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Exponential for formal group of elliptic curve

asked 2020-03-17 18:58:09 +0200

George R gravatar image


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?


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answered 2020-03-18 13:09:37 +0200

FrédéricC gravatar image

You 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)
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answered 2020-06-15 17:45:57 +0200

dan_fulea gravatar image

This is a later answer, hoping that the quick good answer of FrédéricC 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)

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                                                                                                                       

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) )                                                                                         
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Asked: 2020-03-17 18:58:09 +0200

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Last updated: Jun 15 '20