#11980: Improve naive point counting and implement zeta_function for
hyperelliptic
curves over finite fields
-------------------------------------------------+-------------------------
Reporter: dkrenn | Owner: cremona
Type: enhancement | Status:
Priority: major | needs_review
Component: elliptic curves | Milestone: sage-5.12
Keywords: sd35 hyperelliptic curve sd53 | Resolution:
Authors: Daniel Krenn, Jean-Pierre | Merged in:
Flori | Reviewers: Marco
Report Upstream: N/A | Streng
Branch: u/jpflori/ticket/11980 | Work issues:
Dependencies: #15148 | Commit:
| Stopgaps:
-------------------------------------------------+-------------------------
Changes (by {'newvalue': u'Daniel Krenn, Jean-Pierre Flori', 'oldvalue':
u'Daniel Krenn'}):
* status: needs_work => needs_review
* work_issues: correct number of points at infinity, examples for even
degree =>
* author: Daniel Krenn => Daniel Krenn, Jean-Pierre Flori
* branch: => u/jpflori/ticket/11980
Old description:
> The following is not implemented (but can be done for hyperelliptic
> curves):
>
> {{{
> sage: P.<x> = PolynomialRing(GF(9,'a'))
> sage: H = HyperellipticCurve(x^5+x^2+1)
> sage: H.count_points(5)
> Traceback (most recent call last)
> ...
> NotImplementedError: Point counting only implemented for schemes over
> prime fields
> }}}
> Also, having a `count_points` for all cases, the Frobenius polynomial can
> be calculated. At the moment, this is calculated via `frobenius_matrix`,
> which is not always available:
> {{{
> sage: R.<t> = PolynomialRing(GF(8, 'a'))
> sage: H = HyperellipticCurve(t^5 + t + 2, t + 1)
> sage: H.frobenius_polynomial()
> Traceback (most recent call last):
> ...
> NotImplementedError: only implemented for curves y^2 = f(x)
> }}}
>
> Further, the zeta function can be calculated easily when the Frobenius
> polynomial is known.
>
> Patch follows.
New description:
The following is not implemented (but can be done for hyperelliptic
curves):
{{{
sage: P.<x> = PolynomialRing(GF(9,'a'))
sage: H = HyperellipticCurve(x^5+x^2+1)
sage: H.count_points(5)
Traceback (most recent call last)
...
NotImplementedError: Point counting only implemented for schemes over
prime fields
}}}
Also, having a `count_points` for all cases, the Frobenius polynomial can
be calculated. At the moment, this is calculated via `frobenius_matrix`,
which is not always available:
{{{
sage: R.<t> = PolynomialRing(GF(8, 'a'))
sage: H = HyperellipticCurve(t^5 + t + 2, t + 1)
sage: H.frobenius_polynomial()
Traceback (most recent call last):
...
NotImplementedError: only implemented for curves y^2 = f(x)
}}}
Further, the zeta function can be calculated easily when the Frobenius
polynomial is known.
Use git branch.
--
--
Ticket URL: <http://trac.sagemath.org/ticket/11980#comment:11>
Sage <http://www.sagemath.org>
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