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Mathematics of Computation

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Algorithms for the arithmetic of elliptic curves using Iwasawa theory

Authors: William Stein and Christian Wuthrich
Journal: Math. Comp. 82 (2013), 1757-1792
MSC (2010): Primary 11D88, 11G05, 11G40, 11G50, 14G05; Secondary 11Y50, 11Y40, 14G10
Published electronically: September 14, 2012
MathSciNet review: 3042584
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Abstract: We explain how to use results from Iwasawa theory to obtain information about $ p$-parts of Tate-Shafarevich groups of specific elliptic curves over $ \mathbb{Q}$. Our method provides a practical way to compute $ \char93 {\mbox {\Russian {Sh}}}(E/\mathbb{Q})(p)$ in many cases when traditional $ p$-descent methods are completely impractical and also in situations where results of Kolyvagin do not apply, e.g., when the rank of the Mordell-Weil group is greater than 1. We apply our results along with a computer calculation to show that $ {\mbox {\Russian {Sh}}}(E/\mathbb{Q})[p]=0$ for the 1,534,422 pairs $ (E,p)$ consisting of a non-CM elliptic curve $ E$ over $ \mathbb{Q}$ with conductor $ \leq 30,\!000$, rank $ \geq 2$, and good ordinary primes $ p$ with $ 5 \leq p < 1000$ and surjective mod-$ p$ representation.

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Additional Information

William Stein
Affiliation: Department of Mathematics, University of Washington, Seattle, Washington

Christian Wuthrich
Affiliation: School of Mathematical Sciences, University of Nottingham, University Park Nottingham NG7 2RD, United Kingdom

Received by editor(s): July 4, 2011
Received by editor(s) in revised form: November 11, 2011
Published electronically: September 14, 2012
Additional Notes: The first author was supported by NSF grants DMS-0555776 and DMS-0821725.
Article copyright: © Copyright 2012 William Stein and Christian Wuthrich