#9646: Incorrect calculation of elliptic curve formal group law
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Reporter: hlaw | Owner: cremona
Type: defect | Status: new
Priority: major | Milestone: sage-5.0
Component: elliptic curves | Keywords: elliptic curve formal group law
Author: hlaw | Upstream: N/A
Reviewer: | Merged:
Work_issues: |
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Old description:
> If `F(t1, t2)` is a formal group law, then `F(t1, t2) = t1 + t2 (mod
> t1*t2)`. So in particular, the coefficients of `t1^i` and `t2^i` are
> zero for all `i > 1`. However the formal group law of an elliptic curve
> as returned by Sage includes (at least) the terms `-a1^2*t1^3` and
> `-a1^2*t2^2`, as the following example shows:
> {{{
> sage: P.<a1, a2, a3, a4, a6> = PolynomialRing(ZZ, 5)
> sage: E = EllipticCurve(list(P.gens()))
> sage: F = E.formal().group_law(prec = 4)
> sage: t2 = F.parent().gen()
> sage: t1 = F.parent().base_ring().gen()
> sage: F(t1, 0)
> t1 - a1^2*t1^3 # should be t1
> sage: F(0, t2)
> t2 - a1^2*t2^3
> }}}
> Note also that the coefficient of `t1^2*t2 + t1*t2^2` returned by sage is
> `-a1^2 - a2`, whereas, according to Silverman AEC IV.1, it should be
> simply `-a2`.
New description:
If `F(t1, t2)` is a formal group law, then `F(t1, t2) = t1 + t2 (mod
t1*t2)`. So in particular, the coefficients of `t1^i` and `t2^i` are zero
for all `i > 1`. However the formal group law of an elliptic curve as
returned by Sage includes (at least) the terms `-a1^2*t1^3` and
`-a1^2*t2^2`, as the following example shows:
{{{
sage: P.<a1, a2, a3, a4, a6> = PolynomialRing(ZZ, 5)
sage: E = EllipticCurve(list(P.gens()))
sage: F = E.formal().group_law(prec = 4)
sage: t2 = F.parent().gen()
sage: t1 = F.parent().base_ring().gen()
sage: F(t1, 0)
t1 - a1^2*t1^3 # should be t1
sage: F(0, t2)
t2 - a1^2*t2^3
}}}
Note also that the coefficient of `t1^2*t2 + t1*t2^2` returned by sage is
`-a1^2 - a2`, whereas, according to Silverman AEC IV.1, it should be
simply `-a2`.
This was obtained in Sage 4.4.4 on MacOS X 10.5.8 (32 bit).
--
Comment(by hlaw):
Added my Sage version and system info.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/9646#comment:1>
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