Coleman-Gross height pairings and the $p$-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...
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| Format: | Journal article |
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2012
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| _version_ | 1826291536953868288 |
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| author | Balakrishnan, J Besser, A |
| author_facet | Balakrishnan, J Besser, A |
| author_sort | Balakrishnan, J |
| collection | OXFORD |
| description | 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. |
| first_indexed | 2024-03-07T03:00:52Z |
| format | Journal article |
| id | oxford-uuid:b0ed144a-7e6d-4f57-a29c-f6fccaac8268 |
| institution | University of Oxford |
| last_indexed | 2024-03-07T03:00:52Z |
| publishDate | 2012 |
| record_format | dspace |
| spelling | oxford-uuid:b0ed144a-7e6d-4f57-a29c-f6fccaac82682022-03-27T03:59:59ZColeman-Gross height pairings and the $p$-adic sigma functionJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:b0ed144a-7e6d-4f57-a29c-f6fccaac8268Symplectic Elements at Oxford2012Balakrishnan, JBesser, AWe 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. |
| spellingShingle | Balakrishnan, J Besser, A Coleman-Gross height pairings and the $p$-adic sigma function |
| title | Coleman-Gross height pairings and the $p$-adic sigma function |
| title_full | Coleman-Gross height pairings and the $p$-adic sigma function |
| title_fullStr | Coleman-Gross height pairings and the $p$-adic sigma function |
| title_full_unstemmed | Coleman-Gross height pairings and the $p$-adic sigma function |
| title_short | Coleman-Gross height pairings and the $p$-adic sigma function |
| title_sort | coleman gross height pairings and the p adic sigma function |
| work_keys_str_mv | AT balakrishnanj colemangrossheightpairingsandthepadicsigmafunction AT bessera colemangrossheightpairingsandthepadicsigmafunction |