Function: ellpointtoz
Section: elliptic_curves
C-Name: zell
Prototype: GGp
Help: ellpointtoz(E,P): lattice point z corresponding to the point P on the
elliptic curve E.
Doc:
if $E/\C \simeq \C/\Lambda$ is a complex elliptic curve ($\Lambda =
\kbd{E.omega}$), computes a complex number $z$, well-defined modulo the
lattice $\Lambda$, corresponding to the point $P$; i.e.~such that
$P = [\wp_\Lambda(z),\wp'_\Lambda(z)]$ satisfies the equation
$$y^2 = 4x^3 - g_2 x - g_3,$$
where $g_2$, $g_3$ are the elliptic invariants.
If $E$ is defined over $\R$ and $P\in E(\R)$, we have more precisely, $0 \leq
\Re(t) < w1$ and $0 \leq \Im(t) < \Im(w2)$, where $(w1,w2)$ are the real and
complex periods of $E$.
\bprog
? E = ellinit([0,1]); P = [2,3];
? z = ellpointtoz(E, P)
%2 = 3.5054552633136356529375476976257353387
? ellwp(E, z)
%3 = 2.0000000000000000000000000000000000000
? ellztopoint(E, z) - P
%4 = [2.548947057811923643 E-57, 7.646841173435770930 E-57]
? ellpointtoz(E, [0]) \\ the point at infinity
%5 = 0
@eprog
If $E$ is defined over a general number field, the function returns the
values corresponding to the various complex embeddings of the curve
and of the point, in the same order as \kbd{E.nf.roots}:
\bprog
? E=ellinit([-22032-15552*x,0], nfinit(x^2-2));
? P=[-72*x-108,0];
? ellisoncurve(E,P)
%3 = 1
? ellpointtoz(E,P)
%4 = [-0.52751724240790530394437835702346995884*I,
-0.090507650025885335533571758708283389896*I]
? E.nf.roots
%5 = [-1.4142135623730950488016887242096980786, \\ x-> -sqrt(2)
1.4142135623730950488016887242096980786] \\ x-> sqrt(2)
@eprog
If $E/\Q_p$ has multiplicative reduction, then $E/\bar{\Q_p}$ is analytically
isomorphic to $\bar{\Q}_p^*/q^\Z$ (Tate curve) for some $p$-adic integer $q$.
The behavior is then as follows:
\item If the reduction is split ($E.\kbd{tate[2]}$ is a \typ{PADIC}), we have
an isomorphism $\phi: E(\Q_p) \simeq \Q_p^*/q^\Z$ and the function returns
$\phi(P)\in \Q_p$.
\item If the reduction is \emph{not} split ($E.\kbd{tate[2]}$ is a
\typ{POLMOD}), we only have an isomorphism $\phi: E(K) \simeq K^*/q^\Z$ over
the unramified quadratic extension $K/\Q_p$. In this case, the output
$\phi(P)\in K$ is a \typ{POLMOD}.
\bprog
? E = ellinit([0,-1,1,0,0], O(11^5)); P = [0,0];
? [u2,u,q] = E.tate; type(u) \\ split multiplicative reduction
%2 = "t_PADIC"
? ellmul(E, P, 5) \\ P has order 5
%3 = [0]
? z = ellpointtoz(E, [0,0])
%4 = 3 + 11^2 + 2*11^3 + 3*11^4 + 6*11^5 + 10*11^6 + 8*11^7 + O(11^8)
? z^5
%5 = 1 + O(11^9)
? E = ellinit(ellfromj(1/4), O(2^6)); x=1/2; y=ellordinate(E,x)[1];
? z = ellpointtoz(E,[x,y]); \\ t_POLMOD of t_POL with t_PADIC coeffs
? liftint(z) \\ lift all p-adics
%8 = Mod(8*u + 7, u^2 + 437)
@eprog