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arXiv · 1201.6016

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

Jennifer S. Balakrishnan·Amnon Besser
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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.

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Jennifer S. Balakrishnan, Amnon Besser. 2012-07-25. Coleman-Gross height pairings and the $p$-adic sigma function. https://arxiv.org/abs/1201.6016

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