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September 29, 2025Open Access

Mazur's growth number conjecture and congruences

ARAnwesh RayChennai Mathematical Institute

Key Points

Establishing Mazur's conjecture for specific triples demonstrates significant connections between growth of Selmer ranks and Galois representations.
The findings show that the Mordell-Weil ranks in certain extensions remain bounded for elliptic curves, suggesting parallels with modular forms.
Results further indicate that congruences mod p play a crucial role in linking Selmer ranks and Galois representations.
This work builds on and extends existing results related to Greenberg-Vatsal and Emerton-Pollack-Weston, providing new insights into the conjecture.

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 Zₚ-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 Zₚ-extensions of K remain bounded for elliptic curves E' such that Ep and E'p are isomorphic as Galois modules.

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