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

12 JENNIFER S. BALAKRISHNAN, MIRELA ÇIPERIANI, AND WILLIAM STEIN
Observe that if we fix b0 ∈ S(p 2(n+1) DK, N) then Theorem 2.10 implies that there exists a Heegner
point x p n+1 of level N and discriminant p 2(n+1) DK such that Φ(x p n+1) = (b0, O p n+1). Our aim is to
compute
zn = tr K[p n+1 ]/Kn (y p n+1) =
Since the order of Gal(K[p n+1 ]/Kn) equals
σ∈Gal(K[p n+1 ]/Kn)
p −
DK
p
π(σ(x p n+1)).
hK and Gal(K[p n+1 ]/Kn) is the maxi-
mal subgroup of Gal(K[pn+1 ]/K) of order prime to p (this is where we use the assumption that
gcd(hK, p) = 1) and Gal(K[pn+1 ]/K) Cl(Opn+1), using (3.1) we have that
zn =
π(a · xpn+1) and the sum has
Hence we have that
p −
a∈Cl(O p n+1 ), p∤ord(a)
DK
p
hK terms. By Theorem 2.10 we know that
Φ(a · x p n+1) = a · Φ(x p n+1) = (b0, a −1 ).
zn =
a∈Cl(O p n+1 ), p∤ord(a)
π(Ψ −1 (b0, a)).
We know that the Heegner point τ ∈ X0(N) of level N and discriminant p 2(n+1) DK corresponds
to a class (under the action of Γ0(N)) of binary quadratic forms fτ = Ax 2 + Bxy + Cy 2 such that
(i) A, B, C ∈ Z, A > 0, N|A,
(ii) gcd(A, B, C) = gcd(A/N, B, CN) = 1,
(iii) B 2 − 4AC = p 2(n+1) DK.
Since τ = Ψ −1 (b0, a) ∈ X0(N) we have the following additional conditions:
(iv) B ≡ b0 (mod 2N),
(v) ΨIF (a) = fτ .
Finally since ΨIF is a group isomorphism [2, Theorem 5.2.4 and Theorem 5.2.8] the set
{Ψ −1 (b0, a)|a ∈ Cl(O p n+1), p ∤ ord(a)}
corresponds to the set of τ ∈ X0(N) such that fτ satisfies conditions (i)-(iv) listed above and
p ∤ ord(fτ ).
Algorithm 4.1 (Computing Heegner points zn ∈ E(Kn)).
(1) Fix b0 ∈ S(p 2(n+1) DK, N).
(2) Create a set Qb0 of
p −
DK
p
hK binary quadratic forms (A, B, C) that satisfy conditions
(i)-(iv) listed above, p does not divide the order of the equivalence class (under the action
of SL2(Z)) of binary quadratic forms [(A, B, C)], and any two binary quadratic forms in
Qb0 give rise to distinct equivalence classes. Let τf ∈ X0(N) be the Heegner point that
corresponds to the form f = Ax 2 + Bxy + Cy 2 .
(3) Compute zn =
f π(τf ) ∈ E(C) for f ∈ Qb0 , with sufficient numerical precision to satisfy
the natural consistency checks of the following step.
(4) Using lattice basis reduction (LLL), as explained in [14, §2.5] and implemented as the
algebraic_dependency command in [15] (which relies on the algdep command in [16]), algebraically
reconstruct the x-coordinate of zn ∈ E(C) and one of two possible y-coordinates.
Make sure that zn is defined over a Galois dihedral extension of degree 2pn that is ramified
exactly at p and the primes dividing the discriminant of K.