curves and points whose coecients, after few steps, are equal up to hundreds of digits, whereas the dierence between the archimedean heights is still consistent. Making bad choices in the AGM sequence leads to dierent limiting values. Indeed,
in the real setting when bad choices are made the error scales exactly by a quadratic factor as a function of the number of bad choices and at which steps the bad choices are made. Even in the complex computations, for x-coordinates big enough with
respect to a and b, the limit behaves roughly quadratically with respect to the
number and steps of the bad choices. The direct connection between the steps when a bad choice is made and the value of the limit is not clear, but roughly, the greater isn such that then-choice is bad the greater is the limit l.
2.2 Real case
Bost and Mestre described in [BM] an algorithm to compute the local height of a curve embedded in the real numbers. In the previous section we showed a possible gen- eralisation to this algorithm that actually fails. Here we briey recap how the precedure works over the reals, specialising the method introduced for the complex numbers.
2.2.1 Suitable conguration
In the complex case, the chain of isogenous elliptic curves En was dened by the
2-torsion groups. In the real case we need curves with full 2-torsion group. Moreover, we will need the pointP, for which we want to compute the local archimedean height, to lie
in the identity component (the one that contains the point at innityO). Additionally,
we change coordinates in such a way that the biggest x-coordinate of the three real 2-
torsion points be 0. So our aim is reducing the general case to the one above, while keeping track of the local height.
Let (E, ω) be an elliptic curve with its dierential invariant and P a point on
it. Since the characteristic is 0, we can express E with a short Weierstrass equation E : y2 = f(x), where f(x) is polynomial in x of degree 3, which has at least one real
root, we pickx¯the maximum of them and change the model ofE mappingx7→x−x¯and
preserving they-coordinate. We changed the model of the curve twice, but by Corrolary
1.2.8 the local and canonical heights are preserved. There are still two further steps to reach the aimed conguration.
2.2. Real case
In the case the elliptic curveE has just one real point of 2-torsion its equation is E :y2 =x(x2+ux+v),
withu2 <4v. We can map E to the 2-isogenous curve E0 given by E0 =E/h(0,0)i=y2=x(x2−2ux+u2−4v)
that has 3 real 2-torsion points, via the isogenyα that maps (x, y)7→ x 2+ux+v x , y (x2−v) x2 ! .
Sincecα= 1 andker(α) ={(0,0), O} the formula from the Theorem 1.2.7 becomes
λE0(α(P)) = 2λE(P) +
1
2v(x(P)).
If on the new curve E0, the biggestx-coordinate of the 2-torsion real points is dierent
from 0, we can change the coordinates as above preserving λ(since the discriminant is
preserved via translation).
In the case the pointP does not belong to the identity component we can consider 2P that is on the identity component. We keep track of the local height by the Theorem
1.2.7, it follows that
λE0(2P) = 4λE(P) +v(2y(P)).
2.2.2 The chain of real elliptic curves
The problem has been reduced to computing the local height of a point P0 that
belongs to the identity component of a real elliptic curveE0with full 2-torsion. Moreover,
the biggestx-coordinate of the real 2-torsion is 0. Therefore, we have E0 :y2=x(x+a20)(x+b20),
where0< b0 < a0 are real numbers. It follows thatE0[2] ={(0,0),(−a20,0).(−b20,0), O}.
We dene the sequence of curves and the isogenies in a similar way to Section 2.1, where the main dierence is the eld of denition of the coecients involved.
In particular, the real AGM-sequence is dened without choices:
Denition 2.2.1. Let a0 and b0 two positive reals. The real AGM-sequence is dened
2.2. Real case by an= an−1+bn−1 2 bn= p an−1bn−1 ∀n >0,
where for bn we take the positive square root.
We then dene the sequence of isogenous elliptic curves En
En:=x(x+a2n)(x+b2n),
linked by the isogenies αn : En+1 → En which are dened, in the same way as in the
complex case, as (x, y)7→ x(x+b 2 n+1) (x+a2n+1), y (x+anan+1)(x+bnan+1) (x+a2n+1)2 ! .
If we consider a pointPn on the identity component ofEn there is only one point
Pn+1 in the identity component ofEn+1 such that
αn(Pn+1) =Pn,
therefore we can dene a sequence (Pi)i∈N uniquely determined by a point P0 ∈ E0 in
the identity component. In this way, the termszi =x(Pi) +a2i are always non-negative.
By the Theorem 2.1.5 it follows that in the real case
λ(P0) = logz1+ ∞ X i=1 2ilogzi+1 zi +l,
where according to the statements in [BM], [MS16] and [Bra10] lshould be zero.
Speed Test
The canonical height is a key tool of several algorithms, for instance it is used in Zagier's method for nding integral points [Zag87]. In particular most of the algo- rithms need a high precision of the canonical height and therefore a high precision of the archimedean terms is required. Here we compare the algorithm by Silverman imple- mented in Sage and the Bost and Mestre's one. Timing is taken in milliseconds. The test was made with a few elliptic curves dened over the reals.