Coleman-Gross height pairings and the pp-adic sigma function

We give a direct proof that the Mazur-Tate and Coleman-Gross heights on elliptic curves coincide. The main ingredient is to extend the Coleman-Gross height to the case of divisors with non-disjoint support and, doing some $p$-adic analysis, show that, in particular, its component above $p$ gives, in the special case of an ordinary elliptic curve, the $p$-adic sigma function. We use this result to give a short proof of a theorem of Kim characterizing integral points on elliptic curves in some cases under weaker assumptions. As a further application, we give new formulas to compute double Coleman integrals from tangential basepoints.Comment: AMS-LaTeX 17 page

Computing local p-adic height pairings on hyperelliptic curves

We describe an algorithm to compute the local component at p of the Coleman-Gross p-adic height pairing on divisors on hyperelliptic curves. As the height pairing is given in terms of a Coleman integral, we also provide new techniques to evaluate Coleman integrals of meromorphic differentials and present our algorithms as implemented in Sage

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

We state a conjectural criterion for identifying global integral points on a hyperbolic curve over $\mathbb{Z}$ in terms of Selmer schemes inside non-abelian cohomology functors with coefficients in $\mathbb{Q}_p$-unipotent fundamental groups. For $\mathbb{P}^1\setminus \{0,1,\infty\}$ and the complement of the origin in semi-stable elliptic curves of rank 0, we compute the local image of global Selmer schemes, which then allows us to numerically confirm our conjecture in a wide range of cases.Comment: Improvements to the exposition and numerous minor corrections throughou

Constructing genus 3 hyperelliptic Jacobians with CM

Given a sextic CM field $K$, we give an explicit method for finding all genus 3 hyperelliptic curves defined over $\mathbb{C}$ whose Jacobians are simple and have complex multiplication by the maximal order of this field, via an approximation of their Rosenhain invariants. Building on the work of Weng, we give an algorithm which works in complete generality, for any CM sextic field $K$, and computes minimal polynomials of the Rosenhain invariants for any period matrix of the Jacobian. This algorithm can be used to generate genus 3 hyperelliptic curves over a finite field $\mathbb{F}_p$ with a given zeta function by finding roots of the Rosenhain minimal polynomials modulo $p$.Comment: 20 pages; to appear in ANTS XI