The growth of the rank of Abelian varieties upon extensions

Abstract

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, SL 2 (Fp) or PSL 2 (Fp). This implies a restriction on rk E(K(E[p])) − rk E(K(ζp)) 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 ⊗ End Q Res K/Q E contains a number field of degree 2 n . We ask whether every elliptic curve E/K has infinite rank over KQ(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 2 n .

Hyperelliptic modular curves and isogenies of elliptic curves over quadratic fields

“…In particular, since δ is not a square in any of the cases we consider in this paper, E(L) has even rank. The rank of E will also be even over many extensions of L; see [6] for details.…”