#11683: ell_curve_isogeny initialization
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Reporter: saraedum | Owner: cremona
Type: enhancement | Status: new
Priority: minor | Milestone: sage-4.7.2
Component: elliptic curves | Keywords: startup, initialization, sd32
Work_issues: | Upstream: N/A
Reviewer: | Author: Julian Rueth
Merged: | Dependencies:
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Comment(by cremona):
Sorry for the delay. I'll document this better and make sure that the
current documentation and description of the precomputation fits the
existing code -- there are some choice to be made. Meanwhile, here's a
small amount of explanatory theory:
For l=2,3,5,7,13, the curve {{{X_0(l)}}} has genus zero, so its function
field is pure transcendental. The Fricke Module, denoted t, is a
generator of the function field such that t=0 and t=oo lie above j=oo; the
normalization here is completely classical, and the situation is fully
defined by giving j as a degree l+1 rational function in t, which is what
the Fricke_module dict in the code does (though not for l=2 which is
simpler and handled entirely separately).
Next one writes down a "universal elliptic curve with l-isogeny", say
{{{E_t}}}, as a curve over Q(t) with j-invariant j (viewed as an element
of Q(t)). This is not unique! I have used different choices in the past
and this needs sorting out since the later formulas depend on the choice.
A good choice is one such that {{{E_t}}} is defined and with good
reduction at all t except t=0 and t=oo (which are irrelevant since those
values correspond to j=oo); but one is forced to also have bad reduction
at values of t above j=0 and j=1728 when l=1 (mod 3, resp. 4). Once the
model for {{{E_t}}} is chosen, Psi is the monic factor of degree (l-1)/2
of the l-division polynomial of {{{E_t}}} which defined the isogeny, and
which is used in the isogeny code.
I am writing this up -- there's an old preprint of me and Mark Watkins
which never got finished, and now there's more delay since with my student
Kimi I am extending all this to the cases where {{{X_0(l)}}} is
hyperelliptic.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/11683#comment:10>
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