We study the unipotent Albanese map that associates the torsor of paths for p-adic fundamental groups to a point on a hyperbolic curve. It is shown that the map is very transcendental in nature, while standard conjectures about the structure of mixed motives provide control over the image of the map. As a consequence, conjectures of 'Birch and Swinnerton-Dyer type' are connected to finiteness theorems of Faltings-Siegel type.

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Let X denote a hyperbolic curve over Q and let p denote a prime of good reduction. The third author's approach to integral points, introduced in [Kim2] and [Kim3], endows X(Zp) with a nested sequence of subsets X(Zp)n which contain X(Z). These sets have been computed in a range of special cases [Kim4, BKK, DCW2, DCW3]; there is good reason to believe them to be practically computable in general. In 2012, the third author announced the conjecture that for n sufficiently large, X(Z) = X(Zp)n. This conjecture may be seen as a sort of compromise between the abelian confines of the BSD conjecture and the profinite world of the Grothendieck section conjecture. After stating the conjecture and explaining its relationship to these other conjectures, we explore a range of special cases in which the new conjecture can be verified.

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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,

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 be integral, in that it is a Z-basis of the relative tangent space e * T E/Z . It could also be an integral two-torsion point for the arguments of the paper to hold verbatim.The most significant error is in the contribution of the local terms at l = p, that is, Lemma 1.2. The problem is that a point that is integral on X may not be integral on a smooth model over a field of good reduction. As it stands, the lemma will only apply to points that are integral in this stronger sense.However, to get immediate examples, one can replace the lemma by Lemma 1.2 . Suppose the Neron model of E has only one rational component for each prime. (Equivalently, the Tamagawa number is one at each prime.) Then the mapis trivial for every l = p.Therefore, for the functionconstructed via the refined Massey product, we get The assumption can be easily verified if the elliptic curve has square-free minimal discriminant, since the Neron model will then have only one geometric component in the special fiber. We point out that the integral j-invariant hypothesis is no longer necessary in this version.

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