A non-abelian conjecture of Tate–Shafarevich type for hyperbolic curves

Abstract

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.

Quadratic Chabauty and rational points, I: p-adic heights

“…Remark 2.5. The sets X(K p ) n are contained in the set of points which are weakly global of level n, defined in [2]. If the p-primary part of the Shafarevich-Tate group of the Jacobian of X is finite, then the two sets are equal.…”

“…Unless otherwise stated, we will henceforth take U to be a quotient of U 2 surjecting onto V . From the standard presentation of the topological fundamental group of a smooth surface of genus g in terms of 2g generators and 1 relation between commutators, the natural map ∧ 2 V → U [2] gives an exact sequence…”

“…E 1 has type II reduction, which means that the singular point mod 2 does not lift to a Q 2 point. Hence h E1,2 (f 1 (z)) = 2 max{0, −v 2 (x(z))} log p (2).…”

“…The insight of Kim [30] is that, rather than trying to generalise the Jacobian of X, it is easier to generalise its Galois cohomological avatar: the Selmer group. In [31], Kim defined a family of Selmer varieties Sel(U n ) giving a decreasing sequence of subsets [2] X(K p ) 1 ⊃ X(K p ) 2 ⊃ . .…”

Arithmetic gauge theory: A brief introduction

“…Conjecture 8.1. [7] Suppose V is a smooth projective curve of genus ≥ 2. Then A −1 p (Im(loc p )) = V (Q).…”