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A $ p$-adic approach to rational points on curves


Author: Bjorn Poonen
Journal: Bull. Amer. Math. Soc.
MSC (2010): Primary 11G30; Secondary 11G20, 14D07, 14D10, 14G05, 14H25
DOI: https://doi.org/10.1090/bull/1707
Published electronically: September 7, 2020
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Abstract: In 1922 Mordell conjectured the striking statement that, for a polynomial equation $ f(x,y)=0$, if the topology of the set of complex number solutions is complicated enough, then the set of rational number solutions is finite. This was proved by Faltings in 1983 and again by a different method by Vojta in 1991. But neither proof provided a way to provably find all the rational solutions, so the search for other proofs has continued. Recently, Lawrence and Venkatesh found a third proof, relying on variation in families of $ p$-adic Galois representations; this is the subject of the present exposition.


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Additional Information

Bjorn Poonen
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307
MR Author ID: 250625
ORCID: 0000-0002-8593-2792
Email: poonen@math.mit.edu

DOI: https://doi.org/10.1090/bull/1707
Keywords: Mordell conjecture, curve, rational point, genus, $p$-adic number, Galois representation.
Received by editor(s): June 6, 2020
Published electronically: September 7, 2020
Additional Notes: This article is associated with a lecture given January 17, 2020, in the Current Events Bulletin at the 2020 Joint Mathematics Meetings in Denver. The writing of this article was supported in part by National Science Foundation grant DMS-1601946 and Simons Foundation grants #402472 (to Bjorn Poonen) and #550033.
Article copyright: © Copyright 2020 American Mathematical Society