ASKSAGE: Sage Q&A Forum - Individual question feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Mon, 15 Jun 2020 10:45:57 -0500Exponential for formal group of elliptic curvehttps://ask.sagemath.org/question/50285/exponential-for-formal-group-of-elliptic-curve/ Hi,
This is a short and potentially quite simple question - the formal group object for an elliptic curve allows you to compute its logarithm, i.e. the isomorphism from the formal group to the additive formal group. Is it possible to find the exponential as well? If this is not built in, this would just reduce to finding a power series $g$ such that $f(g(T))=T$ for a given $f$. Is this possible with the power series tools?
Thanks!Tue, 17 Mar 2020 12:58:09 -0500https://ask.sagemath.org/question/50285/exponential-for-formal-group-of-elliptic-curve/Answer by FrédéricC for <p>Hi,</p>
<p>This is a short and potentially quite simple question - the formal group object for an elliptic curve allows you to compute its logarithm, i.e. the isomorphism from the formal group to the additive formal group. Is it possible to find the exponential as well? If this is not built in, this would just reduce to finding a power series $g$ such that $f(g(T))=T$ for a given $f$. Is this possible with the power series tools?</p>
<p>Thanks!</p>
https://ask.sagemath.org/question/50285/exponential-for-formal-group-of-elliptic-curve/?answer=50290#post-id-50290You can use .reverse:
sage: x = PowerSeriesRing(QQ, 'x').gen()
sage: y = exp(x) - 1
sage: y.reverse()
x - 1/2*x^2 + 1/3*x^3 - 1/4*x^4 + 1/5*x^5 - 1/6*x^6 + 1/7*x^7 - 1/8*x^8 + 1/9*x^9 - 1/10*x^10 + 1/11*x^11 - 1/12*x^12 + 1/13*x^13 - 1/14*x^14 + 1/15*x^15 - 1/16*x^16 + 1/17*x^17 - 1/18*x^18 + 1/19*x^19 + O(x^20)
Wed, 18 Mar 2020 07:09:37 -0500https://ask.sagemath.org/question/50285/exponential-for-formal-group-of-elliptic-curve/?answer=50290#post-id-50290Answer by dan_fulea for <p>Hi,</p>
<p>This is a short and potentially quite simple question - the formal group object for an elliptic curve allows you to compute its logarithm, i.e. the isomorphism from the formal group to the additive formal group. Is it possible to find the exponential as well? If this is not built in, this would just reduce to finding a power series $g$ such that $f(g(T))=T$ for a given $f$. Is this possible with the power series tools?</p>
<p>Thanks!</p>
https://ask.sagemath.org/question/50285/exponential-for-formal-group-of-elliptic-curve/?answer=52017#post-id-52017This is a later answer, hoping that the quick good answer of [FrédéricC](https://ask.sagemath.org/users/1557/fredericc/) will be accepted.
Let us place the above in the context of a sample elliptic curve, my choice is
E = EllipticCurve(QQ, [-1, 0])
FormalGroupOfE = E.formal_group()
f = FormalGroupOfE.log(prec=20)
print(f)
This gives:
t - 2/5*t^5 + 2/3*t^9 - 20/13*t^13 + 70/17*t^17 + O(t^20)
Here, `t` is a printed variable, that does not exist so far as `t`, so we introduce it.
sage: t = f.parent().gens()[0]
sage: t
t
Now we can associate the `reverse` object, and check the compositions...
sage: g = f.reverse()
sage: g
t + 2/5*t^5 + 2/15*t^9 + 44/975*t^13 + 422/27625*t^17 + O(t^20)
sage: g(f(t))
t + O(t^20)
sage: f(g(t))
t + O(t^20)
If $F$ is the "group law" of the formal group,
sage: F = FormalGroupOfE.group_law()
sage: F
t1 + t2 + 2*t1^4*t2 + 4*t1^3*t2^2 + 4*t1^2*t2^3 + 2*t1*t2^4 - 2*t1^8*t2
+ 8*t1^6*t2^3 + 16*t1^5*t2^4 + 16*t1^4*t2^5 + 8*t1^3*t2^6 - 2*t1*t2^8
+ O(t1, t2)^10
(code was split manually,)
and we have by definition $f(\ F(x,y)\ )=f(x)+f(y)$, so let us check the relation obtained from this one, after applying $g$:
sage: t1, t2 = F.parent().gens()
sage: F(t1, t2) == g( f(t1) + f(t2) )
True
Mon, 15 Jun 2020 10:45:57 -0500https://ask.sagemath.org/question/50285/exponential-for-formal-group-of-elliptic-curve/?answer=52017#post-id-52017