Function: ellpadicbsd
Section: elliptic_curves
C-Name: ellpadicbsd
Prototype: GGLDG
Help: ellpadicbsd(E, p, n, {D = 1}): returns [r,Lp] where
r is the (conjectural) analytic rank of the p-adic L-function attached
to the quadratic twist E_D and Lp is (conjecturally) equal
to the product of the p-adic regulator and the cardinal of the
Tate-Shafarevich group.
Doc: Given an elliptic curve $E$ over $\Q$, its quadratic twist $E_D$
and a prime number $p$, this function is a $p$-adic analog of the complex
functions \tet{ellanalyticrank} and \tet{ellbsd}. It calls \kbd{ellpadicL}
with initial accuracy $p^n$ and may increase it internally;
it returns a vector $[r, L_p]$ where
\item $L_p$ is a $p$-adic number (resp. a pair of $p$-adic numbers if
$E$ has good supersingular reduction) defined modulo $p^N$, conjecturally
equal to $R_p S$, where $R_p$ is the $p$-adic regulator as given by
\tet{ellpadicregulator} (in the basis $(\omega, F \omega)$) and $S$ is the
cardinal of the Tate-Shafarevich group for the quadratic twist $E_D$.
\item $r$ is an upper bound for the analytic rank of the $p$-adic
$L$-function attached to $E_D$: we know for sure that the $i$-th
derivative of $L_p(E_D,.)$ at $\chi^0$ is $O(p^N)$ for all $i < r$
and that its $r$-th derivative is non-zero; it is expected that the true
analytic rank is equal to the rank of the Mordell-Weil group $E_D(\Q)$,
plus $1$ if the reduction of $E_D$ at $p$ is split multiplicative;
if $r = 0$, then both the analytic rank and the Mordell-Weil rank are
unconditionnally $0$.
Recall that the $p$-adic BSD conjecture (Mazur, Tate, Teitelbaum, Bernardi,
Perrin-Riou) predicts an explicit link between $R_p S$ and
$$(1-p^{-1} F)^{-2} \cdot L_p^{(r)}(E_D, \chi^0) / r! $$
where $r$ is the analytic rank of the $p$-adic $L$-function attached to
$E_D$ and $F$ is the Frobenius on $H^1_{dR}$; see \tet{ellpadicL}
for definitions.
\bprog
? E = ellinit("11a1"); p = 7; n = 5; \\ good ordinary
? ellpadicbsd(E, 7, 5) \\ rank 0,
%2 = [0, 1 + O(7^5)]
? E = ellinit("91a1"); p = 7; n = 5; \\ non split multiplicative
? [r,Lp] = ellpadicbsd(E, p, n)
%5 = [1, 2*7 + 6*7^2 + 3*7^3 + 7^4 + O(7^5)]
? R = ellpadicregulator(E, p, n, E.gen)
%6 = 2*7 + 6*7^2 + 3*7^3 + 7^4 + 5*7^5 + O(7^6)
? sha = Lp/R
%7 = 1 + O(7^4)
? E = ellinit("91b1"); p = 7; n = 5; \\ split multiplicative
? [r,Lp] = ellpadicbsd(E, p, n)
%9 = [2, 2*7 + 7^2 + 5*7^3 + O(7^4)]
? ellpadicregulator(E, p, n, E.gen)
%10 = 2*7 + 7^2 + 5*7^3 + 6*7^4 + 2*7^5 + O(7^6)
? [rC, LC] = ellanalyticrank(E);
? [r, rC]
%12 = [2, 1] \\ r = rC+1 because of split multiplicative reduction
? E = ellinit("53a1"); p = 5; n = 5; \\ supersingular
? [r, Lp] = ellpadicbsd(E, p, n);
? r
%15 = 1
? Lp
%16 = [3*5 + 2*5^2 + 2*5^5 + O(5^6), \
5 + 3*5^2 + 4*5^3 + 2*5^4 + 5^5 + O(5^6)]
? R = ellpadicregulator(E, p, n, E.gen)
%17 = [3*5 + 2*5^2 + 2*5^5 + O(5^6), 5 + 3*5^2 + 4*5^3 + 2*5^4 + O(5^5)]
\\ expect Lp = R*#Sha, hence (conjecturally) #Sha = 1
? E = ellinit("84a1"); p = 11; n = 6; D = -443;
? [r,Lp] = ellpadicbsd(E, 11, 6, D) \\ Mordell-Weil rank 0, no regulator
%19 = [0, 3 + 2*11 + O(11^6)]
? lift(Lp) \\ expected cardinal for Sha is 5^2
%20 = 25
? ellpadicbsd(E, 3, 12, D) \\ at 3
%21 = [1, 1 + 2*3 + 2*3^2 + O(3^8)]
? ellpadicbsd(E, 7, 8, D) \\ and at 7
%22 = [0, 4 + 3*7 + O(7^8)]
@eprog