Implemented the (local) Hilbert symbol for global function fields #37554

S17A05 · 2024-03-06T13:49:57Z

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 #37173, which will be done once both this PR and the referenced PR have been merged.

Amends: Changed some documentation, adapted characteristic check to check ==2 instead of <= 2

Amend: Added missing multiline continuation markers

- Simplified return command, removed `ZZ` from import again - Return `0` instead of an error for zero inputs

src/sage/rings/function_field/function_field.py

- Fixed errors caused by unexpected behavior of reduction map - Added examples for bimultiplicativity - Corrected docstring

Amend: Fixed weird spacing

src/sage/rings/function_field/function_field.py

- Fixed Lint issue - Implemented reviewer feedback - Added checks for symmetry

kwankyu · 2024-03-27T11:06:20Z

I fixed typos and refined the docstring. If you pardon me, please consider:

diff --git a/src/sage/rings/function_field/function_field.py b/src/sage/rings/function_field/function_field.py
index d1652873c71..4f91e51c111 100644
--- a/src/sage/rings/function_field/function_field.py
+++ b/src/sage/rings/function_field/function_field.py
@@ -1229,23 +1229,10 @@ class FunctionField(Field):
 
     def hilbert_symbol(self, a, b, P):
         r"""
-        Return the Hilbert symbol `(a,b)_{F_P}` (where `F_P` is the
-        completion of this function field at the place `P`).
+        Return the Hilbert symbol `(a,b)_{F_P}` for the local field `F_P`.
 
-        The Hilbert symbol `(a,b)_{F_P}` is `0` if one of `a` or `b` is
-        zero. Otherwise it takes the value `1` if  the quaternion algebra
-        defined by `(a,b)` over `F_P` is split, and `-1` if it is division.
-
-        For the valuation `\nu` belonging to the completion of the function
-        field at `P`, we compute the valuations `\nu(a)` and `\nu(b)` as well
-        as elements `a_0` and `b_0` such that, for a uniformizer `\pi` of
-        `\nu`, the elememts `a*\pi^{-\nu(a))}` and `a_0` respectively the
-        elements `b*\pi^{-\nu(b)}` and `b_0` are congruent modulo `\pi`.
-        Motivated by formula 12.4.10 in [Voi2021]_.
-
-        Currently only tested for function fields separable over their base
-        since places are not fully supported for other function fields. Only
-        implemented for global function fields of odd characteristic.
+        The local field `F_P` is the completion of this function field `F` at
+        the place `P`.
 
         INPUT:
 
@@ -1253,6 +1240,23 @@ class FunctionField(Field):
 
         - ``P`` -- a place of this function field
 
+        The Hilbert symbol `(a,b)_{F_P}` is `0` if `a` or `b` is zero.
+        Otherwise it takes the value `1` if  the quaternion algebra defined by
+        `(a,b)` over `F_P` is split, and `-1` if it is a division ring.
+
+        ALGORITHM:
+
+        For the valuation `\nu = \nu_P` of `F`, we compute the valuations
+        `\nu(a)` and `\nu(b)` as well as elements `a_0` and `b_0` of the
+        residue field such that for a uniformizer `\pi` at `P`,
+        `a\pi^{-\nu(a)}` and `a_0` respectively `b\pi^{-\nu(b)}` and `b_0` are
+        congruent modulo `\pi`.  Then the Hilbert symbol is computed by the
+        formula 12.4.10 in [Voi2021]_.
+
+        Currently only implemented for global function fields of odd
+        characteristic since places are not fully supported for other function
+        fields.
+
         EXAMPLES::

S17A05 · 2024-03-27T11:16:23Z

I fixed typos and refined the docstring. If you pardon me, please consider:

diff --git a/src/sage/rings/function_field/function_field.py b/src/sage/rings/function_field/function_field.py
index d1652873c71..4f91e51c111 100644
--- a/src/sage/rings/function_field/function_field.py
+++ b/src/sage/rings/function_field/function_field.py
@@ -1229,23 +1229,10 @@ class FunctionField(Field):
 
     def hilbert_symbol(self, a, b, P):
         r"""
-        Return the Hilbert symbol `(a,b)_{F_P}` (where `F_P` is the
-        completion of this function field at the place `P`).
+        Return the Hilbert symbol `(a,b)_{F_P}` for the local field `F_P`.
 
-        The Hilbert symbol `(a,b)_{F_P}` is `0` if one of `a` or `b` is
-        zero. Otherwise it takes the value `1` if  the quaternion algebra
-        defined by `(a,b)` over `F_P` is split, and `-1` if it is division.
-
-        For the valuation `\nu` belonging to the completion of the function
-        field at `P`, we compute the valuations `\nu(a)` and `\nu(b)` as well
-        as elements `a_0` and `b_0` such that, for a uniformizer `\pi` of
-        `\nu`, the elememts `a*\pi^{-\nu(a))}` and `a_0` respectively the
-        elements `b*\pi^{-\nu(b)}` and `b_0` are congruent modulo `\pi`.
-        Motivated by formula 12.4.10 in [Voi2021]_.
-
-        Currently only tested for function fields separable over their base
-        since places are not fully supported for other function fields. Only
-        implemented for global function fields of odd characteristic.
+        The local field `F_P` is the completion of this function field `F` at
+        the place `P`.
 
         INPUT:
 
@@ -1253,6 +1240,23 @@ class FunctionField(Field):
 
         - ``P`` -- a place of this function field
 
+        The Hilbert symbol `(a,b)_{F_P}` is `0` if `a` or `b` is zero.
+        Otherwise it takes the value `1` if  the quaternion algebra defined by
+        `(a,b)` over `F_P` is split, and `-1` if it is a division ring.
+
+        ALGORITHM:
+
+        For the valuation `\nu = \nu_P` of `F`, we compute the valuations
+        `\nu(a)` and `\nu(b)` as well as elements `a_0` and `b_0` of the
+        residue field such that for a uniformizer `\pi` at `P`,
+        `a\pi^{-\nu(a)}` and `a_0` respectively `b\pi^{-\nu(b)}` and `b_0` are
+        congruent modulo `\pi`.  Then the Hilbert symbol is computed by the
+        formula 12.4.10 in [Voi2021]_.
+
+        Currently only implemented for global function fields of odd
+        characteristic since places are not fully supported for other function
+        fields.
+
         EXAMPLES::

Thanks, I'll integrate your changes. Just one small thing: Technically speaking there is a difference to formula 12.4.10 in [Voi2021]_, because the element $a_0$ computed in the code is only congruent modulo $\pi$ to $a \pi^{-\nu(a)}$ (which is the $a_0$ defined in the book) - effectively, this of course doesn't make a difference. Should I still keep the last sentence of the first ALGORITHM paragpraph as you wrote it?

kwankyu · 2024-03-27T11:41:34Z

Anyone reading the paragraph and the formula will understand that. Not a big deal. Also you explained the point.

src/sage/rings/function_field/function_field.py

- Rewrote docstring accordings to suggestions - Fixed typo in comment

S17A05 · 2024-03-28T03:06:59Z

I implemented a slightly modified version of your suggested docstring, mainly because I wanted to emphasize that $a_0$ and $b_0$ are elements of the residue field here, rather than of the discrete valuation ring.

github-actions · 2024-03-28T03:42:38Z

Documentation preview for this PR (built with commit 4f1e025; changes) is ready! 🎉

kwankyu

Thanks. LGTM.

sagemathgh-37173: Implemented `.ramified_places` and modified further methods to extend quaternion algebra functionality to number fields 1. Implemented method `.ramified_places` for quaternion algebras over number fields. Integrated `.ramified_primes()` into it in the process. 2. Modified `.is_division_algebra()`, `.is_matrix_ring()` and `.is_isomorphic` to use `.ramified_places` instead of `.discriminant()`, thus extending them to base number fields. 3. Rerouted `.discriminant()` through `.ramified_places` since the original call to `.hilbert_conductor` also computed all finite ramified places. 4. Added `.is_totally_definite()` and moved `is_definite()`. Some more detail: 1. 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 sagemath#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. 2. `.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 $\mathbb{Q}(\sqrt{5})$. Possible future work: - Extend functionality to all global fields (of characteristic not equal to $2$) [UPDATE: Will be done once both this PR and sagemath#37554 have been merged] URL: sagemath#37173 Reported by: Sebastian A. Spindler Reviewer(s): AurelPage, Frédéric Chapoton, grhkm21, Matthias Köppe, Sebastian A. Spindler

S17A05 added t: feature s: needs review c: algebra labels Mar 6, 2024

S17A05 force-pushed the fun_field_hilbert branch from 6bf63ef to 54611e1 Compare March 6, 2024 13:55

Implemented the (local) Hilbert symbol for global function fields

be7c5f1

Amends: Changed some documentation, adapted characteristic check to check ==2 instead of <= 2

S17A05 force-pushed the fun_field_hilbert branch from 54611e1 to be7c5f1 Compare March 6, 2024 14:02

Corrected code in examples

b7a9614

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S17A05 added s: needs work and removed s: needs review labels Mar 6, 2024

Small optimization and docstring modifications

02c46a9

S17A05 mentioned this pull request Mar 6, 2024

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

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Docstring corrections

c12a9de

Amend: Added missing multiline continuation markers

S17A05 force-pushed the fun_field_hilbert branch from cb29b63 to c12a9de Compare March 8, 2024 06:24

S17A05 added 3 commits March 8, 2024 08:38

Fixed docstrings and small modifications

0db2eb1

- Simplified return command, removed `ZZ` from import again - Return `0` instead of an error for zero inputs

Fixed LINT and docstring issues

b8f8056

Small input adaption, finally fixed docstrings

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- Fixed errors caused by unexpected behavior of reduction map - Added examples for bimultiplicativity - Corrected docstring

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S17A05 force-pushed the fun_field_hilbert branch from 7d7bcdd to 894a6b4 Compare March 26, 2024 12:59

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