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

14 JENNIFER S. BALAKRISHNAN, MIRELA ÇIPERIANI, AND WILLIAM STEIN
If the point P ∈ E(F ) reduces
• to the identity in E(k℘) for all primes ℘ | p, where k℘ is the residue field of F at ℘, and
• to a non-singular point at all primes of bad reduction,
then we have the following formula (see Mazur-Stein-Tate [12]) for computing the cyclotomic p-adic
height of P :
(5.1) hp,F (P ) = 1
p ·
⎛
⎝
logp(NF℘/Qp (σ℘(P ))) −
⎞
ordv(dv(P )) · logp(#kv) ⎠ ,
℘|p
where σ℘ is the p-adic sigma function at the prime ℘ and kv is the residue field of F at v. Note that
this assumes that we are working with a minimal model of E/F .
Suppose now that F has class number 1. Then since OF is a principal ideal domain, there is a
global choice of denominator d(P ) and the above formula (5.1) simplifies to the following:
(5.2) hp,F (P ) = 1
p · log ⎛
⎝
p
σ℘(P )
NF℘/Qp d(P )
⎞
⎠ .
℘|p
Note that our point P does not necessarily satisfy the two conditions listed above; in order to
use the above formulas we compute the height of mP , where m ∈ Z is such that mP does reduce
appropriately. Then we use that the height pairing is a quadratic form to recover the height of P .
We now need to compute the denominator d(mP ) of mP . We start by computing the denominator
d(P ) of P . If α is an algebraic number an integer denominator of α is some positive integer d such
that dα is an algebraic integer. Naturally d is not unique, since any positive multiple of d is also an
integer denominator of α. The notion of integer denominator is computationally useful and easy to
compute, since we represent algebraic numbers in terms of a power basis.
Algorithm 5.1 (Denominator d(P ) of P ∈ E(F ) with F of class number 1).
(1) Input P = (x, y), where P ∈ E(F ).
(2) Read off an integer denominator d := d(x) of x, and consider the ideal (x) = (d·x)
(d)
, where
(d · x) and (d) denote OF -ideals.
(3) Simplify (x) = (d·x)
(d) by canceling common prime ideals in the numerator and denominator
ideals.
(4) What is left in the factored denominator ideal is a perfect square of prime ideals in OF , and
the square root of this ideal is generated by the desired denominator d(P ).
One could repeat the above process for mP ∈ E(F ), but this may be infeasible due to the
numerical explosion in the coordinates of mP . Instead, we make use of m-division polynomials to
write d(mP ) in terms of d(P ). Using Proposition 1 of [19], we easily deduce the following:
Proposition 5.2. Let F be a number field of class number one, fm the m-th division polynomial
of an elliptic curve E/F , and P ∈ E(F ) a non-torsion point that reduces to a non-singular point in
E(kv) for every bad reduction prime v. Then the denominators d(P ), d(mP ) ∈ OF are related as
follows:
d(mP ) = fm(P )d(P ) m2
.
Proof. By Proposition 1 of [19] we know that
v∤p
d(mP ) = ufm(P )d(P ) m2
,
where u ∈ F is a unit in the completion of F at every finite prime. Since F has class number 1 it
follows that u is a unit in OF . Then as d(mP ) is only defined up to units the result follows.