Applications

Since their introduction by N´eron and Tate, canonical heights have found numerous applications. In this section we highlight a few, concentrating on those that require the actualcomputationof the canonical height. However, the canonical height also figures prominently in several important theorems in arithmetic geometry, including Faltings’ Theorem, stating that a curve of genus greater than one defined over a number field has only finitely many rational points (especially in the proof that uses Vojta’s inequality which is phrased in terms of the canonical height with respect to the theta divisor on its Jacobian, see [49, Chapter E]).

Furthermore, the canonical height substantially simplifies the proof of the Mordell-Weil Theorem for abelian varieties, more precisely the step that the so-called “weak” Mordell-Weil Theorem implies the full Mordell-Weil Theorem (see [49, Chapter C]).

Suppose that A = J is the Jacobian of a smooth projective curve C of positive genus g defined over a number field k. In many cases one is interested in actually determining a set of generators for the Mordell-Weil groupJ(k).

This is an interesting problem in its own right, but also useful for other purposes as well. If, for instance,k=Q, then a method for the computation of the Z-rational points on a hyperelliptic curve itself is discussed in [14].

This approach uses linear forms in logarithms and the Mordell-Weil sieve described in [13], but requires, in addition, a set of generators for the full Mordell-Weil group.

So suppose that we have already computed the rank r of J(k). If the dimension ofJ is small, then in many cases this can be done using 2-descent

(see [93] for an implementation-oriented description); more generally one usesn-descent forn≥2 or descent by isogeny (see [85] for a general concep-tual framework). Even if one cannot calculate the rank exactly using these approaches, it is often possible to give an upper bound. There are methods to search fork-rational points on the Jacobian, for example Stoll’s program j-points [96] for the genus 2 case if k = Q. Suppose we have found r pointsP₁, . . . , P_r ∈ J(k) that are independent (see below). Then we know that H = hP1, . . . , P_ri is a subgroup of J(k) of finite index. It turns out that in order to saturate this subgroup, it suffices to have

(a) a method to compute the canonical height ˆhD with respect to an ample symmetric divisor class [D] on J,

(b) a method to list all points inJ(k) with height ˆh_D bounded by a con-stant B.

Step (b) can be split up into two steps if we can

(b’) list all points inJ(k) with height h_D bounded by a given constantB, where h_D is some choice of Weil height function on J with respect to D,

(b”) bound the difference ˆh_D−h_D.

An algorithm that uses (a), (b’) and (b”) to compute generators of the Mordell-Weil group from the knowledge of r and an independent set of r points P₁, . . . , P_r is presented in [94, §7]. In this thesis we are mostly concerned with (a), although we shall discuss (b”) occasionally. Step (b’) is possible usingj-pointsin the genus 2 case.

Using part (iii) of Theorem 1.3 it is easy to decide for any abelian variety A/k whether a given point P ∈A(¯k) has finite order. How can we decide whether elements of the Mordell-Weil group are independent?

Definition 1.42. Letkbe a number field andA/kan abelian variety defined over k. LetD∈Div(A)(k) such that [D]∈Pic(A) is ample and symmetric and let ˆhD be the canonical height on A with respect toD.

(i) Thecanonical height pairing or N´eron-Tate pairing on A with respect to D is the bilinear pairing

(·,·)D :A(¯k)×A(¯k) −→ R

(P, Q) 7→ ˆh_D(P +Q)−ˆh_D(P)−ˆh_D(Q)

2 .

(ii) LetP₁, . . . , P_n ∈A(¯k). Theregulator of P₁, . . . , P_n with respect to D is the quantity

Reg_D(P₁, . . . , P_n) := det (((P_i, P_j)_D)_1≤i,j≤n).

(iii) Theregulator of A(k) with respect to Dis the regulator ofP₁, . . . , P_r, whereP₁, . . . , P_r is any independent set of generators of the free part ofA(k). We denote it by Reg_D(A/k).

From Theorem 1.3 we see that the canonical height pairing is a positive definite quadratic form on A(¯k)/A(¯k)_tors and that the regulator of any set of points in A(¯k) is nonnegative. Therefore, the regulator of a set of points vanishes if and only if that set is dependent, leading to an effective method to check independence.

The regulator of an abelian variety also appears in the formulation of the Birch and Swinnerton-Dyer conjecture. We only introduce the conjecture overQ, but it can be formulated over general number fields; furthermore we introduce most terms without explaining them, see [49, §F.4.1] for a more elaborate (but still concise) discussion.

Instead of allowing the regulator with respect to arbitrary ample sym-metric divisors which would introduce some ambiguity, one looks at the so-calledcanonical regulator, that is the regulator with respect to the Poincar´e divisor on the product A×A, where ˆˆ A is the dual abelian variety to A.

The relation between this quantity and the regulator with respect to a fixed ample symmetric divisor on A is explained in [49, Remark F.4.1.3], but we are not concerned with this difficulty, as Jacobians are self-dual. Let L(A, s) denote theL-series of A whose convergence for Re(s)>3/2 follows by definition, Ω_A the real period of a certain differential, called the N´eron differential by Hindry and Silverman, c_p the Tamagawa number #Φ_p(F_p) for any p∈M_Q⁰ and ^X(A/Q) the Shafarevich-Tate group.

Conjecture 1.43. (Birch and Swinnerton-Dyer)

(i) L(A, s) has a zero ats= 1 of order equal to the rankr of A(Q).

(ii) The Taylor expansion ofL(A/Q, s)arounds= 1has leading coefficient

s→1lim

In order for the conjecture to make sense, we need to assume thatL(A, s) has a suitable analytic continuation; in fact it is conjectured to have an ana-lytic continuation to all of C. Moreover, finiteness of^Xis conjectured, but not known. However, under the assumption that^Xis indeed finite, its order must be a nonnegative integer, so if we can compute the other terms, some of which are transcendental, we can gather experimental evidence for the conjecture. This has been done for specific elliptic curves by several authors, starting with Birch and Swinnerton-Dyer in [5] - in fact the numerical results led them to the formulation of the conjecture, which was originally phrased

for elliptic curves and only later extended by Tate to general abelian varieties in [101]. Other examples include [15] and [27], see also the recent thesis of Miller [71], where the conjecture is verified for all elliptic curves of small conductor except for a few exceptions.

For abelian varieties of higher dimension much less has been done. The only two examples that the author is aware of are the work by Yoshida [104] dealing with a specific modular Jacobian of dimension 2 and another paper [44] by Flynn et al. where they undertake a more systematic study of a number of modular Jacobians of dimension 2. Here ”modular” means that the Jacobian is a quotient of the Jacobian of a modular curveX₀(N) for some level N; it is natural to first consider such Jacobians because the analytic continuation of L-series associated to modular forms – and hence the analytic continuation ofL(A, s) – is known.

Notice that the ability to compute canonical heights explicitly comes in twice if we wish to collect evidence for the Birch and Swinnerton-Dyer conjecture: We need it to find a basis for the Mordell-Weil group ofA and we need it - and this basis - to compute the regulator. Since up to now no method has been found to compute canonical heights for Jacobians of curves of genus at least 3, it has not been possible to check the plausibility of the conjecture in such cases (unless the Mordell-Weil rank vanishes). We develop a method for the computation of canonical heights on arbitrary Jacobians in Chapter 5.

Elliptic curves

25

2.1 Heights on elliptic curves

Letlbe a field. We consider elliptic curves given by a Weierstrass equation y²+a₁xy+a₃y=x³+a₂x²+a₄x+a₆, a_i ∈l, (2.1) where the discriminant ∆ of the equation is nonzero.

For now E denotes an elliptic curve given by a Weierstrass equation (2.1) over a number field or one-dimensional function fieldk with algebraic closure ¯k. Without loss of generality we may assume that a_i ∈ Ok for all i, so that the given Weierstrass equation is anO_k-integral model ofE in the sense of Definition 1.12.

In order to develop a reasonable height theory on E it is necessary to pick an ample symmetrick-rational divisor class onE. A natural choice is a multiple of the class of the divisor (O), whereO is the identity of the group law on E. We choose the class of D = 2(O) and retain this notation for the remainder of this chapter. The linear system associated to this divisor is base point free and a map to projective space corresponding to a basis of L(2(O)) is given by

κ: E −→ P¹

(x, y) 7→ (x: 1) O 7→ (1 : 0).

Definition 2.1. Let E/k be an elliptic curve defined by an O_k-integral Weierstrass equation (2.1). The functionh:E(¯k)−→Rgiven by

h(P) :=h_D(P) := 1 d_k

X

v∈Mk

−Nvmin{v(x(P)),0}

is called thenaive height on E. Thecanonical height onE is the function ˆh:E(¯k) −→ R

P 7→ lim

n→∞

1

4ⁿh(2ⁿP).

Note that this is in accordance with our constructions from Section 1.2.