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On 3-adic heights on elliptic curves. (English) Zbl 1400.11115

Summary: In 2006, B. Mazur, W. Stein and J. Tate [Doc. Math., J. DMV Extra Vol., 577–614 (2006; Zbl 1135.11034)] gave an algorithm for computing \(p\)-adic heights on elliptic curves over \(\mathbb{Q}\) for good, ordinary primes \(p \geq 5\). In this paper, we extend their algorithm to the case of \(p = 3\). We also discuss the 3-adic precision that must be maintained throughout the computation, following the work of D. Harvey [LMS J. Comput. Math. 11, 40–59 (2008; Zbl 1222.11149)]. We conclude by giving examples of 3-adic regulators and their compatibility with the 3-adic Birch and Swinnerton-Dyer conjecture, computed using Sage.

MSC:

11G50 Heights
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11Y40 Algebraic number theory computations

Software:

SageMath
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Full Text: DOI

References:

[1] Balakrishnan, J. S.; Çiperiani, M.; Stein, W., \(p\)-adic heights of Heegner points and Λ-adic regulators, Math. Comp., 84, 292, 923-954 (2015) · Zbl 1316.11116
[2] Harvey, D., Efficient computation of \(p\)-adic heights, LMS J. Comput. Math., 11, 40-59 (2008) · Zbl 1222.11149
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[5] Mazur, B.; Stein, W.; Tate, J., Computation of \(p\)-adic heights and log convergence, Doc. Math., Extra Vol., 577-614 (2006), (electronic) · Zbl 1135.11034
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[8] Miller, R. L., Proving the Birch and Swinnerton-Dyer conjecture for specific elliptic curves of analytic rank zero and one, LMS J. Comput. Math., 14, 327-350 (2011) · Zbl 1300.11074
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[10] van den Bogaart, T., About the choice of a basis in Kedlaya’s algorithm, extract from the author’s Leiden, PhD thesis
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