Computing canonical heights with little (or no) factorization

Abstract

Let E / Q E/ \mathbb {Q} be an elliptic curve with discriminant  Δ \Delta , and let P ∈ E ( Q ) P\in E( \mathbb {Q}) . The standard method for computing the canonical height h ^ ( P ) \hat h(P) is as a sum of local heights h ^ ( P ) = λ ^ ∞ ( P ) + ∑ p λ ^ p ( P ) \hat h (P)= \hat \lambda _{\infty }(P)+\sum _{p} \hat \lambda _{p}(P) . There are well-known series for computing the archimedean height  λ ^ ∞ ( P ) \hat \lambda _{\infty }(P) , and the non-archimedean heights  λ ^ p ( P ) \hat \lambda _{p}(P) are easily computed as soon as all prime factors of  Δ \Delta have been determined. However, for curves with large coefficients it may be difficult or impossible to factor  Δ \Delta . In this note we give a method for computing the non-archimedean contribution to  h ^ ( P ) \hat h (P) which is quite practical and requires little or no factorization. We also give some numerical examples illustrating the algorithm.