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How to do a $p$-descent on an elliptic curve

Authors: Edward F. Schaefer and Michael Stoll
Journal: Trans. Amer. Math. Soc. 356 (2004), 1209-1231
MSC (2000): Primary 11G05; Secondary 14H25, 14H52, 14Q05
Published electronically: October 27, 2003
MathSciNet review: 2021618
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Abstract: 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

Michael Stoll
Affiliation: School of Engineering and Science, International University Bremen, P.O. Box 750561, 28725 Bremen, Germany

Keywords: Elliptic curve over number field, $p$-descent, Selmer group, Mordell-Weil rank, Shafarevich-Tate group
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
Article copyright: © Copyright 2003 American Mathematical Society