#14189: Extend modular degree and congruence modulus of elliptic curves over QQ
to
arbitrary level.
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Reporter: spice | Owner: spice
Type: enhancement | Status:
needs_review
Priority: major | Milestone:
sage-5.8
Component: elliptic curves | Resolution:
Keywords: modular degree, congruence modulus | Work issues:
Report Upstream: N/A | Reviewers:
Authors: | Merged in:
Dependencies: | Stopgaps:
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Description changed by spice:
Old description:
> 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
> [http://wstein.org/papers/ars-congruence/current.pdf 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 Peterssen 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 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.
>
> For more precise definitions of the two concepts see
New description:
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
[http://wstein.org/papers/ars-congruence/current.pdf 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 Peterssen 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 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.
--
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/14189#comment:2>
Sage <http://www.sagemath.org>
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