How to do a -descent on an elliptic curve |
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Authors: | Edward F. Schaefer Michael Stoll |
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Affiliation: | Department of Mathematics and Computer Science, Santa Clara University, Santa Clara, California 95053 ; School of Engineering and Science, International University Bremen, P.O. Box 750561, 28725 Bremen, Germany |
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Abstract: | In this paper, we describe an algorithm that reduces the computation of the (full) -Selmer group of an elliptic curve over a number field to standard number field computations such as determining the (-torsion of) the -class group and a basis of the -units modulo th powers for a suitable set of primes. In particular, we give a result reducing this set of `bad primes' to a very small set, which in many cases only contains the primes above . As of today, this provides a feasible algorithm for performing a full -descent on an elliptic curve over , but the range of our algorithm will certainly be enlarged by future improvements in computational algebraic number theory. When the Galois module structure of is favorable, simplifications are possible and -descents for larger are accessible even today. To demonstrate how the method works, several worked examples are included. |
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Keywords: | Elliptic curve over number field $p$-descent Selmer group Mordell-Weil rank Shafarevich-Tate group |
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