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Coleman's theory of p-adic integration figures prominently in several number-theoretic applications, such as finding torsion and rational points on curves, and computing p-adic regulators in K-theory (including p-adic heights on elliptic curves). We describe an algorithm for computing Coleman integrals on hyperelliptic curves, and its implementation in Sage.

Let E be an elliptic curve defined over Q. The aim of this paper is to make it possible to compute Heegner L-functions and anticyclotomic Λ-adic regulators of E, which were studied by Mazur-Rubin and Howard. We generalize results of Cohen and Watkins and thereby compute Heegner points of non-fundamental discriminant. We then prove a relationship between the… (More)

We give an overview of some p-adic algorithms for computing with el-liptic and hyperelliptic curves, starting with Kedlaya's algorithm. While the original purpose of Kedlaya's algorithm was to compute the zeta function of a hyperel-liptic curve over a finite field, it has since been used in a number of applications. In particular, we describe how to use… (More)

- Jennifer Mun, Young Park, Bjorn Poonen, Matt Baker, Jennifer S Balakrishnan, Manjul Bhargava +30 others
- 2014

Faltings' theorem states that curves of genus g > 2 have finitely many rational points. Using the ideas of Faltings, Mumford, Parshin and Raynaud, one obtains an upper bound on the upper bound on the number of rational points [Szp85], XI, §2, but this bound is too large to be used in any reasonable sense. In 1985, Coleman showed [Col85] that Chabauty's… (More)

- Jacqueline Anderson, Jennifer S Balakrishnan, Kristin Lauter, Jennifer Park
- 2013

Bruinier and Yang conjectured a formula for intersection numbers on an arithmetic Hilbert modular surface, and as a consequence obtained a conjectural formula for CM(K).G 1 under strong assumptions on the ramification in K. Yang later proved this conjecture under slightly stronger assumptions on the ramification. In recent work, Lauter and Viray proved a… (More)

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