On 3-adic heights on elliptic curves

In 2006, Mazur, Stein, and Tate [5] gave an algorithm for computing p  -adic heights on elliptic curves over QQ for good, ordinary primes p≥5p≥5. In this paper, we extend their algorithm to the case of p=3p=3. We also discuss the 3-adic precision that must be maintained throughout the computation, following the work of Harvey [2]. We conclude by giving examples of 3-adic regulators and their compatibility with the 3-adic Birch and Swinnerton-Dyer conjecture, computed using Sage [9].