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

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

2008-03-07

arxiv.org/abs/0803.1122v1

Mathematics

Number Theory

7 pages

Scientific paper

10.1112/blms/bdp079

Let E be an elliptic curve defined over a number field K. Michael Larsen conjectured that for any finitely generated subgroup G of Gal(\bar 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-infinity 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 K has a real place or E has non-integral j-invariant.

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