Function: ellpadicregulator
Section: elliptic_curves
C-Name: ellpadicregulator
Prototype: GGLG
Help:ellpadicregulator(E,p,n,S): E elliptic curve/Q, S a vector of
 points in E(Q), p prime, n an integer; returns the p-adic
 cyclotomic regulator of the points of S at precision p^n.
Doc: Let $E/\Q$ be an elliptic curve. Return the determinant of the Gram
 matrix of the vector of points $S=(S_1,\cdots, S_r)$  with respect to the
 ``canonical'' cyclotomic $p$-adic height on $E$, given to $n$ ($p$-adic)
 digits.

 When $E$ has ordinary reduction at $p$, this is the expected Gram
 deteterminant in $\Q_p$.

 In the case of supersingular reduction of $E$ at $p$, the definition
 requires care: the regulator $R$ is an element of
 $D := H^1_{dR}(E) \otimes_\Q \Q_p$, which is a two-dimensional
 $\Q_p$-vector space spanned by $\omega$ and $\eta = x \omega$
 (which are defined over $\Q$) or equivalently but now over $\Q_p$
 by $\omega$ and $F\omega$ where $F$ is the Frobenius endomorphism on $D$
 as defined in \kbd{ellpadicfrobenius}. On $D$ we
 define the cyclotomic height $h_E = f \omega + g \eta$
 (see \tet{ellpadicheight}) and a canonical alternating bilinear form
 $[.,.]_D$ such that $[\omega, \eta]_D = 1$.

 For any $\nu \in D$, we can define a height $h_\nu := [ h_E, \nu ]_D$
 from $E(\Q)$ to $\Q_p$ and $\langle \cdot, \cdot \rangle_\nu$ the attached
 bilinear form. In particular, if $h_E = f \omega + g\eta$, then
 $h_\eta = [ h_E, \eta ]_D$ = f and $h_\omega = [ h_E, \omega ]_D = - g$
 hence $h_E = h_\eta \omega - h_\omega \eta$.
 Then, $R$ is the unique element of $D$ such that
 $$[\omega,\nu]_D^{r-1} [R, \nu]_D = \det(\langle S_i, S_j \rangle_{\nu})$$
 for all $\nu \in D$ not in $\Q_p \omega$. The \kbd{ellpadicregulator}
 function returns $R$ in the basis $(\omega, F\omega)$, which was chosen
 so that $p$-adic BSD conjectures are easy to state, see \kbd{ellpadicbsd}.

 Note that by definition
 $$[R, \eta]_D = \det(\langle S_i, S_j \rangle_{\eta})$$
 and
 $$[R, \omega+\eta]_D =\det(\langle S_i, S_j \rangle_{\omega+\eta}).$$