2026-09-20T03:08:14Zhttps://dspace.mit.edu/server/oai/requestoai:dspace.mit.edu:1721.1/901892026-06-17T14:47:15Zcom_1721.1_7582com_1721.1_7581col_1721.1_131022
Bjorn Poonen.
Park, Jennifer Mun Young
Massachusetts Institute of Technology. Department of Mathematics.
Massachusetts Institute of Technology. Department of Mathematics
2014-09-19T21:45:01Z
2014-09-19T21:45:01Z
2014
2014
http://hdl.handle.net/1721.1/90189
890211552
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2014.
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 75-76).
Faltings' theorem states that curves of genus g > 2 have finitely many rational points. Using the ideas of Faltings, Mumford, Parshin and Raynaud, one obtains an upper bound on the upper bound on the number of rational points, XI, [paragraph]2, but this bound is too large to be used in any reasonable sense. In 1985, Coleman showed that Chabauty's method, which works when the Mordell-Weil rank of the Jacobian of the curve is smaller than g, can be used to give a good effective bound on the number of rational points of curves of genus g > 1. We draw ideas from nonarchimedean geometry to show that we can also give an effective bound on the number of rational points outside of the special set of Symd X, where X is a curve of genus g > d, when the Mordell-Weil rank of the Jacobian of the curve is at most g > d.
by Jennifer Mun Young Park.
Ph.D.
76 pages
eng
Massachusetts Institute of Technology
M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission.
http://dspace.mit.edu/handle/1721.1/7582
Mathematics.
Effective Chabauty for symmetric powers of curves
Thesis
application/pdf
Publication
unknown
unknown
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<Abstract>Faltings' theorem states that curves of genus g > 2 have finitely many rational points. Using the ideas of Faltings, Mumford, Parshin and Raynaud, one obtains an upper bound on the upper bound on the number of rational points, XI, [paragraph]2, but this bound is too large to be used in any reasonable sense. In 1985, Coleman showed that Chabauty's method, which works when the Mordell-Weil rank of the Jacobian of the curve is smaller than g, can be used to give a good effective bound on the number of rational points of curves of genus g > 1. We draw ideas from nonarchimedean geometry to show that we can also give an effective bound on the number of rational points outside of the special set of Symd X, where X is a curve of genus g > d, when the Mordell-Weil rank of the Jacobian of the curve is at most g > d.</Abstract>
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