In this thesis we are interested in practical methods to compute canonical heights on certain abelian varieties. However, it is not a very good idea to use Definition 1.2 for this purpose, since the size of the coordinates that we need to compute - assuming we can represent them somehow - grows exponentially. Fortunately, there are other methods. The definition of the canonical height given in the previous section is due to Tate, but at the same time N´eron constructed the canonical height as the sum of local con-tributions in [78]. It was later reformulated in the language we use below by Lang, see [59, Chapter 11]. Although the construction is more complicated than Tate’s construction, which allows for a rather short proof of Theorem 1.3, it has both theoretical and practical merits. We shall split the canoni-cal height into a sum of certain functions which Lang canoni-calls N´eron functions and Hindry-Silverman call canonical local heights. We shall see that this decomposition allows us to compute canonical heights for abelian varieties of dimension one and two.
LetA be an abelian variety defined over a field lwith an absolute value v. Let D∈Div(A)(l).
Definition 1.4. A Weil function associated with D and v is a function λD,v:A(l)\supp(D)−→R
with the following property: Suppose D is represented locally by (U, f), whereU ⊂A(l) is an open subset and f is a rational function. Then there exists a locally bounded continuous function α :U −→ R such that for all P ∈U \supp(D) we have
λD,v(P) =−log|f(P)|v+α(P), where the normalization of|.|v has been fixed in Section 1.1.
In this context, ‘locally bounded’ means bounded on bounded subsets and ‘bounded’ and ‘continuous’ refer to thev-adic topology, see [59, §10.1].
Next we define N´eron functions, which are Weil functions having some special properties.
Definition 1.5. We call an association D 7→ λD associating to each l-rational divisorD on A a Weil functionλD a N´eron family if the following conditions are satisfied.
(1) If D, E ∈Div(A)(l), then λD+E,v=λD,v+λE,v+c1 for somec1 ∈R.
(2) If D= div(f)∈Div(A)(l) is principal, thenλD,v=−log|f|v+c2 for somec2 ∈R.
(3) For all D ∈ Div(A)(l) we have λ[2]∗(D),v = λD,v ◦[2] +c3 for some c3 ∈R.
We call the imageλD,vunder such an association aN´eron function associated withD and v.
Lang shows in [59,§11.1] that for anyl-rational divisorDon an abelian variety A there exists a N´eron function λD,v associated with D and v that is unique up to constants. In the process he shows how N´eron functions can be constructed. This also gives a method of verifying whether a given Weil function associated with a divisor on an abelian variety is a N´eron function when the linear equivalence class of the divisor is symmetric.
Proposition 1.6. (Lang) Let D ∈ Div(A)(l) be a divisor whose class in Pic(A) is symmetric and letλ be a Weil function associated with D and v.
Letf ∈l(A) be a rational function such that[2]∗(D) = 4D+ div(f), and let ε:A(l)−→Rbe the unique bounded continuous function on A(l) such that
λ(2P) = 4λ(P)−log|f(P)|v−ε(P) for allP outside a suitable Zariski closed subset of A(l).
Let µ(P) :=P∞
n=04−n−1ε(2nP) and letλˆ:=λ−µ. Thenµ:A(l)−→R is bounded and continuous. Furthermore, λˆ is the unique N´eron function associated with D and v that satisfies
ˆλ(2P) = 4ˆλ(P)−log|f(P)|v. (1.2) Proof. A similar result is proved in [59, Chapter 11, Proposition 1.1]. The following proof is a generalization of the discussion preceding [43, Theo-rem 4].
Existence and uniqueness of εare obvious becauseλis a Weil function.
Note that although λ is only defined on A(l)\supp(D), the function ε is defined on all of A(l), because it is a Weil function associated with 0 ∈ Div(A) and v. See Proposition 2.3 and Corollary 2.4 of [59, Chapter 10].
It follows from this that µ converges and is defined on A(l). It is also bounded and continuous, since multiplication by 2n is continuous. A straightforward calculation reveals that we have
ε(P) = 4µ(P)−µ(2P);
this is known as Tate’s telescoping trick.
Hence we get
λ(2Pˆ )−4ˆλ(P) =λ(2P)−µ(2P)−4λ(P) + 4µ(P)
=−log|f(P)|v.
Therefore ˆλsatisfies property (3) of a N´eron function. The verifications that it also satisfies (1) and (2) are immediate; this proves the Proposition.
In particular it follows that any Weil function satisfying (1.2) will auto-matically be a N´eron function. The crucial point is that we can fix a specific N´eron function in its class modulo constants by fixing the functionf. Definition 1.7. Let f ∈ l(A) be a rational function such that [2]∗(D) = 4D+ div(f). We call the unique N´eron function that satisfies (1.2) the canonical local height on A associated with D, v and f and denote it by ˆλD,v,f.
We now relate canonical local heights to canonical heights. The following theorem tells us that if we pick somef as above consistently for all placesv, then the sum of all canonical local heights associated withDandf coincides with the canonical height.
Theorem 1.8. (N´eron) Let k be a number field or a one-dimensional func-tion field and letAbe an abelian variety defined overk. LetDbe ak-rational
divisor on A whose class is ample and symmetric and letf ∈k(A) be a ra-tional function such that[2]∗(D) = 4D+ div(f). For each v∈Mk letλˆD,v,f denote the canonical local height associated with D, v andf. Then we have
ˆhD(P) = 1 dk
X
v∈Mk
nvλˆD,v,f(P) for allP ∈A(k)\supp(D).
Proof. Although this theorem is not proved there directly in this form, it follows almost immediately from the results of [59,§11.1].
It is worth noting that when the condition P /∈ supp(D) fails we can repair the situation easily; we can use the moving lemma (cf. [49, Lemma A.2.2.5 (ii)]) to find someD′ ∈[D] such thatP /∈supp(D′) and use suitable canonical local heights forD′.
Remark 1.9. The canonical local heights are defined not only on A(k), but also onA(kv). Therefore we may and shall pass to the completion whenever we only deal with one place at a time.
Remark 1.10. We have not defined canonical heights for anti-symmetric divisor classes [D]. This is possible, but leads to a linear form, as opposed to a quadratic form. It can also be decomposed into a sum of canonical local height and the only difference is that we have to take a functionf satisfying [2]∗D= 2D+div(f) in the preceding theorem. It is also possible to construct canonical heights for general divisors on A as a sum of a quadratic and a linear form. All of this is done in [49,§B.5 ].
Remark 1.11. For an exposition of canonical local heights in terms of line bundles see [8, Chapter 9].