Opened 8 years ago
Closed 8 years ago
#14189 closed enhancement (fixed)
Extend modular degree and congruence modulus of elliptic curves over QQ to arbitrary level.
Reported by: | spice | Owned by: | spice |
---|---|---|---|
Priority: | major | Milestone: | sage-5.8 |
Component: | elliptic curves | Keywords: | modular degree, congruence modulus |
Cc: | was, aly.deines | Merged in: | sage-5.8.beta4 |
Authors: | Simon Spicer | Reviewers: | Aly Deines |
Report Upstream: | N/A | Work issues: | |
Branch: | Commit: | ||
Dependencies: | Stopgaps: |
Description (last modified by )
The modular degree of m an elliptic curve is defined as the degree of the map from X_0(N) down to E, when N is the conductor of E. See Agashe/Ribet/Stein's paper: The Modular Degree, Congruence Primes and Multiplicity One.
The congruence number for E is defined as the largest integer r such that there exists a cusp form of level N that is orthogonal to the cusp form attached to E w.r.t. the Petersson inner product, but congruent to it modulo r.
One of the major results known connecting the two values is that for elliptic curves over QQ, the modular degree divides the congruence number.
Both of these invariants generalize to higher level. Maps also exist from X_0(M*N) down to E for any positive integer M, so one can define modular degrees for any multiple of N. Likewise, we can ask, for any M*N, what the largest integer r is such that the space of cusp forms at level M*N forms generated by E for which mod r congruences exist between orthogonal cusp forms of level M*N.
This enhancement would allow us to compute the generalized notions of modular degree and congruence modulus and investigate whether divisibility still holds.
Attachments (1)
Change History (22)
comment:1 Changed 8 years ago by
- Cc aly.deines added
- Status changed from new to needs_review
comment:2 Changed 8 years ago by
- Description modified (diff)
comment:3 Changed 8 years ago by
- Description modified (diff)
comment:4 Changed 8 years ago by
- Description modified (diff)
comment:5 Changed 8 years ago by
- Reviewers set to aly.deines
- Status changed from needs_review to positive_review
comment:6 Changed 8 years ago by
- Status changed from positive_review to needs_work
comment:7 Changed 8 years ago by
- Status changed from needs_work to needs_review
comment:8 Changed 8 years ago by
- Status changed from needs_review to positive_review
comment:9 Changed 8 years ago by
- Reviewers changed from aly.deines to Aly Deines
comment:10 Changed 8 years ago by
- Status changed from positive_review to needs_work
The documentation doesn't build properly:
[plane_cur] /release/merger/sage-5.8.beta3/local/lib/python2.7/site-packages/sage/schemes/elliptic_curves/ell_rational_field.py:docstring of sage.schemes.elliptic_curves.ell_rational_field.EllipticCurve_rational_field.modular_degree:96: WARNING: Literal block expected; none found.
comment:11 Changed 8 years ago by
- Status changed from needs_work to needs_review
Documentation should now build.
comment:12 Changed 8 years ago by
Removed unnecessary asterisks: M*N
becomes
MN
.
comment:13 Changed 8 years ago by
Removes unnecessary asterisks for real this time.
comment:14 Changed 8 years ago by
- Status changed from needs_review to positive_review
comment:15 Changed 8 years ago by
- Status changed from positive_review to needs_work
There is an obvious doctest failure:
********************************************************************** File "/release/merger/sage-5.8.beta3/devel/sage-main/sage/schemes/elliptic_curves/ell_rational_field.py", line 3412: sage: E.congruence_number() # long time (4s on sage.math, 2011) Expected: 176 # Higher level cases Got: 176 **********************************************************************
comment:16 Changed 8 years ago by
Also: this patch seems to add about 20 seconds to the total test time of ell_rational_field.py
. Either it slows down some things or the added tests should be marked # long time
.
comment:17 Changed 8 years ago by
- Status changed from needs_work to needs_review
Doctest mentioned above has been fixed, and I've marked the relevant tests with # long time
as per suggestion.
comment:18 Changed 8 years ago by
- Status changed from needs_review to positive_review
Thanks Simon. It now only takes 2 seconds longer on my mac.
comment:19 Changed 8 years ago by
- Milestone changed from sage-5.8 to sage-5.9
comment:20 Changed 8 years ago by
- Milestone changed from sage-5.9 to sage-5.8
comment:21 Changed 8 years ago by
- Merged in set to sage-5.8.beta4
- Resolution set to fixed
- Status changed from positive_review to closed
I've modified
self._generalized_modcong_numbers()
to allow computation of just the modular degree or the congruence number singly; previously the were both computed and cached. Also, the function now returns a dictionary with one or both of the above invariants as entries as well as populating the cache; previously the values were just cached by the function.self.modular_degree()
andself.congruence_number()
have also been modified slightly to incorporate this.