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

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

2012-01-29

arxiv.org/abs/1201.6016v1

Mathematics

Number Theory

AMS-LaTeX 12 pages

Scientific paper

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 supports 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.

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