Coleman–Gross height pairings and the p-adic sigma function

Abstract

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.

Quadratic Chabauty for modular curves: Algorithms and examples

“…An early version of the method appeared in work of Kim [Kim10,BKK11], where Massey products were used to construct a locally analytic function, vanishing on the set of integral points of an elliptic curve of rank 1. These functions were interpreted as p-adic height functions, extending the method, in Balakrishnan-Besser [BB15] and Balakrishnan-Besser-Müller [BBM16]. It was extended to its current form in Balakrishnan-Dogra [BD18], where a systematic use of Nekovář's theory of p-adic heights suggested a streamlined approach towards a very general class of curves allowing an abundance of geometric correspondences.…”

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

“…This section is concerned with relating the mixed extensions A(b, z) defined above to the mixed extensions arising from the theory of motivic height pairings as developed by Nekovář [36] and Scholl [41]. Such relations have been established in the case of fundamental groups of affine elliptic curves in work of Balakrishnan and Besser [3] and Balakrishnan, Dan-Cohen, Kim and Wewers [2] and in the case of affine hyperelliptic curves in work of Balakrishnan, Besser and Müller [5]. 6.1.…”

Principal bundles and reciprocity laws in number theory

“…The main applications so far of the theory of the previous section are to explicit reciprocity laws on hyperbolic curves [3,6,10], and we give a brief survey of illustrative examples. Some other works with further developments which we do not discuss include [4] and [5]. It should be noted that most of these examples do not make direct use of the reciprocity laws described here.…”