Computing canonical heights with little (or no) factorization

Let E/Q be an elliptic curve with discriminant ∆, and let P ∈ E(Q). The standard method for computing the canonical height ĥ(P ) is as a sum of local heights ĥ(P ) = λ̂∞(P ) + ∑ p λ̂p(P ). There are well-known series for computing the archimedean height λ̂∞(P ), and the non-archimedean heights λ̂p(P ) are easily computed as soon as all prime factors of ∆ have been determined. However, for curves with large coefficients it may be difficult or impossible to factor ∆. In this note we give a method for computing the nonarchimedean contribution to ĥ(P ) which is quite practical and requires little or no factorization. We also give some numerical examples illustrating the algorithm. Let E be an elliptic curve defined over a number field K, say given by a Weierstrass equation E : y + a1xy + a3y = x 3 + a2x 2 + a4x+ a6. (1) The canonical height on E is a quadratic form ĥ : E(K) −→ R. The canonical height is an extremely important theoretical and computational tool in the arithmetic study of elliptic curves. See [18, Chapter VIII, Section 9] for the definition and basic properties of ĥ, and [20], [21], and [23] for some discussion of how to compute ĥ in practice. In this paper, which may be considered as a continuation of our earlier note [20], we will discuss the computation of the canonical height for curves E whose coefficients a1, . . . , a6 are large. We note that this is not a mere intellectual exercise, since curves with huge integer coefficients have already made their appearance in the search for curves whose Mordell-Weil group E(Q) has large rank [5], [11], [12], [13], [14], and the standard tool for proving that a set of points P1, . . . , Pr ∈ E(Q) is linearly independent is to check the non-vanishing of the height regulator matrix det ( 〈Pi, Pj〉 ) . Here the height pairing 〈 · , ·〉 is defined (up to a normalizing factor) by the formula 〈P,Q〉 = ĥ(P +Q)− ĥ(P )− ĥ(Q). Tate’s definition ĥ(P ) = limn→∞ 4 −nh ( x(2P ) ) of the canonical height is not practical for numerical computations. Instead, one uses the Néron-Tate decomposition of the canonical height into a sum of local heights, one for each distinct Received by the editor October 24, 1995. 1991 Mathematics Subject Classification. Primary 11G05, 11Y50.