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Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie

IMPACT FACTOR 2018: 1.859

CiteScore 2018: 1.14

SCImago Journal Rank (SJR) 2018: 2.554
Source Normalized Impact per Paper (SNIP) 2018: 1.411

Mathematical Citation Quotient (MCQ) 2018: 1.55

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Volume 2015, Issue 698


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

Jennifer S. Balakrishnan / Amnon Besser
  • School of Mathematical and Statistical Sciences, Arizona State University, PO Box 871804, Tempe, AZ 85287-1804, USA; and Department of Mathematics, Ben-Gurion University of the Negev, P.O.B. 653, Be'er Sheva 84105, Israel
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Published Online: 2013-01-09 | DOI: https://doi.org/10.1515/crelle-2012-0095


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.

About the article

Received: 2012-02-09

Revised: 2012-07-29

Published Online: 2013-01-09

Published in Print: 2015-01-01

Funding Source: NSF

Award identifier / Grant number: DMS-1103831

Funding Source: Israel Science Foundation

Award identifier / Grant number: 1129/08

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2015, Issue 698, Pages 89–104, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: https://doi.org/10.1515/crelle-2012-0095.

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