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

10 JENNIFER S. BALAKRISHNAN, MIRELA ÇIPERIANI, AND WILLIAM STEIN
(3.2)
(3.3)
(3.4)
Since Gal(K[p n+1 ]/Kn) Gal(K[p]/K) for every n ≥ 0, it follows that
trKn+2/Kn+1 (zn+2) = apzn+1 − zn for n ≥ 1;
apz1 − apz0 if p is inert,
trK2/K1 (z2) =
apz1 − (ap − 2)z0 if p splits;
(ap − 1)(ap + 1) − p
trK1/K(z1) =
z0 if p is inert,
(ap − 1) 2 − p z0 if p splits.
We can now see that for every n ≥ 0, we have that tr Kn+1/Kn (zn+1) = unzn for some unit in
un ∈ Zp[Gal(K∞/K)] under the following conditions:
(3.5)
p does not divide (ap − 1)ap(ap + 1)
p does not divide (ap − 1)ap
if p is inert,
if p splits.
More precisely, if the above conditions hold, then we have
u0 =
(ap − 1)(ap + 1) − p
(ap − 1)
if p is inert,
2 − p if p splits;
u1 =
ap − apu −1
0 tr K1/K if p is inert,
ap − (ap − 2)u −1
0 tr K1/K if p splits;
un = ap − u −1
n−1 tr Kn/Kn−1
for n ≥ 2.
Throughout the paper we assume that the conditions (3.5) hold and hence E has good ordinary
non-anomalous reduction at p. Following Mazur and Rubin [11] we consider the anticyclotomic
universal norm module
U = lim E(Kn) ⊗ Zp,
←−
n
where the transition maps are the trace maps. Then the cyclotomic p-adic height pairing
is τ-Hermitian, i.e.
h : U ⊗Λ U τ → Γcycl ⊗Zp Λ ⊗Zp Qp
h(u ⊗ v) = h(u ⊗ v) τ = h(τu ⊗ τv),
for all universal norms u, v ∈ U. Observe that since p is a prime of ordinary non-anomalous reduction
which does not divide the product of the Tamagawa numbers, the cyclotomic p-adic height pairing
takes values in Γcycl ⊗Zp Λ. We know that U is free of rank one over Λ = Zp[Gal(K∞/K)]. This
implies that the image of the cyclotomic p-adic height pairing is generated by the Λ-adic regulator 1
R ∈ Γcycl ⊗Zp Λ. We would like to compute R, and to do so we use Heegner points.
Our assumption of the conditions (3.5) implies that Heegner points give rise to the Heegner
submodule H ⊆ U. In particular, the points
c0 = z0 and cn =
n−1
i=0
ui
−1
zn for n ≥ 1
are trace compatible and correspond to an element c ∈ U. Mazur and Rubin define the Heegner
L-function
L := h(c ⊗ c τ ) ∈ Γcycl ⊗Zp Λ.
One can easily see that L = R char(U/H) 2 .
1 Note that this definition of the Λ-adic regulator differs slightly from that of Howard [9].