gh-38174: New algorithms to compute the characteristic polynomial of …

…the Frobenius endomorphism of a Drinfeld module

    
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This pull request implements two new algorithms to compute the
characteristic polynomial of the Frobenius endomorphism of a Drinfeld
$\mathbb F_q[T]$-module over a finite field $K$. Previously, only the
algorithms based on crystalline cohomology (see [Musleh-Schost
2023](https://dl.acm.org/doi/10.1145/3597066.3597080)) or on Anderson
motives (see [Caruso-Leudière 2023](https://arxiv.org/abs/2307.02879))
were implemented. We propose two new algorithms:

- The algorithm based on central simple algebras described in Chapter 4
of [Caruso-Leudière 2023](https://arxiv.org/abs/2307.02879).

- The algorithm described by Gekeler in [Gekeler 1991](https://www.scien
cedirect.com/science/article/pii/002186939190211P).

**Acknowledgement.** This implementation was originally due to @xcaruso
(see [here](https://github.com/xcaruso/sage/blob/d2e36bd18b51c93806b7a3b
5c8261da7dc98c494/src/sage/rings/function_field/drinfeld_modules/finite_
drinfeld_module.py)), and after a private discussion, I took the liberty
of creating this PR.

I also propose to change the formula computed by `frobenius_norm`.
Before, it computed

$$(-1)^n \mathrm{N}_{K/\mathbb F_q}(\Delta) p^{n / \mathrm{deg}(p)},$$

where $K$ is the ground field, $n$ is the degree of $K$ over $\mathbb
F_q$, and $p$ is the monic generator of the $\mathbb
F_q[T]$-characteristic of $K$. The docstring claimed this was $(-1)^r$
times the constant coefficient of the characteristic polynomial of the
Frobenius endomorphism, $r$ being the rank of the Drinfeld module. I
believe this was a mistake, and instead changed the formula, following
Theorem 4.2.7 of [Papikian's
book](https://link.springer.com/book/10.1007/978-3-031-19707-9), to
$$(-1)^{nr - n - r} \mathrm{N}_{K/\mathbb F_q}(\Delta) p^{n /
\mathrm{deg}(p)}.$$

P.S. @DavidAyotte, given that you now work in the industry, please tell
us if you still want to be tagged on these Drinfeld module stuff. Same
question for you @ymusleh given that you defended your thesis
(congratulations!).

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preview.


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(Dédicace à ABLCCN)
    
URL: #38174
Reported by: Antoine Leudière
Reviewer(s): Antoine Leudière, David Ayotte, Xavier Caruso

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

@@ -1716,6 +1716,9 @@ REFERENCES:
             Yokonuma-Hecke algebras and the HOMFLYPT
             polynomial*. (2015) :arxiv:`1204.1871v4`.
 
+.. [CL2023] Xavier Caruso and Antoine Leudière.
+            *Algorithms for computing norms and characteristic polynomials on general Drinfeld modules*, (2023) :arxiv:`2307.02879`.
+
 .. [Cle1872] Alfred Clebsch, *Theorie der binären algebraischen Formen*,
              Teubner, 1872.
 
@@ -2814,6 +2817,10 @@ REFERENCES:
 .. [Gek1991] \E.-U. Gekeler. On finite Drinfeld modules. Journal of
              algebra, 1(141):187–203, 1991.
 
+.. [Gek2008] \E.-U. Gekeler. Frobenius Distributions of Drinfeld Modules over
+             Finite Fields. Transactions of the American Mathematical Society,
+             Volume 360 (2008), no. 4.
+
 .. [Gek2017] \E.-U. Gekeler. On Drinfeld modular forms of higher rank.
              Journal de théorie des nombres de Bordeaux,
              Volume 29 (2017) no. 3, pp. 875-902. :doi:`10.5802/jtnb.1005`