#11980: hyperelliptic curves: count_points and frobenius_polynomial do not work
in
all cases, not zeta_function available
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Reporter: dkrenn |
Owner: cremona
Type: enhancement |
Status: needs_work
Priority: major |
Milestone: sage-4.8
Component: elliptic curves |
Keywords: sd35 hyperelliptic curve
Work_issues: correct number of points at infinity, examples for even degree |
Upstream: N/A
Reviewer: Marco Streng |
Author: Daniel Krenn
Merged: |
Dependencies:
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Changes (by mstreng):
* keywords: => sd35 hyperelliptic curve
* reviewer: => Marco Streng
* work_issues: => correct number of points at infinity, examples for
even degree
Comment:
In case f has even degree, the points at infinity are counted incorrectly
by this patch. To get the correct characteristic polynomial of Frobenius,
we must consider the smooth model of the curve. This is what you get when
you desingularize at infinity. This smooth curve has 0, 1 or 2 points at
infinity defined over the base field, so instead of just with 1, the count
must start with 0, 1 or 2. (The count will not always agree with the count
from the base class, as the base class thinks about the singular curve.
See also #11800.)
In case {{{H : y^2 = f(x)}}} with degree(f) > 2, there are three cases:
* degree(f) is odd: there is a unique point at infinity over the base
field (a rational Weierstrass point at infinity)
* degree(f) is even and its leading coefficient is a square: there are
two points at infinity defined over the base field
* degree(f) is even and its leading coefficient is not a square: there is
no point at infinity defined over the base field (there are two points
over a quadratic extension, and they are conjugate to each other)
A similar list of cases can be written down in greater generality
(including characteristic 2).
Alternative to get the patch in quickly: raise an error if the curve is
not of the form {{{y^2 = f(x)}}} with f of odd degree > 2.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/11980#comment:5>
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