From 41937687bb001d5eaa836eb9c66ca15217a26acc Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Antoine=20Leudi=C3=A8re?= Date: Mon, 18 Mar 2024 12:57:45 +0100 Subject: [PATCH 1/2] Remove typo in DrinfeldModule.j_invariant docstring In DrinfeldModule.j_invariant docstring, there was a typo in the first `.. MATH::` environment: the exponent on the denominator was $d_q$ instead of $d_r$. --- .../rings/function_field/drinfeld_modules/drinfeld_module.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py index b0f177dfcfa..d395784995b 100644 --- a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py +++ b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py @@ -1489,7 +1489,7 @@ def j_invariant(self, parameter=None, check=True): .. MATH:: j_{k_1, \ldots, k_n}^{d_1, \ldots, d_n, d_r}(\phi) - := \frac{1}{g_r^{d_q}}\prod_{i = 1}^n g_{k_i}^{d_i} + := \frac{1}{g_r^{d_r}}\prod_{i = 1}^n g_{k_i}^{d_i} where `1\leqslant k_1 < k_2 < \ldots < k_n \leqslant r - 1` and the integers `d_i` satisfy the *weight-0 condition*: From b4121bad8a713640a7c785a7945f2d8f2878d463 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Antoine=20Leudi=C3=A8re?= Date: Mon, 18 Mar 2024 21:49:15 +0100 Subject: [PATCH 2/2] Add comments on of Potemine's j-invariants We believe there was an ambiguity between our naming of j-invariants, basic j-invariants and jk-invariants, as we follow Potemine's convention, which differs from Papikian's. Consequently, we remove the ambiguity in a note in the docstring. --- src/doc/en/reference/references/index.rst | 4 +++ .../drinfeld_modules/drinfeld_module.py | 25 +++++++++++++++---- 2 files changed, 24 insertions(+), 5 deletions(-) diff --git a/src/doc/en/reference/references/index.rst b/src/doc/en/reference/references/index.rst index 09b78514e1c..5f854c44c58 100644 --- a/src/doc/en/reference/references/index.rst +++ b/src/doc/en/reference/references/index.rst @@ -5110,6 +5110,10 @@ REFERENCES: B46f, https://eudml.org/doc/121696 +.. [Pap2023] Mihran Papikian, *Drinfeld modules*. + Graduate Texts in Mathematics, 206, + Springer International Publishing, Cham, 2023. + .. [PALP] Maximilian Kreuzer, Harald Skarke: "PALP: A Package for Analyzing Lattice Polytopes with Applications to Toric Geometry" omput.Phys.Commun. 157 (2004) 87-106 diff --git a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py index d395784995b..02f2d0be604 100644 --- a/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py +++ b/src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py @@ -1510,6 +1510,26 @@ def j_invariant(self, parameter=None, check=True): :meth:`basic_j_invariant_parameters` for computing the list of all basic `j`-invariant parameters. + .. NOTE:: + + In [Pap2023]_, Papikian follows a slightly different + convention: + + - His `j`-invariants (see Definition 3.8.7) correspond to + our basic `j`-invariants, as defined above. + - His *basic* `j`-invariant (see Example 3.8.10) correspond + to our `j_k`-invariants, as implemented in + :meth:`jk_invariants`. + + We chose to follow Potemine's convention, as he introduced + those objects in [Pot1998]_. Theorem 2.2 of [Pot1998]_ or + Theorem 3.8.11 of [Pap2023]_ assert that two Drinfeld + `\mathbb F_q[T]`-modules over `K` are isomorphic over the + separable closure of `K` if and only if their basic + `j`-invariants (as implemented here) coincide for any + well-defined couple of tuples `((k_1, k_2, \ldots, k_n), + (d_1, d_2, \ldots, d_n, d_r))`, . + INPUT: - ``parameter`` (tuple or list, integer or NoneType; default: @@ -1536,11 +1556,6 @@ def j_invariant(self, parameter=None, check=True): OUTPUT: the `j`-invariant of ``self`` for the given parameter. - REFERENCE: - - The notion of basic `j`-invariant was introduced by Potemine in - [Pot1998]_. - EXAMPLES:: sage: Fq = GF(25)