Mazur's growth number conjecture and congruences

2025/05/26 by Anwesh Ray, Ray, Anwesh
Mathematics · #11G05 #11R23 #Analytic Number Theory Research #FOS: Mathematics #Mathematics and Applications #Meromorphic and Entire Functions #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2505.19542

openalex publication_date 2025/05/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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

Citations

• Mazur's Growth Number Conjecture in the Rank One Case
• Rank stability in quadratic extensions and Hilbert's tenth problem for the ring of integers of a number field
• Hilbert's tenth problem via additive combinatorics
• Base change and Iwasawa Main Conjectures for \rm GL₂

Related

• On a conjecture of Mazur predicting the growth of Mordell--Weil ranks in ℤ_p-extensions
• Mazur's Growth Number Conjecture in the Rank One Case
• Selmer stability in families of congruent Galois representations
• Growth of Selmer rank in nonabelian extensions of number fields
• Mazur's isogeny theorem