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How to do a $p$-descent on an elliptic curve.
- Source :
-
Transactions of the American Mathematical Society . Mar2004, Vol. 356 Issue 3, p1209-1231. 23p. - Publication Year :
- 2004
-
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. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 00029947
- Volume :
- 356
- Issue :
- 3
- Database :
- Academic Search Index
- Journal :
- Transactions of the American Mathematical Society
- Publication Type :
- Academic Journal
- Accession number :
- 19934848
- Full Text :
- https://doi.org/10.1090/S0002-9947-03-03366-X