p-ADIC HEIGHTS OF HEEGNER POINTS AND ANTICYCLOTOMIC ...

26 JENNIFER S. BALAKRISHNAN, MIRELA ÇIPERIANI, AND WILLIAM STEIN
(2) We know that x(zn) ∈ Ln and Ln = K 〈τ〉
n
is totally ramified at p. Let pn be the unique
prime of Ln above p and Lpn the completion of Ln at pn. Since p splits in K/Q we know
that there are two primes ℘n, ℘ ′ n of Kn that divide pn. Hence, K℘n = K℘ ′ n = Lpn and
res℘n zn, res℘ ′ n zn ∈ E(Lpn). Since z τ n = −zn it follows that res℘ ′ n zn = − res℘n zn.
We p-adically construct res℘n zn ∈ E(Lpn ). Observe that while we need to choose the
sign of the y-coordinate, this choice is irrelevant in the end since res℘ ′ n zn = − res℘n zn and
the sigma function is odd.
(3) Compute m. Use it to compute res℘n (mzn) ∈ E(Lpn ), fm(x(zn)), tm = −
Lpn .
(4) Recover
hp,Kn (zn) = 1
p · m2
2
σp(tm)
logp −NLpn /Qp
− m
fm(x(zn))
2
logp(D(zn)) = 1
p · m2
2
σp(tm)
logp NLpn /Qp
− m
fm(x(zn))
2
logp(D(zn)) .
x(res℘n (mzn))
y(res℘n
(mzn)) ∈
Algorithm 7.5 (The p-adic height hp,Kn(zn) of a Heegner point zn ∈ E(Kn)). Assume that
a) E(Q)tors = O,
b) the analytic rank of E/Q equals 0, and
c) p is inert in K/Q.
It follows that mo = 1.
(1) Compute x(zn) ∈ R using Algorithm 4.1. (If the x-coordinate of the first zn is not real then
we use an element of Gal(Kn/K) to find a conjugate that is. This is the zn that we want.)
Save the leading coefficient D(zn) of the minimal polynomial h(x) of x(zn) over Z.
(2) We know that x(zn) ∈ Ln. As before Ln = K 〈τ〉
n is totally ramified at p, pn is the unique
prime of Ln above p, and Lpn the completion of Ln at pn. Since p is inert in K/Q there is
a unique prime ℘n of Kn above p and K℘n = Lpn[ √ DK].
We p-adically construct res℘n zn ∈ E(Lpn [√DK]). Observe that while we need to choose
the sign of the y-coordinate, this choice is irrelevant since the sigma function is odd.
(3) Compute m. Use it to compute res℘n(mzn) ∈ E(Lpn[ √ DK]), fm(x(zn)) ∈ Lpn, and tm =
x(res℘n (mzn))
− y(res℘n (mzn)) ∈ Lpn [√DK]. (4) Recover
hp,Kn(zn) = 1
p · m 2
log p
NK℘n /Qp (σp(tm))
2
NLpn /Qp (fm(x(zn)))
− m 2 log p(D(zn))
8. Computing p-adic height pairings of Heegner points
In this section, we give an algorithm to compute the p-adic height pairing 〈zn, σzn〉 Kn for σ ∈
Gal(Kn/K) and then illustrate it in an example. Recall that ɛ denotes the sign of the functional
equation of E/Q. Then since hp,Kn (σzn) = hp,Kn (zn) for every σ ∈ Gal(Kn/K), we have that
〈zn, σzn〉 Kn = hp,Kn (zn − ɛσzn) − hp,Kn (zn) − hp,Kn (−ɛσzn)
= hp,Kn (zn − ɛσzn) − 2hp,Kn (zn).
It remains to discuss the auxiliary computation of hp,Kn (zn − ɛσzn).
We will assume that E has trivial rational torsion which implies that there exist a Heegner point
zn ∈ Kn such that z τ n = −ɛzn. It then follows that
(σzn − ɛσ −1 zn) τ = −ɛσ −1 zn + σzn.
.