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Bulletin of the London Mathematical Society Advance Access originally published online on September 13, 2009
Bulletin of the London Mathematical Society 2009 41(6):1002-1008; doi:10.1112/blms/bdp079
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© 2009 London Mathematical Society

A note on Larsen's conjecture and ranks of elliptic curves

Tim Dokchitser

Robinson College, Cambridge
CB3 9AN
United Kingdom
t.dokchitser@dpmms.cam.ac.uk

Vladimir Dokchitser

Gonville & Caius College
Cambridge
CB2 1TA
United Kingdom

Received 17 March 2008. Revision received 2 December 2008.

Let E be an elliptic curve defined over a number field K. Michael Larsen conjectured that, for any finitely generated subgroup G of Gal(K/K), the Mordell–Weil rank of E is unbounded in number fields fixed by G. We prove that the conjecture holds over K = Q for both the analytic rank and the p{infty}-Selmer rank of E for every odd prime p. For arbitrary E/K, we show that Larsen's conjecture follows from the standard conjectures for ranks of elliptic curves, provided that K has a real place or E has non-integral j-invariant.

2000 Mathematics Subject Classification 11G05 (primary), 11G40, 14G25 (secondary).

The first author is supported by a Royal Society University Research Fellowship.


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