The growth of the rank of Abelian varieties upon extensions

2012/10/22 by Peter Bruin, Bruin, Peter, Filip Najman +1
Mathematics · #11G05 #11G10 #14K15 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11G05 #msc:11G10 #msc:14K15

paper · pdf · doi:10.48550/arxiv.1210.6085

9 pages

arxiv created 2012/10/22 · arxiv updated 2012/10/24

Abstract

We study the growth of the rank of elliptic curves and, more generally, Abelian varieties upon extensions of number fields. First, we show that if L/K is a finite Galois extension of number fields such that \Gal(L/K) does not have an index 2 subgroup and A/K is an Abelian variety, then \rk A(L)-\rk A(K) can never be 1. We obtain more precise results when \Gal(L/K) is of odd order, alternating, \SL₂(\F_p) or \PSL₂(\F_p). This implies a restriction on \rk E(K(E[p]))-\rk E(K(ζ_p)) when E/K is an elliptic curve whose mod p Galois representation is surjective. Similar results are obtained for the growth of the rank in certain non-Galois extensions. Second, we show that for every n≥2 there exists an elliptic curve E over a number field K such that \Q⊗_\Q\Res_K/\Q E contains a number field of degree 2ⁿ. We ask whether every elliptic curve E/K has infinite rank over K\Q(2), where \Q(2) is the compositum of all quadratic extensions of \Q. We show that if the answer is yes, then for any n≥2, there exists an elliptic curve E/K admitting infinitely many quadratic twists whose rank is a positive multiple of 2ⁿ.

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