Computing canonical heights with little (or no) factorization

Joseph H. Silverman
1997 Mathematics of Computation  
Let E/Q be an elliptic curve with discriminant ∆, and let P ∈ E(Q). The standard method for computing the canonical heightĥ(P ) is as a sum of local heightsĥ(P ) =λ∞(P) + pλ p(P ). There are well-known series for computing the archimedean heightλ∞(P ), and the non-archimedean heightsλp(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
more » ... method for computing the nonarchimedean contribution toĥ(P ) which is quite practical and requires little or no factorization. We also give some numerical examples illustrating the algorithm.
doi:10.1090/s0025-5718-97-00812-0 fatcat:56lvvt5e7rf5rfpo5v4nxx64m4