Implemented .ramified_places and modified further methods to extend quaternion algebra functionality to number fields
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…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
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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! |
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After fixing some typos and some errors in the docstrings, I believe this should be ready for review now. |
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… `.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
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- 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
5 tasks
vbraun
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Jan 30, 2024
… `.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
vbraun
pushed a commit
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Feb 1, 2024
… `.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
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- 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
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- 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
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Amend: Even more refactoring of `.is_definite()` and `.is_totally_definite()` Amend 2: Corrected backtick error
- 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
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…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
vbraun
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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
mkoeppe
reviewed
Apr 6, 2024
mkoeppe
reviewed
Apr 6, 2024
mkoeppe
reviewed
Apr 6, 2024
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Documentation preview for this PR (built with commit 7c04029; changes) is ready! 🎉 |
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The current test failures seem unrelated (I'm unable to reproduce the failure in |
- Avoids possible future issues caused by rounding precision - Seems to be more efficient Amend: Docstring fix
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There's quite a few failures in the Build & Test using Conda (macos, 3.11), but as far as I can tell they are all not related to this PR. |
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.ramified_placesfor quaternion algebras over number fields. Integrated.ramified_primes()into it in the process..is_division_algebra(),.is_matrix_ring()and.is_isomorphicto use.ramified_placesinstead of.discriminant(), thus extending them to base number fields..discriminant()through.ramified_placessince the original call to.hilbert_conductoralso computed all finite ramified places..is_totally_definite()and movedis_definite().Some more detail:
The new method
.ramified_placesreturns all places at which the quaternion algebraselframifies; this includes the infinite places by default, but can be reduced to only the finite places with the optional parameterinf. The old version of.ramified_primes()from Fixes and simplifications for.ramified_primes(),.discriminant()and.is_isomorphicof quaternion algebras #37164 has been integrated into.ramified_places, thus setting the former up for possible future deprecation; currently it callsself.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_isomorphicnow compares the set of finite ramified places and, unless working over.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 invariantsPossible future work: