Massey products for elliptic curves of rank 1

Abstract

The author must begin with an apology for writing on a topic so specific, so elementary, and so well-understood as the study of elliptic curves of rank 1. Nevertheless, it is hoped that a contribution not entirely without value or novelty is to be found within the theory of Selmer varieties for hyperbolic curves, applied to the complement X = E \ {e} of the origin inside an elliptic curve E over Q with Mordell-Weil rank 1. Assume throughout this paper that p is an odd prime of good reduction such that X(E)[p ∞ ] is finite and that E has integral j-invariant. All of these assumptions will hold, for example, if E has complex multiplication and ord s=1 L(E, s) = 1.Let E be a regular Z-model for E and X the complement in E of the closure of e. The main goal of the present inquiry is to find explicit analytic equations defining X (Z) inside X (Z p ). The approach of this paper makes use of a rigidified Massey product in Galois cohomology.1 That is, theétale local unipotent Albanese map2 ) to the level-two local Selmer variety (recalled below) associates to point z a nonabelian cocycle a(z), which can be broken canonically into two components a(z) = a 1 (z) + a 2 (z), with a 1 (z) taking values in to construct a function on the local points of X . Recall that Massey products are secondary cohomology products arising in connection with associative differential graded algebras (A,

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

“…If it were, we would get the same quotients even for rational points in this rank one situation, which the computations show not to be the case. In fact, as noted already in [6], for a fixed y, the equality…”

“…That is, torsors for U DR can be classified by U DR /F 0 or F 0 \U DR . Now, in [6], Section 3, we described p cr ∈ R x as the power series…”

“…

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.

…”

Iterated Coleman integration for hyperelliptic curves

“…We follow the convention of Kim [14] and define our integrals as follows: for a collection of dummy parameters R 1 ; : : : ; R n 1 and 1-forms 1 ; : : : ; n .…”

“…Having algorithms to compute Coleman integrals allows one to compute p-adic regulators in K-theory [8; 7], carry out the method of Chabauty-Coleman for finding rational points on higher genus curves [15], and utilize Kim's nonabelian analogue of the Chabauty method [14].…”