes()` where rational invariants caused errors 4. Removed `.hilbert_conductor` from `sage.arith.misc` import list and added `.hilbert_symbol` 5. Reduced restriction of `.ramified_primes()` to rational quaternion algebras in docstring in preparation for planned `.ramified_places()` function over number fields (currently being worked on) In more detail: 1. The original workflow for `.ramified_primes()` went as follows: - Call `.discriminant()`, which calls `.hilbert_conductor` - Inside `.hilbert_conductor`, the ramified primes are computed and their product (the discriminant of the quaternion algebra) is returned - Finally, factor the discriminant back into its prime factors Hence we have a redundant product and, more crucially, a redundant prime factorization. This fix modifies `.ramified_primes()` to instead directly build the list computed in `.hilbert_conductor()` (up to a bug fix described in 3.) and return it; the list might not always be sorted by magnitude of primes, so an optional argument `sorted` (set to `False` by default) allows to enforce this (small to large) sorting. Furthermore, `.discriminant()` has been adapted to directly take the product of the list returned by `.ramified_primes()` (only in the rational case, for now - see 5.) 2. Since the `.discriminant()`-function needs to compute all (finite) ramified primes (this was also true before this PR, it was just hidden inside `.hilbert_conductor()` instead), the function `.is_isomorphic()` now compares the unsorted lists of finite ramified primes to decide whether two rational quaternion algebras are isomorphic. 3. The function `sage.arith.misc.hilbert_conductor` requires its arguments to be integers (to create certain lists of prime divisors); since it was originally used to determine the discriminant (and, as explained in 1., the ramified primes), it raises an error when the invariants are proper rational numbers. To get around the analogous error for the method `.hilbert_symbol`, we instead look at the numerators and denominators of both invariants separately, using the fact that we can (purely on a mathematical level) rescale both invariants by the squares of their respective denominators without leaving the isomorphism class of the algebra. 4. The only call to `sage.arith.misc.hilbert_conductor` in quaternion_algebra.py was given in the old computation of the discriminant (the other `.hilbert_conductor` in the code, also in `.discriminant()`, refers to the one in `sage.rings.number_field`), so it was removed from the import list. The new approach to `.ramified_primes()` requires `sage.arith.misc.hilbert_symbol`, which was added to the import list. 5. As of now the `.ramified_primes()`-method is only supported for rational quaternion algebras. I'm currently working on a version over number fields, but once it works correctly this will be implemented as a new function `.ramified_places` (Update: see sagemath#37173) ~~to distinguish between different formats (prime numbers vs ideals) over $\mathbb{Q}$~~ (Update: this wasn't really feasible, see the issues discussed in sagemath#7596; thanks to @yyyyx4 for pointing me towards this discussion) ~~and, furthermore,~~ to not cause confusion using the term "primes" for the Archimedean real places where a quaternion algebra might ramify. Hence the implementation restriction in the docstring of `.ramified_primes()` was removed, but the method still throws a ValueError if not called with a quaternion algebra defined over the rational numbers. #sd123 URL: sagemath#37164 Reported by: Sebastian Spindler Reviewer(s): grhkm21, Sebastian Spindler

- Removed greek letter alpha in docstrings in hopes of this fixing infinite build loops - Updated `.is_definite()` to fit PR sagemath#37173 (up to the reference to Voight's book) - Other small modifications of docstrings, comments and error warnings - Slightly cleaned up code with respect to intermediately defined variables

Amend: Refactored `.is_definite()`

Amend: Even more refactoring of `.is_definite()` and `.is_totally_definite()` Amend 2: Corrected backtick error

grhkm21

some reviews

src/sage/algebras/quatalg/quaternion_algebra.py

- Rewrote some docstring descriptions - Broke apart examples into multiple blocks, with added flavor text - Modified docstring examples to emphasize certain features more - Refactored some larger descriptions to use bullet point lists Amend 1: Fixed missing space before NOTE Amend 2: Fixed indent of second line in bullet point list in `.discriminant()`

- Added missing check that the base field of a totally definite algebra needs to be totally real - Moved note in `.ramified_places` below input and moved the interal info on initial choice of finite primes to code comments

…unction fields Implemented the computation of the (local) Hilbert symbol at a place of a global function field of odd characteristic following formula 12.4.10 in John Voight's "Quaternion Algebras" (in preparation of extending some quaternion algebra functionality to these global function fields). The global Hilbert symbol will be implemented via a call to `.is_matrix_ring()` of the appropriate quaternion algebra once the latter method has been extended using an extension of PR sagemath#37173, which will be done once both this PR and the referenced PR have been merged. URL: sagemath#37554 Reported by: Sebastian A. Spindler Reviewer(s): Kwankyu Lee, Sebastian A. Spindler

Updated details for John Voight's book "Quaternion Algebras" in the list of references and modified some docstrings in `quaternion_algebra.py`. Split off from sagemath#37173. URL: sagemath#37557 Reported by: Sebastian A. Spindler Reviewer(s): Travis Scrimshaw

src/sage/algebras/quatalg/quaternion_algebra.py

github-actions · 2024-04-06T17:01:53Z

Documentation preview for this PR (built with commit 6426296; changes) is ready! 🎉
This preview will update shortly after each push to this PR.

S17A05 · 2024-04-13T09:03:47Z

The current test failures seem unrelated (I'm unable to reproduce the failure in graphs.generators.random, though); also Build&Test/build seems to test all files as new, at least on merging the new beta version.

- Avoids possible future issues caused by rounding precision - Seems to be more efficient Amend: Docstring fix

S17A05 mentioned this pull request Jan 27, 2024

Fixes and simplifications for .ramified_primes(), .discriminant() and .is_isomorphic of quaternion algebras #37164

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Fixed lint issues and refernce formatting

97f1257

yyyyx4 added p: major / 3 t: enhancement s: needs review c: algebra labels Jan 27, 2024

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Fixed typo and corrected/modified docstrings

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Eloitor mentioned this pull request Jan 29, 2024

Constructor quaternion algebra over number fields from ramification #37189

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Merge branch 'develop' into ramified_places

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Moved code in .is_isomorphic, edited some docstrings

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Merge branch 'develop' into ramified_places

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Merge branch 'develop' into ramified_places

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Merge branch 'develop' into ramified_places

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Merge branch 'develop' into ramified_places

796cd1d

Adapted .ramified_places slightly, removed docstring modifications

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S17A05 mentioned this pull request Mar 6, 2024

Modified quaternion algebra documentation #37557

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Small doctest fix

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Merge branch 'develop' into ramified_places

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Remove redundant copy of .is_definite()

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Some doctest refactoring

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Amend: Refactored `.is_definite()`

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More small refactoring

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Amend: Even more refactoring of `.is_definite()` and `.is_totally_definite()` Amend 2: Corrected backtick error

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Small docstring modifications

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Corrected .is_totally_definite(), moved .. NOTE

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- Added missing check that the base field of a totally definite algebra needs to be totally real - Moved note in `.ramified_places` below input and moved the interal info on initial choice of finite primes to code comments

Added missing type descriptor for inf argument

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- Avoids possible future issues caused by rounding precision - Seems to be more efficient Amend: Docstring fix

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Implemented `.ramified_places` and modified further methods to extend quaternion algebra functionality to number fields #37173

Implemented `.ramified_places` and modified further methods to extend quaternion algebra functionality to number fields #37173

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Implemented .ramified_places and modified further methods to extend quaternion algebra functionality to number fields #37173

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Implemented .ramified_places and modified further methods to extend quaternion algebra functionality to number fields #37173

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Implemented `.ramified_places` and modified further methods to extend quaternion algebra functionality to number fields #37173

Implemented `.ramified_places` and modified further methods to extend quaternion algebra functionality to number fields #37173

S17A05 commented Jan 27, 2024 •

edited

github-actions bot commented Apr 6, 2024 •

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