1Function: ellpadicbsd
2Section: elliptic_curves
3C-Name: ellpadicbsd
4Prototype: GGLDG
5Help: ellpadicbsd(E, p, n, {D = 1}): returns [r,Lp] where
6 r is the (conjectural) analytic rank of the p-adic L-function attached
7 to the quadratic twist E_D and Lp is (conjecturally) equal
8 to the product of the p-adic regulator and the cardinal of the
9 Tate-Shafarevich group.
10Doc: Given an elliptic curve $E$ over $\Q$, its quadratic twist $E_D$
11 and a prime number $p$, this function is a $p$-adic analog of the complex
12 functions \tet{ellanalyticrank} and \tet{ellbsd}. It calls \kbd{ellpadicL}
13 with initial accuracy $p^n$ and may increase it internally;
14 it returns a vector $[r, L_p]$ where
15
16 \item $L_p$ is a $p$-adic number (resp. a pair of $p$-adic numbers if
17 $E$ has good supersingular reduction) defined modulo $p^N$, conjecturally
18 equal to $R_p S$, where $R_p$ is the $p$-adic regulator as given by
19 \tet{ellpadicregulator} (in the basis $(\omega, F \omega)$) and $S$ is the
20 cardinal of the Tate-Shafarevich group for the quadratic twist $E_D$.
21
22 \item $r$ is an upper bound for the analytic rank of the $p$-adic
23 $L$-function attached to $E_D$: we know for sure that the $i$-th
24 derivative of $L_p(E_D,.)$ at $\chi^0$ is $O(p^N)$ for all $i < r$
25 and that its $r$-th derivative is nonzero; it is expected that the true
26 analytic rank is equal to the rank of the Mordell-Weil group $E_D(\Q)$,
27 plus $1$ if the reduction of $E_D$ at $p$ is split multiplicative;
28 if $r = 0$, then both the analytic rank and the Mordell-Weil rank are
29 unconditionnally $0$.
30
31 Recall that the $p$-adic BSD conjecture (Mazur, Tate, Teitelbaum, Bernardi,
32 Perrin-Riou) predicts an explicit link between $R_p S$ and
33 $$(1-p^{-1}  F)^{-2} \cdot L_p^{(r)}(E_D, \chi^0) / r! $$
34 where $r$ is the analytic rank of the $p$-adic $L$-function attached to
35 $E_D$ and $F$ is the Frobenius on $H^1_{dR}$; see \tet{ellpadicL}
36 for definitions.
37 \bprog
38 ? E = ellinit("11a1"); p = 7; n = 5; \\ good ordinary
39 ? ellpadicbsd(E, 7, 5) \\ rank 0,
40 %2 = [0, 1 + O(7^5)]
41
42 ? E = ellinit("91a1"); p = 7; n = 5; \\ non split multiplicative
43 ? [r,Lp] = ellpadicbsd(E, p, n)
44 %5 = [1, 2*7 + 6*7^2 + 3*7^3 + 7^4 + O(7^5)]
45 ? R = ellpadicregulator(E, p, n, E.gen)
46 %6 = 2*7 + 6*7^2 + 3*7^3 + 7^4 + 5*7^5 + O(7^6)
47 ? sha = Lp/R
48 %7 = 1 + O(7^4)
49
50 ? E = ellinit("91b1"); p = 7; n = 5; \\ split multiplicative
51 ? [r,Lp] = ellpadicbsd(E, p, n)
52 %9 = [2, 2*7 + 7^2 + 5*7^3 + O(7^4)]
53 ? ellpadicregulator(E, p, n, E.gen)
54 %10 = 2*7 + 7^2 + 5*7^3 + 6*7^4 + 2*7^5 + O(7^6)
55 ? [rC, LC] = ellanalyticrank(E);
56 ? [r, rC]
57 %12 = [2, 1]  \\ r = rC+1 because of split multiplicative reduction
58
59 ? E = ellinit("53a1"); p = 5; n = 5; \\ supersingular
60 ? [r, Lp] = ellpadicbsd(E, p, n);
61 ? r
62 %15 = 1
63 ? Lp
64 %16 = [3*5 + 2*5^2 + 2*5^5 + O(5^6), \
65        5 + 3*5^2 + 4*5^3 + 2*5^4 + 5^5 + O(5^6)]
66 ? R = ellpadicregulator(E, p, n, E.gen)
67 %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)]
68 \\ expect Lp = R*#Sha, hence (conjecturally) #Sha = 1
69
70 ? E = ellinit("84a1"); p = 11; n = 6; D = -443;
71 ? [r,Lp] = ellpadicbsd(E, 11, 6, D) \\ Mordell-Weil rank 0, no regulator
72 %19 = [0, 3 + 2*11 + O(11^6)]
73 ? lift(Lp)  \\ expected cardinal for Sha is 5^2
74 %20 = 25
75 ? ellpadicbsd(E, 3, 12, D)  \\ at 3
76 %21 = [1, 1 + 2*3 + 2*3^2 + O(3^8)]
77 ? ellpadicbsd(E, 7, 8, D)   \\ and at 7
78 %22 = [0, 4 + 3*7 + O(7^8)]
79 @eprog
80