The ABC Conjecture Implies Vojta's Height Inequality for Curves
Following Elkies (Internat. Math. Res. Notices7 (1991) 99–109) and Bombieri (Roth's theorem and the abc-conjecture, preprint, ETH Zürich, 1994), we show that the ABC conjecture implies the one-dimensional case of Vojta's height inequality. The main geometric tool is the construction of a B...
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| Veröffentlicht in: | Journal of number theory 2002-08, Vol.95 (2), p.289-302 |
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| container_title | Journal of number theory |
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| creator | Van Frankenhuysen, Machiel |
| description | Following Elkies (Internat. Math. Res. Notices7 (1991) 99–109) and Bombieri (Roth's theorem and the abc-conjecture, preprint, ETH Zürich, 1994), we show that the ABC conjecture implies the one-dimensional case of Vojta's height inequality. The main geometric tool is the construction of a Belyǐ function. We take care to make explicit the effectivity of the result: we show that an effective version of the ABC conjecture would imply an effective version of Roth's theorem, as well as giving an (in principle) explicit bound on the height of rational points on an algebraic curve of genus at least two. |
| doi_str_mv | 10.1006/jnth.2001.2769 |
| format | Article |
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We take care to make explicit the effectivity of the result: we show that an effective version of the ABC conjecture would imply an effective version of Roth's theorem, as well as giving an (in principle) explicit bound on the height of rational points on an algebraic curve of genus at least two.</description><identifier>ISSN: 0022-314X</identifier><identifier>EISSN: 1096-1658</identifier><identifier>DOI: 10.1006/jnth.2001.2769</identifier><language>eng</language><publisher>Elsevier Inc</publisher><subject>ABC conjecture ; Diophantine approximation ; effective Mordell ; Mordell's conjecture ; Roth's theorem ; the error term in the ABC conjecture ; type of an algebraic number ; Vojta's height inequality</subject><ispartof>Journal of number theory, 2002-08, Vol.95 (2), p.289-302</ispartof><rights>2002 Elsevier Science (USA)</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c213t-37fcd9e44d22879e5bc878ca1b1b9b46de2461ae9c81f1465e4b5049e751bd503</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://dx.doi.org/10.1006/jnth.2001.2769$$EHTML$$P50$$Gelsevier$$Hfree_for_read</linktohtml><link.rule.ids>314,780,784,3549,27923,27924,45994</link.rule.ids></links><search><creatorcontrib>Van Frankenhuysen, Machiel</creatorcontrib><title>The ABC Conjecture Implies Vojta's Height Inequality for Curves</title><title>Journal of number theory</title><description>Following Elkies (Internat. Math. Res. Notices7 (1991) 99–109) and Bombieri (Roth's theorem and the abc-conjecture, preprint, ETH Zürich, 1994), we show that the ABC conjecture implies the one-dimensional case of Vojta's height inequality. The main geometric tool is the construction of a Belyǐ function. 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Math. Res. Notices7 (1991) 99–109) and Bombieri (Roth's theorem and the abc-conjecture, preprint, ETH Zürich, 1994), we show that the ABC conjecture implies the one-dimensional case of Vojta's height inequality. The main geometric tool is the construction of a Belyǐ function. We take care to make explicit the effectivity of the result: we show that an effective version of the ABC conjecture would imply an effective version of Roth's theorem, as well as giving an (in principle) explicit bound on the height of rational points on an algebraic curve of genus at least two.</abstract><pub>Elsevier Inc</pub><doi>10.1006/jnth.2001.2769</doi><tpages>14</tpages><oa>free_for_read</oa></addata></record> |
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| source | ScienceDirect Journals (5 years ago - present); EZB-FREE-00999 freely available EZB journals |
| subjects | ABC conjecture Diophantine approximation effective Mordell Mordell's conjecture Roth's theorem the error term in the ABC conjecture type of an algebraic number Vojta's height inequality |
| title | The ABC Conjecture Implies Vojta's Height Inequality for Curves |
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