Major extensions to local components code

[loefflerd]

Some while ago, I contributed to Sage some code for computing the local component (at a given prime) of the automorphic representation attached to a modular form, based on algorithms from a paper I wrote with Jared Weinstein.

The existing implementation is incomplete in several ways:

• it throws an error if the given modular form doesn't have minimal level at p among its character twists;
• if the local component is supercuspidal, induced from a character of a ramified quadratic extension, then it doesn't completely compute the character, it just returns a string saying which extension it comes from;
• in supercuspidal 2-adic cases where the level is divisible by 64, it fails because a certain quotient group that comes up in the computation isn't cyclic.

I have now reworked the code to fill in all of the above gaps. It should now work for all newforms and all primes, with the exception of certain nasty 2-adic cases where no nice parametrisation exists. In particular, with the new code the following example works, fulfilling a request from Ariel Pacetti:

sage: E = EllipticCurve("575c1")
sage: Pi = E.newform().local_component(5)
sage: Pi.characters()
[
Character of Q_5*, of level 1, mapping 2 |--> 1/2*a, 5 |--> 1/2*a + 2,
Character of Q_5*, of level 1, mapping 2 |--> -1/2*a, 5 |--> -1/2*a + 2
]

Component: modular forms

Keywords: local-components

Author: David Loeffler

Branch/Commit: c165835

Reviewer: David Roe

Issue created by migration from https://trac.sagemath.org/ticket/26635

[loefflerd]

Branch: u/davidloeffler/minimal_twist_new

[loefflerd]

Commit: 37ce61f

[loefflerd]

New commits:

1b55638	Add a minimal twist function for newform objects
37ce61f	Local components: handle many previously-unimplemented cases

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

482e00a

Fix non-ascii chars, non-Py3 code etc picked up by patchbot

[sagetrac-git]

Changed commit from 37ce61f to 482e00a

[loefflerd]

comment:6

I just committed a fix for some minor coding-style violations picked up by the patchbot.

[roed314]

Changed branch from u/davidloeffler/minimal_twist_new to u/roed/minimal_twist_new

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

f8160a3

Fix trivial test failures

[sagetrac-git]

Changed commit from 482e00a to f8160a3

[roed314]

comment:9

I hopefully fixed the merge failure and a couple of deprecation warnings. There seems to be one actual doctest failure:

File "src/sage/modular/local_comp/local_comp.py", line 704, in sage.modular.local_comp.local_comp.PrimitiveSupercuspidal.characters
Failed example:
    Newforms(DirichletGroup(64, QQ).1, 2, names='a')[0].local_component(2).characters() # long time
Expected:
    [
    Character of unramified extension Q_2(s)* (s^2 + s + 1 = 0), of level 3, mapping s |--> 1, 2*s + 1 |--> -1/4*a0, 4*s + 1 |--> -1, -1 |--> 1, 2 |--> 1,
    Character of unramified extension Q_2(s)* (s^2 + s + 1 = 0), of level 3, mapping s |--> 1, 2*s + 1 |--> -1/4*a0, 4*s + 1 |--> 1, -1 |--> 1, 2 |--> 1
    ]
Got:
    [
    Character of unramified extension Q_2(s)* (s^2 + s + 1 = 0), of level 3, mapping s |--> 1, 2*s + 1 |--> 1/2*a0, 4*s + 1 |--> 1, -1 |--> 1, 2 |--> 1,
    Character of unramified extension Q_2(s)* (s^2 + s + 1 = 0), of level 3, mapping s |--> 1, 2*s + 1 |--> 1/2*a0, 4*s + 1 |--> -1, -1 |--> 1, 2 |--> 1
    ]

[roed314]

comment:10

Nope, merge still failed. I'll try a more recent version of Sage soon.

[loefflerd]

comment:11

Nice to see some movement on this at long last, thanks for taking a look at it! The doctest failure looks harmless to me, my guess is that Sage has changed its mind about which generator to use for the coeff field of the newform. Let me know if you have trouble getting the merge to work, I can have a go myself (if I still remember how).

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

0cef4b0

Merge branch 'u/roed/minimal_twist_new' of git://trac.sagemath.org/sage into t/26635/minimal_twist_new

[sagetrac-git]

Changed commit from f8160a3 to 0cef4b0

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

1681b79

Fix doctest failure

[sagetrac-git]

Changed commit from 0cef4b0 to 1681b79

[roed314]

comment:14

I got the merge to work (and have adjusted the doctest to what's currently returned). I'll try to read through the code soon, and let you know if I have questions!

[loefflerd]

comment:15

I checked about the failing long doctest. It is indeed unrelated to the code at hand, and arises because sage is using a random algorithm to choose a generator of the coeff field of the newform. Shall we just flag it as # random?

[sagetrac-git]

Changed commit from 1681b79 to c2267ce

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

c2267ce

Mostly style changes, one potential speedup

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

d0c9d58

Mark fragile doctest as random

[sagetrac-git]

Changed commit from c2267ce to d0c9d58

[sagetrac-git]

Changed commit from d0c9d58 to 7d4dd1b

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

7d4dd1b

Some more style fixes

[roed314]

comment:19

I read through the code and it looks good. I made various minor changes to better conform to Sage's style guide. The one substantive change I made was to move the line t = self.discrete_log(n, p) outside a loop.

I'm going to give this a positive review now, but we should keep an eye on the patchbot results. I'm also hopeful that I can help with improving speed eventually using #26849 (which still needs work). Finally, what's the theoretical status on the missing 2-adic cases?

I'm sorry this took so long to get reviewed!

[roed314]

Reviewer: David Roe

[loefflerd]

comment:20

Thanks for the review!

Your ticket for p-adic unit groups looks interesting & obviously it would be nice to unify it with the local-component code; but I doubt it would lead to much of a speed increase. In local-component computations, the computation time is overwhelmingly dominated by computing modular-symbol spaces and Hecke operators on them. The smoothchar.py stuff is just there to give a user-friendly way of interacting with the results of the modsym computation, it is computationally trivial in itself.

[vbraun]

comment:22

PDF docs don't build

[loefflerd]

comment:23

Replying to @vbraun:

PDF docs don't build

Volker: I cannot reproduce this failure on my machine (since pdf doc building doesn't work on my system for unrelated reasons). Can you post the relevant log file?

[loefflerd]

comment:24

Wait, I found the error: exactly two characters missing in a single docstring in smoothchar.py. Testing, will upload shortly.

[loefflerd]

Changed branch from u/roed/minimal_twist_new to u/davidloeffler/minimal_twist_new

[loefflerd]

Changed commit from 7d4dd1b to c165835

[loefflerd]

comment:26

Right, that should fix it. Given the microscopic size of the change, I presume it is valid to reinstate the positive review.

---

New commits:

c165835

Fix broken latex in docstring

[roed314]

comment:27

Yep, I agree with the positive review.

[vbraun]

Changed branch from u/davidloeffler/minimal_twist_new to c165835