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
 determinant 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
 $$[R, \omega+\eta]_{D} =\det(\langle S_{i}, S_{j} \rangle_{\omega+\eta}).$$
