Rank growth of abelian varieties over certain finite Galois extensions

Abstract

Let $A/K$ be an abelian variety over a number field $K$. We prove that a finite automorphism group $G \subseteq \mathrm{Aut}_K(X)$ of a smooth projective variety $X/K$ such that $X/G \cong \mathbb{P}_K^d$ can force the rank growth of $A$ over infinitely many mutually linearly disjoint $G$-extensions $L_i/K$. The proof is based on Hilbert irreducibility and N\'{e}ron specialization. We then combine the theorem with finite group representations to obtain explicit lower bounds for rank growth. As applications, we obtain rank growth results for Jacobian varieties and construct explicit examples. We further prove arbitrarily large rank growth of abelian varieties over symmetric extensions. Finally, we study the connection with infinite rank conjectures.