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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.

The paper [6] contains a few errors in the basic assumptions as well as in the formula of corollary 0.2. First of all, it should have been made clear at the outset that the regular model E for the elliptic curve E must be the minimal regular model, and X the complement of the origin in the regular minimal model. Similarly, the tangential base-point b must… (More)

The Coleman integral is a p-adic line integral. Double Coleman integrals on elliptic curves appear in Kim's nonabelian Chabauty method, the first numerical examples of which were given by the author, Kedlaya, and Kim [3]. This paper describes the algorithms used to produce those examples, as well as techniques to compute higher iterated integrals on… (More)

- Jennifer S. Balakrishnan, J. Steffen Müller, William Stein
- Math. Comput.
- 2016

In 2006, Mazur, Stein, and Tate [4] gave an algorithm for computing p-adic heights on elliptic curves E over Q for good, ordinary primes p ≥ 5. Their work makes essential use of Kedlaya's algorithm [3], where the action of Frobenius is computed on a certain basis of the first de Rham cohomology of E, with E given by a " short " Weierstrass model. Kedlaya's… (More)

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, which enable us to compute Heegner points of non-fundamental discriminant. We then prove a relationship… (More)

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)

Let E/Q be an elliptic curve and p a rational prime of good ordinary reduction. For every imaginary quadratic field K/Q satisfying the Heegner hypothesis for E we have a corresponding line in E(K) ⊗ Qp, known as a shadow line. When E/Q has analytic rank 2 and E/K has analytic rank 3, shadow lines are expected to lie in E(Q) ⊗ Qp. If, in addition, p splits… (More)

- Jennifer S. Balakrishnan
- Advances on Superelliptic Curves and their…
- 2015

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)