sagemathgh-36538: Implement Drinfeld modular forms
    
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This PR aims to implement Drinfeld modular forms into SageMath. These
special types of modular forms live in the function field theory and are
linked with Drinfeld modules.

#### Definitions

Without going too much into the details, we give some definitions and
insight of what is going on here. Let $K = \mathbb{F}_q(T)$ together
with the $1/T$-adic norm and $A = \mathbb{F}_q[T]$. We define
$K\_{\infty} = \mathbb{F}_q((1/T))$ be the completion of $K$ and let
$\mathbb{C}\_{\infty}$ be the completion of an algebraic closure of
$K\_{\infty}$. In this specific setting, the *Drinfeld period domain* of
rank $r$ is

$$
\Omega^r(\mathbb{C}\_{\infty}) := \lbrace (w_1, \ldots, w_{r-1}, 1)\in
\mathbb{C}\_{\infty}^r : \text{the }w_i \text{'s are
}K\_{\infty}\text{-linearly independant} \rbrace.
$$

(this period domain is to be seen as the analogue of the complex upper
half plane)

So, for each $w = (w_1, \ldots, w_{r-1}, 1)\in
\Omega^r(\mathbb{C}\_{\infty})$, we have an associated $A$-lattice (i.e.
a discrete projective $A$-module of rank r): $\Lambda_w = A w_1 + \ldots
A w_{r-1} + A$.

The analytic theory of Drinfeld modules tells us that, for each such
$w$, there exist a Drinfeld module

$$
\phi^w_T : T \mapsto T + g\_1(w) \tau + \ldots + g\_{r-1}(w) \tau^{r-1}
+ g\_r(w) \tau^r
$$

associated with this lattice. This association is what we call the
universal Drinfeld module over $\Omega^r(\mathbb{C}\_{\infty})$. We see
the coefficients as functions $g_i : \Omega^r(\mathbb{C}\_{\infty})
\rightarrow \mathbb{C}\_{\infty}$. These functions satisfy modular
transformation property under the action of the group
$\mathrm{GL}_r(A)$:

$$
g_i(\gamma(w)) = j(\gamma, w)^i g_i(w)
$$

where $j$ is some automorphic factor.

The ring of Drinfeld modular forms for $\mathrm{GL}_r(A)$ is

$$
\mathcal{M}\_{\bullet} = \mathbb{C}\_{\infty}[g_1, \ldots g_{r-1}, g_r].
$$

#### Implementations details

In this implementation, we create a new parent `DrinfeldModularForms`
which represents the polynomial ring $K[g_1,\ldots, g_{r-1}, g_r]$ and a
new element class `DrinfeldModularFormsElement`. Then, for arbitrary
rank, we only see a Drinfeld modular form symbolically as a multivariate
polynomial in $g_1, \ldots, g_r$, the coefficients forms. In the special
case when the rank is two, we aim to implement algorithms for computing
the expansion at infinity of any Drinfeld modular forms. These
expansions are analogues of the usual $q$-expansion of classcial modular
forms. This will be achieved via the Lopez-Petrov nonstandard expansion
theory.

We note that this implementation was initially done as a pip-installable
package:

https://davidayotte.github.io/drinfeld_modular_forms

This work has been an ongoing project for almost two years now and since
the addition of Drinfeld modules in SageMath, the author has seen an
interest in the implementation of Drinfeld modular forms to SageMath. In
the rank two case, the ability to compute their expansion at infinity
shows some interesting computational properties of these objects.

We are aware that this is a quite specialized topic, so we aim to make
this implementation as convivial as possible for the user and well
documented. Any comments and suggestions are welcome!

CC: @xcaruso @kryzar @ymusleh

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### ⌛ Dependencies

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<!-- If you're unsure about any of these, don't hesitate to ask. We're
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URL: sagemath#36538
Reported by: David Ayotte
Reviewer(s): Antoine Leudière, David Ayotte, Xavier Caruso

sagemathgh-36538: Implement Drinfeld modular forms
    
<!-- ^^^^^
Please provide a concise, informative and self-explanatory title.
Don't put issue numbers in there, do this in the PR body below.
For example, instead of "Fixes sagemath#1234" use "Introduce new method to
calculate 1+1"
-->
<!-- Describe your changes here in detail -->

<!-- Why is this change required? What problem does it solve? -->
<!-- If this PR resolves an open issue, please link to it here. For
example "Fixes sagemath#12345". -->
<!-- If your change requires a documentation PR, please link it
appropriately. -->

This PR aims to implement Drinfeld modular forms into SageMath. These
special types of modular forms live in the function field theory and are
linked with Drinfeld modules.

#### Definitions

Without going too much into the details, we give some definitions and
insight of what is going on here. Let $K = \mathbb{F}_q(T)$ together
with the $1/T$-adic norm and $A = \mathbb{F}_q[T]$. We define
$K\_{\infty} = \mathbb{F}_q((1/T))$ be the completion of $K$ and let
$\mathbb{C}\_{\infty}$ be the completion of an algebraic closure of
$K\_{\infty}$. In this specific setting, the *Drinfeld period domain* of
rank $r$ is

$$
\Omega^r(\mathbb{C}\_{\infty}) := \lbrace (w_1, \ldots, w_{r-1}, 1)\in
\mathbb{C}\_{\infty}^r : \text{the }w_i \text{'s are
}K\_{\infty}\text{-linearly independant} \rbrace.
$$

(this period domain is to be seen as the analogue of the complex upper
half plane)

So, for each $w = (w_1, \ldots, w_{r-1}, 1)\in
\Omega^r(\mathbb{C}\_{\infty})$, we have an associated $A$-lattice (i.e.
a discrete projective $A$-module of rank r): $\Lambda_w = A w_1 + \ldots
A w_{r-1} + A$.

The analytic theory of Drinfeld modules tells us that, for each such
$w$, there exist a Drinfeld module

$$
\phi^w_T : T \mapsto T + g\_1(w) \tau + \ldots + g\_{r-1}(w) \tau^{r-1}
+ g\_r(w) \tau^r
$$

associated with this lattice. This association is what we call the
universal Drinfeld module over $\Omega^r(\mathbb{C}\_{\infty})$. We see
the coefficients as functions $g_i : \Omega^r(\mathbb{C}\_{\infty})
\rightarrow \mathbb{C}\_{\infty}$. These functions satisfy modular
transformation property under the action of the group
$\mathrm{GL}_r(A)$:

$$
g_i(\gamma(w)) = j(\gamma, w)^i g_i(w)
$$

where $j$ is some automorphic factor.

The ring of Drinfeld modular forms for $\mathrm{GL}_r(A)$ is

$$
\mathcal{M}\_{\bullet} = \mathbb{C}\_{\infty}[g_1, \ldots g_{r-1}, g_r].
$$

#### Implementations details

In this implementation, we create a new parent `DrinfeldModularForms`
which represents the polynomial ring $K[g_1,\ldots, g_{r-1}, g_r]$ and a
new element class `DrinfeldModularFormsElement`. Then, for arbitrary
rank, we only see a Drinfeld modular form symbolically as a multivariate
polynomial in $g_1, \ldots, g_r$, the coefficients forms. In the special
case when the rank is two, we aim to implement algorithms for computing
the expansion at infinity of any Drinfeld modular forms. These
expansions are analogues of the usual $q$-expansion of classcial modular
forms. This will be achieved via the Lopez-Petrov nonstandard expansion
theory.

We note that this implementation was initially done as a pip-installable
package:

https://davidayotte.github.io/drinfeld_modular_forms

This work has been an ongoing project for almost two years now and since
the addition of Drinfeld modules in SageMath, the author has seen an
interest in the implementation of Drinfeld modular forms to SageMath. In
the rank two case, the ability to compute their expansion at infinity
shows some interesting computational properties of these objects.

We are aware that this is a quite specialized topic, so we aim to make
this implementation as convivial as possible for the user and well
documented. Any comments and suggestions are welcome!

CC: @xcaruso @kryzar @ymusleh

### 📝 Checklist

<!-- Put an `x` in all the boxes that apply. -->
<!-- If your change requires a documentation PR, please link it
appropriately -->
<!-- If you're unsure about any of these, don't hesitate to ask. We're
here to help! -->
<!-- Feel free to remove irrelevant items. -->

- [x] The title is concise, informative, and self-explanatory.
- [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

<!-- List all open PRs that this PR logically depends on
- sagemath#12345: short description why this is a dependency
- sagemath#34567: ...
-->

<!-- If you're unsure about any of these, don't hesitate to ask. We're
here to help! -->
    
URL: sagemath#36538
Reported by: David Ayotte
Reviewer(s): Antoine Leudière, David Ayotte, Xavier Caruso