Mazur's growth number conjecture and congruences

TL;DR

This paper explores how Mazur's conjecture on Selmer rank growth in $Z_p$-extensions of imaginary quadratic fields behaves under $p$-congruences between Galois representations, establishing new cases and bounds.

Contribution

It proves Mazur's conjecture for specific triples and extends results to Greenberg Selmer groups of modular forms congruent mod $p$, including Hida family specializations.

Findings

Mazur's conjecture verified for certain triples $(E,K,p)$

Bounded Mordell-Weil ranks in non-anticyclotomic $Z_p$-extensions for congruent elliptic curves

Extension of results to Greenberg Selmer groups and Hida families

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 $E'$ such that $E[p]$ and $E'[p]$ are isomorphic as Galois modules.