Explicit Coleman Integration for Hyperelliptic Curves

Abstract

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. Q P ω i } 2g−1 i=0 .

Appendix and erratum to “Massey products for elliptic curves of rank 1”

“…Sage includes an implementation of single Coleman integration for hyperelliptic curves developed by Balakrishnan, Bradshaw, and Kedlaya [3]; the computations here rely on a generalization to multiple integrals suggested in op. cit.…”

“…They also depend on Cremona's tables of elliptic curves [5], which are included in Sage. See [1] for the full Sage source code used in the computations, which will appear in a future release of Sage. An integral two-torsion base-point is used in each of the first three examples.…”

Coleman integration for even-degree models of hyperelliptic curves

“…In joint work with Bradshaw and Kedlaya [3], we gave explicit methods to compute single Coleman integrals for hyperelliptic curves with odd-degree models. The key theoretical input was Kedlaya's algorithm [12] to compute the action of Frobenius on Monsky-Washnitzer cohomology, formulated for odd-degree models of hyperelliptic curves.…”

Iterated Coleman integration for hyperelliptic curves

“…This integration theory relies on locally defined antiderivatives that are extended analytically by the principle of Frobenius equivariance. In joint work with Bradshaw and Kedlaya [1], we made this construction explicit and gave algorithms to compute single Coleman integrals for hyperelliptic curves.…”

“…Our methods for computing iterated integrals are similar in spirit to those detailed in [1]. We begin with algorithms for tiny iterated integrals, use Frobenius equivariance to write down a linear system yielding the values of integrals between points in different residue disks, and, if needed, use basic properties of integration to correct endpoints.…”