Let the elliptic curve E be defined by the equation
formula here
with a1, …, a6 ∈ ℤ.
Define a finite set of places S =
{q1, …, qs−1,
qs = ∞} of ℚ and put
Q = max {q1, …, qs−1}. Let
E(ℚ) denote the set of (x, y) ∈ ℚ2
satisfying (1) and the infinite point [Oscr ].
The multiplicative height of a rational point P =
(x, y) ∈ E(ℚ) is defined as the following product over all
places q of ℚ (including q = ∞):
formula here
where the [mid ]x[mid ]qs are the normalized multiplicative
absolute values of ℚ corresponding to the places q.