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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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How to do a $p$-descent on an elliptic curve
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by Edward F. Schaefer and Michael Stoll PDF
Trans. Amer. Math. Soc. 356 (2004), 1209-1231 Request permission


In this paper, we describe an algorithm that reduces the computation of the (full) $p$-Selmer group of an elliptic curve $E$ over a number field to standard number field computations such as determining the ($p$-torsion of) the $S$-class group and a basis of the $S$-units modulo $p$th powers for a suitable set $S$ of primes. In particular, we give a result reducing this set $S$ of ‘bad primes’ to a very small set, which in many cases only contains the primes above $p$. As of today, this provides a feasible algorithm for performing a full $3$-descent on an elliptic curve over $\mathbb Q$, but the range of our algorithm will certainly be enlarged by future improvements in computational algebraic number theory. When the Galois module structure of $E[p]$ is favorable, simplifications are possible and $p$-descents for larger $p$ are accessible even today. To demonstrate how the method works, several worked examples are included.
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Additional Information
  • Edward F. Schaefer
  • Affiliation: Department of Mathematics and Computer Science, Santa Clara University, Santa Clara, California 95053
  • Email:
  • Michael Stoll
  • Affiliation: School of Engineering and Science, International University Bremen, P.O. Box 750 561, 28 725 Bremen, Germany
  • Email:
  • Received by editor(s): January 24, 2003
  • Published electronically: October 27, 2003
  • Additional Notes: We are indebted to Claus Fieker for his invaluable help in getting KANT to produce a basis for the group $A(S,5)^{(1)}$ needed in the example in Section 8.2. We thank John Cremona, Zafer Djabri, Everett Howe, Hendrik W. Lenstra Jr., Karl Rubin, Nigel Smart and Don Zagier for useful and interesting discussions. The first author was supported by National Security Agency grant MSPF-02Y-033
  • © Copyright 2003 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 356 (2004), 1209-1231
  • MSC (2000): Primary 11G05; Secondary 14H25, 14H52, 14Q05
  • DOI:
  • MathSciNet review: 2021618