Ranks of abelian varieties over infinite extensions of the rationals

Abstract

Let S be an infinite set of rational primes and, for some p ∈ S, let ${\mathbb{Q}_S^{(p)}}$ be the compositum of all extensions unramified outside S of the form ${\mathbb{Q}(\mu_p,\sqrt[p]{d})}$, for ${d \in \mathbb{Q}^\times}$. If ${(\sigma) = (\sigma_{1},\ldots,\sigma_{n}) \in {\rm Gal} {(\overline{\mathbb{Q}}/\mathbb{Q})}^n}$, let ${(\mathbb{Q}_S^{(p)})^{(\sigma)}}$ be the intersection of the fixed fields by ${{\langle\sigma_{i}\rangle}}$, for i = 1, . . , n. We provide a wide family of elliptic curves ${E/\mathbb{Q}}$ such that the rank of ${E((\mathbb{Q}_S^{(p)})^{(\sigma)})}$ is infinite for all n ≥ 0 and all ${(\sigma) \in {\rm Gal}(\overline{\mathbb{Q}}/\mathbb{Q})^n}$, subject to the parity conjecture. Similarly, let ${(A/\mathbb{Q},\phi)}$ be a polarized abelian variety, let K be a quadratic number field fixed by ${(\sigma) \in {\rm Gal}(\overline{\mathbb{Q}}/\mathbb{Q})^n}$, let S be an infinite set of primes of ${\mathbb{Q}}$ and let $K^{p-{\rm dihe}}_S$ be the maximal abelian p-elementary extension of K unramified outside primes of K lying over S and dihedral over ${\mathbb{Q}}$. We show that, under certain hypotheses, the ${\mathbb{Z}_p}$ -corank of sel _p ∞(A/F) is unbounded over finite extensions F/K contained in ${(K^{p-{\rm dihe}}_S)^{(\sigma)}/K}$. As a consequence, we prove a strengthened version of a conjecture of M. Larsen in a large number of cases.