sagemathgh-37630: Enhance documentation of Drinfeld module Potemine j…

…-invariants

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The definition of $j$-invariants, *basic* $j$-invariants and
$j_k$-invariants defer in [Potemine's
article](https://link.springer.com/article/10.1023/A:1009724323513) and
[Papikian's
book](https://link.springer.com/book/10.1007/978-3-031-19707-9). Thanks
to @DavidAyotte, we follow Potemine's convention. Seeing however the
growing popularity of Papikian's book, I reckon it is a good idea to add
a note in the documentation addressing those differences.

Also, I fixed a typo.

### Potemine:

![image](https://github.com/sagemath/sage/assets/31312212/5e09f13c-782d-
4665-b5a6-c0c8947cb74f)
![image](https://github.com/sagemath/sage/assets/31312212/65c6e812-3f93-
4a4c-9bb7-ce84ca1dcf10)

### Papikian:

![image](https://github.com/sagemath/sage/assets/31312212/eb811d6f-5a3c-
4764-b9f8-d12d562309b3)
![image](https://github.com/sagemath/sage/assets/31312212/d100c393-dd6d-
47b0-b39b-edfd02cda7a6)

### 📝 Checklist

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- [x] The description explains in detail what this PR is about.
- [x] I have linked a relevant issue or discussion.
- [x] I have created tests covering the changes.
- [x] I have updated the documentation accordingly.

### ⌛ Dependencies

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URL: sagemath#37630
Reported by: Antoine Leudière
Reviewer(s): David Ayotte

build/pkgs/configure/checksums.ini

@@ -1,4 +1,4 @@
 tarball=configure-VERSION.tar.gz
-sha1=16786ab29c5625c411bee47d25d62d628477e9dc
-md5=f37ca4623241cad2846b45122d180cc3
-cksum=536361291
+sha1=25a360e4892317526249e50eaa51297891511d61
+md5=0fd34b91fc06199e6c3672ad969472ee
+cksum=1226224895

build/pkgs/configure/package-version.txt

@@ -1 +1 @@
-b0343d194ea320a6ea030d6b1274fbacdac0d7ac
+d13f6164a314f330071fe2a99112e78de03347f0

src/doc/en/reference/references/index.rst

@@ -5133,6 +5133,10 @@ 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

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*:
@@ -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)