2022-11-21 02:07:09 +0100 received badge ● Popular Question (source) 2022-10-10 05:28:34 +0100 marked best answer Formal Group Law on Elliptic Curves It is very likely I am misunderstanding something, in which case I apologize in advance. Here's my question. Start with an elliptic curve $E/\mathbb Z_p$ and $M=(p)$ the maximal ideal of $\mathbb Z_p$. I am considering $\hat E$ the formal group and say $F$ its group law. I am trying to take two points in $M$ and add them using the group law, something that should be possible and converge (see Silverman AEC pg 119). On Sage, this is easy to set up: E=EllipticCurve('61a1') Ep=E.change_ring(Qp(5)) Ef=Ep.formal_group() F=Ef.group_law(16) a=Qp(5, prec=16)(5+3*5^2) b=Qp(5, prec=16)(5^2-2*5^3) X=Ef.x(16) Y=Ef.y(16) I=Ef.inverse(16)  Now the command F(a,b) pops an error saying that a,b need to have positive valuation. The issue is that it is working over the ring $\mathbb Z_p[[t]]$ over which, indeed a,b have valuation 0. But, unless I am misunderstanding something, since a and b have positive valuation in $\mathbb Z_p$, we should be allowed to add them, right? The same issue does not exist with the other functions, like X(a), Y(a), I(a) etc. I have found a way to get around this issue by taking exp(log(a)+log(b)) where exp and log are the formal group logarithmic and exponential functions, but surely this is not a nice way. So is this indeed a bug or am I misunderstanding something? And if it is a bug, is there an easy/good way to fix/get around it? 2022-10-10 05:28:34 +0100 received badge ● Scholar (source) 2022-10-10 05:28:34 +0100 received badge ● Supporter (source) 2022-09-30 08:06:11 +0100 received badge ● Student (source) 2022-09-28 10:09:25 +0100 asked a question Formal Group Law on Elliptic Curves Formal Group Law on Elliptic Curves It is very likely I am misunderstanding something, in which case I apologize in adva