Implement the logarithm and the exponential of a Drinfeld module

[DavidAyotte]

📚 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.

📝 Checklist

• I have made sure that the title is self-explanatory and the description concisely explains the PR.
• I have linked an issue or discussion.
• I have created tests covering the changes.
• I have updated the documentation accordingly.

⌛ Dependencies

Depends on #35026 because this PR implements Drinfeld modules.

CC: @xcaruso @ymusleh @kryzar

[DavidAyotte]

It should be possible to implement the exponential of a Drinfeld module once issue #35261 is fixed.

EDIT (2023-03-13): I found a workaround to the issue mentioned above. Essentially, it consists in computing explicitely the coefficients of the exponential using the the fact that it is the compositional inverse of the logarithm.

[codecov-commenter]

Codecov Report

Patch coverage: 100.00% and project coverage change: -0.03 ⚠️

Comparison is base (f449b14) 88.62% compared to head (01d620d) 88.60%.

❗ Current head 01d620d differs from pull request most recent head 643cfde. Consider uploading reports for the commit 643cfde to get more accurate results

Additional details and impacted files

@@             Coverage Diff             @@
##           develop   #35260      +/-   ##
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- Coverage    88.62%   88.60%   -0.03%     
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- Hits        353308   352512     -796     
- Misses       45345    45348       +3     

Impacted Files	Coverage Δ
...function_field/drinfeld_modules/drinfeld_module.py	97.05% <100.00%> (+1.02%)	⬆️

... and 67 files with indirect coverage changes

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[kryzar]

Great! I'm not an expert but I'd be happy to review the code!

[DavidAyotte]

Great! I'm not an expert but I'd be happy to review the code!

If you want a good grasp of these objects, I suggest that you read the chapter on the Carlitz module of Goss's book (chapter 3) and then section 4.6 for the general case.

[mantepse]

Since #35261 is fixed, could you please recheck? The method from #35261 should be faster, or at least not slower.

[DavidAyotte]

Since #35261 is fixed, could you please recheck? The method from #35261 should be faster, or at least not slower.

Yes it works now, thanks! Beforehand I used your workaround by using the option valuation=1 in the element constructor. I was wondering if it's best to keep it or not? Looking at the documention of Lazy series, I'm not sure what this parameter does exactly.

[mantepse]

Ok, thanks for the feedback, I have to explain the parameter better then in the Dichtring.

For now: it is any lower bound on the minimal degree (aka valuation) of the series. It is mostly optional, in many cases the code will figure it out by itself. But if it is known it does not hurt to supply it.

[mantepse]

Do I understand correctly that the exponential of a Drinfeld module is not a Drinfeld module (but a sequence of Drinfeld modules)?

[mantepse]

Can you see a relation to sage.rings.lazy_series_ring.LazyCompletionGradedAlgebra?

[DavidAyotte]

Do I understand correctly that the exponential of a Drinfeld module is not a Drinfeld module (but a sequence of Drinfeld modules)?

Yes and no. The exponential of a Drinfeld module is not a Drinfeld module, but it is not a sequence of Drinfeld module either. Let $F$ be a $\mathbb{F}_q[T]$-algebra which is a field. A Drinfeld $\mathbb{F}_q[T]$-module over $F$ is really just a morphism of $\mathbb{F}_q$-algebra determined by $\phi : T \mapsto \phi_T:=\iota(T) + g_1 \tau + \cdots + g_r \tau^r$ where $\tau:x \mapsto x^q$ is the $q$-Frobenius, $g_i\in F$, $g_r\neq 0$ and $\iota : \mathbb{F}_q[T]\rightarrow F$ is the algebra structure morphism.

In particular a Drinfeld $\mathbb{F}_q[T]$-module is determined by the list $[\iota(T), g_1, \ldots, g_r]$ and that's how they are represented in sagemath.

Suppose that $F = C$ is an algebraically closed and complete extension of the rational function field $\mathbb{F}_q(T)$, then there exists a map $e_{\phi} : C \rightarrow C$ which is $\mathbb{F}_q$-linear, entire and surjective. Moreover, it satisfy the functional equation: $e_{\phi}(ax) = \phi_a(e_{\phi}(x))$. This map is the exponential of the Drinfeld module $\phi$ and, by its properties, it is a power series: $\sum_{i\geq 0} a_i z^{q^i}$. The logarithm is the compositional inverse of the exponential.

However, I did explained the things a bit backward here, because given a Drinfeld module described by $[\iota(T), g_1, \ldots, g_r]$, it is actually easier to first compute the logarithm using a recurrence relation on the list elements and then compute the compositional inverse of the logarithm to obtain the exponential.

I hope this gives you a better understanding of what's going on here!

[DavidAyotte]

What do you mean with "Ore power series"?

It's a bit unknown, but I mean the power series version of an Ore polynomial. However, I think this would require a bit of work to implement in SageMath.

Yes, it is exactly what I had in mind when I wrote: "I think that there are still some place for optimization but it's not obvious and will be for another ticket." :-)

What I don't really like about using this form $1 + \sum_{i} \alpha_i \tau^i$ ("Ore power series") over this one $z + \sum_i \alpha_i z^{q^i}$ ("standard") is that manipulating the power series will give wrong result as SageMath won't treat the variable $\tau$ as the Frobenius. So, just to be sure that I understand correctly: do you suggest to keep the standard form in this ticket?

[xcaruso]

Yes!
And for another ticket (maybe): implement Ore powers series and add an option to logarithm and exponential to let them return Ore powers series instead of actual series.

[mantepse]

What do you mean with "Ore power series"?

It's a bit unknown, but I mean the power series version of an Ore polynomial. However, I think this would require a bit of work to implement in SageMath.

Am I mistaken that the only thing you'd need to do is to put OrePolynomialRing into GradedAlgebrasWithBasis?

[DavidAyotte]

What do you mean with "Ore power series"?

It's a bit unknown, but I mean the power series version of an Ore polynomial. However, I think this would require a bit of work to implement in SageMath.

Am I mistaken that the only thing you'd need to do is to put OrePolynomialRing into GradedAlgebrasWithBasis?

Can you expand a bit on this idea? Do you mean to do something like this:

sage: A = GF(3)['T']
sage: K.<T> = Frac(A)
sage: L = OrePolynomialRing(K, K.frobenius_endomorphism())
sage: GradedAlgebrasWithBasis(L)
...

[mantepse]

Can you expand a bit on this idea? Do you mean to do something like this:

sage: A = GF(3)['T']
sage: K.<T> = Frac(A)
sage: L = OrePolynomialRing(K, K.frobenius_endomorphism())
sage: GradedAlgebrasWithBasis(L)
...

What I meant is to use the generic completion of a graded algebra with basis. However, this does not work out of the box, because OrePolynomialRing is not in the category GradedAlgebrasWithBasis.

Then you can just say

sage: R.formal_series_ring()
Lazy completion of Ore Polynomial Ring in x over Finite Field in a of size 5^3 twisted by a |--> a^5

However, I am not sure whether this would be against some unwritten policy in sage, because the polynomial rings aren't in this category either.

On the other hand, it would be very easy to provide a specialised implementation, for example by adapting LazyCompletionGradedAlgebra.

[xcaruso]

They were my last two comments.
After this, I'm happy with the ticket and ready to give a positive review.

[xcaruso]

All tests pass.

@kryzar Do you agree for giving a positive review to this ticket?

[kryzar]

Apologies for delaying my review.

I approve the changes. I commented on two things which I think could be better next time. Those are important, but it is also important not to delay this PR any longer.

[DavidAyotte]

I refactored logarithm and exponential following Antoine's suggestion and put back the need review status of the ticket. I hope the code is a bit clearer now.

[github-actions]

Documentation preview for this PR is ready! 🎉
Built with commit: 2b242fb

[xcaruso]

OK!