Drinfeld modules #35026
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Drinfeld modules #35026
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This commit is a bit dirty, apologies. There are new getters (see
below), and the commit includes some refactoring of the doctests.
The added getters are:
- characteristic,
- constant_term,
- delta (rank two specific),
- g (rank two specific),
- height,
- j (rank two specific).
The added methods are:
- frobenius_trace,
- frobenius_norm,
- invert (required by frobenius_norm),
- is_ordinary,
- is_supersingular,
- characteristic_polynomial.
The added methods are:
- is_morphism,
- is_endomorphism,
- is_automorphism,
- is_isogeny,
- velu.
Rank two finite Drinfeld modules have specific enough properties to have
their own class. Therefore, we created the class
`FiniteDrinfeldModule_rank_two`.
The following methods were moved from `FiniteDrinfeldModule` to
`FiniteDrinfeldModule_rank_two`:
- characteristic_polynomial,
- delta,
- frobenius_norm,
- frobenius_trace,
- g,
- is_ordinary,
- is_supersingular.
Before this change, those methods were regular methods of just
`FiniteDrinfeldModule`; an exception was raised when the rank of the
Drinfeld module was not two.
Furthermore, the methods `_latex_` and `_repr_` were slightly changed.
…s-category_object
- Comment constructor - Check that codomain of input morphism is a field - Delete various useless docstrings (they will be written later, at a more final stage of development) - Change gamma to morphism - Various small enhancements
Codecov ReportBase: 88.59% // Head: 88.60% // Increases project coverage by
Additional details and impacted files@@ Coverage Diff @@
## develop #35026 +/- ##
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+ Coverage 88.59% 88.60% +0.01%
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3 tasks
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pdf docs do not build |
This would lead the pdf build to crash.
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Thanks for the comment. I think the issue is fixed. Here is the end of the output of I also fixed a few typos. |
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Documentation preview for this PR is ready! 🎉 |
4 tasks
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🍾 🍾 🍾 Many thanks to all coders and reviewers who made this contribution merged! |
vbraun
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May 22, 2023
…odule
<!-- ^^^^^
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Don't put issue numbers in there, do this in the PR body below.
For example, instead of "Fixes #1234" use "Introduce new method to
calculate 1+1"
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### 📚 Description
The goal of this PR is to implement the logarithm and the exponential of
a Drinfeld module.
### Background material
Let $\phi:T \mapsto \gamma(T) + g_1 \tau + \cdots + g_r \tau^r$ with
$g_r\neq 0$ be a rank $r$ Drinfeld module and let $\Lambda\subset
\mathbb{C}\_{\infty}$ be the associated rank $r$ lattice. Then recall
that the exponential $e\_{\phi}:\mathbb{C}\_{\infty} \rightarrow
\mathbb{C}\_{\infty}$ of the Drinfeld module $\phi$ is defined by:
$$
e_{\phi}(z) := z\prod_{\lambda\in \Lambda\setminus \{0\}} \left( 1 -
z/\lambda \right).
$$
It can be viewed as a power series in $z$:
$$
e_{\phi}(z) = z + \alpha_1 z^{q} + \alpha_2 z^{q^2} + \alpha_3 z^{q^3} +
\cdots
$$
Moreover, it satisfies the following functional equation:
$$
e_{\phi}(az) = \phi_a(e_{\phi}(z)).
$$
The logarithm of $\phi$ is defined to be the compositional inverse of
$e_{\phi}$, denoted $\mathrm{log}\_{\phi}$. By applying
$\mathrm{log}\_{\phi}$ on both side of the functional equation, we
obtain the following recurrence relation:
$$
\beta_k = \frac{1}{\gamma(T) - \gamma(T)^{q^i - 1}} \sum_{i = 0}^{k-1}
\beta_i g_{k-i}^{q^i}
$$
where $\beta_k$ is the $q^k$-th coefficient of $\mathrm{log}_{\phi}$ and
$\beta_0 = 1$.
### Examples
```
sage: A = GF(2)['T']
sage: K.<T> = Frac(A)
sage: phi = DrinfeldModule(A, [T, 1])
sage: q = A.base_ring().cardinality()
sage: log = phi.logarithm(); log
z + ((1/(T^2+T))*z^2) + ((1/(T^6+T^5+T^3+T^2))*z^4) + O(z^7)
sage: log[q] == -1/((T**q - T))
True
sage: log[q**2] == 1/((T**q - T)*(T**(q**2) - T))
True
sage: log[q**3] == -1/((T**q - T)*(T**(q**2) - T)*(T**(q**3) - T))
True
```
### Ideas
The idea here is to first compute the logarithm of a Drinfeld module
using lazy power series and then compute its compositional inverse to
find its exponential.
<!-- Describe your changes here in detail -->
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example "Closes #1337" -->
### 📝 Checklist
<!-- Put an `x` in all the boxes that apply. -->
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appropriately -->
<!-- If you're unsure about any of these, don't hesitate to ask. We're
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- [ ] I have linked an issue or discussion.
- [x] I have created tests covering the changes.
- [x] I have updated the documentation accordingly.
### ⌛ Dependencies
<!-- List all open pull requests that this PR logically depends on -->
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- #xyz: short description why this is a dependency
- #abc: ...
-->
Depends on #35026 because this PR implements Drinfeld modules.
CC: @xcaruso @ymusleh @kryzar
URL: #35260
Reported by: David Ayotte
Reviewer(s): Antoine Leudière, David Ayotte, Xavier Caruso
vbraun
pushed a commit
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Jul 30, 2023
<!-- ^^^^^ 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 #1234" use "Introduce new method to calculate 1+1" --> ### 📚 Description The goal of this PR is to implement basic j-invariants of Drinfeld module as defined by Potemine in Potemine, I.Y. Minimal Terminal ℚ-Factorial Models of Drinfeld Coarse Moduli Schemes. Mathematical Physics, Analysis and Geometry 1, 171–191 (1998). https://doi.org/10.1023/A:1009724323513 These j-invariants allows to determine whether two Drinfeld `Fq[T]`-modules of arbitrary ranks are isomorphic or not. @kryzar @ymusleh @xcaruso <!-- Describe your changes here in detail --> <!-- Why is this change required? What problem does it solve? --> <!-- If it resolves an open issue, please link to the issue here. For example "Closes #1337" --> ### 📝 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! --> - [x] I have made sure that the title is self-explanatory and the description concisely explains the PR. - [] I have linked an issue or discussion. - [x] I have created tests covering the changes. - [x] I have updated the documentation accordingly. ### ⌛ Dependencies <!-- List all open pull requests that this PR logically depends on --> <!-- - #xyz: short description why this is a dependency - #abc: ... --> Depends on #35026 because this PR implements Drinfeld modules. URL: #35057 Reported by: David Ayotte Reviewer(s): Antoine Leudière, David Ayotte, Xavier Caruso
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📚 Description
Fixes #33713.
We implement Drinfeld$\mathbb{F}_q[T]$ -modules with a focus towards finite Drinfeld modules. We also implement isogenies of Drinfeld modules. More details in the issue.
📝 Checklist
⌛ Dependencies
None.