MAGMA Computational Algebra System

The formal group

These functions are for elliptic curves over any (exact) field.

FormalGroupLaw(E, prec) : CrvEll, RngIntElt -> RngMPolElt

This returns a polynomial in two variables, T₁ + T₂ + ..., which expresses the formal group law associated to addition on E, up to precision prec. (More precisely, it contains the terms which have total degree less than or equal to prec.)

The formal variables T₁ and T₂ may be identified with the function -x/y on E, where x and y are the standard affine coordinates on E. (Note that this function is a local parameter in a neighbourhood of O_E.)

FormalGroupHomomorphism(phi, prec) : MapSch, RngIntElt -> RngSerPowElt

This gives the homomomorphism of formal groups associated to the isogeny phi, returned as a power series in one variable, up to precision prec. As in FormalGroupLaw, this is in terms of the parameter -x/y on each curve.

FormalLog(E) : CrvEll -> RngSerPowElt, PtEll

    Precision: RngIntElt                Default: 10

This returns the formal logarithm for the elliptic curve E as a power series f(T), where the parameter T is the function -x/y on E. (This is the same parameter used in FormalGroupLaw).

The function also returns a point P(T) on E, with coordinates in a Laurent series ring with generator T, which again corresponds to -x/y. Thus P(T) is a formal parametrization of E in a neighbourhood of O_E.