How to do a 𝑝-descent on an elliptic curve

Schaefer, Edward; Stoll, Michael

doi:10.1090/S0002-9947-03-03366-X

How to do a $p$-descent on an elliptic curve
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by Edward F. Schaefer and Michael Stoll

Trans. Amer. Math. Soc. 356 (2004), 1209-1231

DOI: https://doi.org/10.1090/S0002-9947-03-03366-X

Published electronically: October 27, 2003

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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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