Mazur's growth number conjecture and congruences

Abstract

Motivated by the work of Greenberg-Vatsal and Emerton-Pollack-Weston, I investigate the extent to which Mazur's conjecture on the growth of Selmer ranks in ${\mathbb{Z}}_p$-extensions of an imaginary quadratic field persists under $p$-congruences between Galois representations. As a first step, I establish Mazur's conjecture for certain triples $(E,K,p)$ under explicit hypotheses. Building on this, I prove analogous results for Greenberg Selmer groups attached to modular forms that are congruent mod $p$ to $E$, including all specializations arising from Hida families of fixed tame level. In particular, I show that the Mordell-Weil ranks in non-anticyclotomic ${\mathbb{Z}}_p$-extensions of $K$ remain bounded for elliptic curves $E^{\prime }$ such that $E[p]$ and $E^{\prime }[p]$ are isomorphic as Galois modules.