Class polynomial for Drinfeld modules

[xcaruso]

This PR implements an algorithm for computing the class polynomial (that is, the Fitting ideal of the class module) of a Drinfeld modules over $K = \text{Frac}(A)$.

We recall that the class module of a Drinfeld module $E : A \to A\{\tau\}$ is defined as the quotient
$$H(E) := \frac{E(K_\infty)}{E(A) + \exp_E(K_\infty)}$$
where:

• for an $A$-algebra $R$, the notation $E(R)$ refers to $R$ equipped with the structure of $A$-module coming from $E$
• $K_\infty$ is the completion of $\text{Frac}(A)$ at infinity
• $\exp_E : K_\infty \to K_\infty$ is the exponential associated to $E$.

The implementation provided in this PR is based on the following remark: for $s$ large enough, $H(E)$ is also the quotient of $E(K_\infty/A)$ by the $A$-submodule generated by $T^{-n}$ with $n \geq s$.
This is due to the fact that, when $n$ is large, $T^{-n}$ is in the domain of convergence of the logarithm of $E$.

📝 Checklist

• The title is concise and informative.
• The description explains in detail what this PR is about.
• I have linked a relevant issue or discussion.
• I have created tests covering the changes.
• I have updated the documentation and checked the documentation preview.

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[xcaruso]

I'm not entirely sure that my algorithm for computing Taelman's unit is correct. I need to think more about this.

[xcaruso]

Finally, I prefer to remove things about Taelman's unit here and keep it for another PR.

[kryzar]

So again here, it seems the failing doctest is independent from this PR. Could you please merge from develop?

[xcaruso]

Here is a proof that the algorithm used in this PR is correct.

Let $\varpi = 1/T$; it is a uniformizer of the completion as infinity $K_\infty$.
For each positive integer $n$, let $I_n$ be the set of elements of $K_\infty$ of valuation at least $n$, i.e. $I_n$ is the $\mathbb F_q$-subvector space generated by $\varpi^m$ with $m \geq n$.
Let $s$ be a positive integer such that the $E$-logarithm converges on $I_s$ and induces a bijection on this space.
We notice that $\exp_E(\varpi^{s-1}) \equiv \varpi^{s-1} \pmod {\varpi^s}$.

Consider now $x \in K_\infty$ and decompose it as $x = y + a_{s-1} \varpi^{s-1} + \cdots + a_1 \varpi + z$ with $y \in I_s$ and $z \in A$.
We deduce that
$$\exp_E(x) \equiv a_{s-1} \varpi^{s-1} + a_{s-2} \phi_T(\varpi^{s-1}) + \cdots + a_1 \phi_T^{s-2}(\varpi) \pmod {I_s + A}.$$
Since the action of $E_T$ corresponds to the multiplication by $T$ in $E(K_\infty)$, we obtain the claim of the description of this PR.

In the implementation, we choose a specific value of $s$, which is
$$s = \max_i \Big\lfloor \frac{\deg g_i }{q^i-1} \Big\rfloor$$
where $g_i$ is the $i$-th coefficient of $E_T$. This comes from the fact that $\phi_T(\varpi^n) \equiv \varpi^n \pmod {I_{n-1}}$ for all $n \geq s$. Hence, taking $s$ as above does not change the final result of the computation.

[kryzar]

Thank you very much for the comment and the explanation of your algorithm on this PR. I think it's very valuable work.

I only have very minor comments (mainly on the Taelman reference you added which appears unused, and on two lines which are reported untested by the testing-coverage tool).

[kryzar]

The failing test is also unrelated to this ticket. So, again, I approve the PR anyway.

Thanks @xcaruso!