Visualising Sha[2] in Abelian surfaces

Author:
Nils Bruin

Translated by:

Journal:
Math. Comp. **73** (2004), 1459-1476

MSC (2000):
Primary 11G05; Secondary 14G05, 14K15

DOI:
https://doi.org/10.1090/S0025-5718-04-01633-3

Published electronically:
January 8, 2004

MathSciNet review:
2047096

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Given an elliptic curve over a number field and an element in its -Selmer group, we give two different ways to construct infinitely many Abelian surfaces such that the homogeneous space representing occurs as a fibre of over another elliptic curve . We show that by comparing the -Selmer groups of , and , we can obtain information about and we give examples where we use this to obtain a sharp bound on the Mordell-Weil rank of an elliptic curve.

As a tool, we give a precise description of the -Selmer group of an Abelian surface that is -isogenous to a product of elliptic curves .

One of the constructions can be applied iteratively to obtain information about . We give an example where we use this iterated application to exhibit an element of order in .

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Additional Information

**Nils Bruin**

Affiliation:
Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6

Email:
bruin@member.ams.org

DOI:
https://doi.org/10.1090/S0025-5718-04-01633-3

Keywords:
Visualisation,
Shafarevich-Tate group,
elliptic curve,
two-descent,
Mordell-Weil group

Received by editor(s):
February 2, 2002

Received by editor(s) in revised form:
September 13, 2002

Published electronically:
January 8, 2004

Additional Notes:
The research in this paper was funded by the Pacific Institute for the Mathematical Sciences, Simon Fraser University, the University of British Columbia, and the School of Mathematics of the University of Sydney

Article copyright:
© Copyright 2004
American Mathematical Society