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Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

ISSN 1088-9485 (online) ISSN 0273-0979 (print)

The 2024 MCQ for Bulletin of the American Mathematical Society is 0.84.

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A $p$-adic approach to rational points on curves
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by Bjorn Poonen;
Bull. Amer. Math. Soc. 58 (2021), 45-56
DOI: https://doi.org/10.1090/bull/1707
Published electronically: September 7, 2020

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.
References
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Bibliographic 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
  • 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.
  • © Copyright 2020 American Mathematical Society
  • Journal: Bull. Amer. Math. Soc. 58 (2021), 45-56
  • MSC (2010): Primary 11G30; Secondary 11G20, 14D07, 14D10, 14G05, 14H25
  • DOI: https://doi.org/10.1090/bull/1707
  • MathSciNet review: 4188807