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Mazur's growth number conjecture and congruences

Anwesh Ray

TL;DR

This work investigates the persistence of Mazur's Growth Number Conjecture for $p^ abla$-Selmer groups in $ abla$-extensions of imaginary quadratic fields under $p$-congruences of Galois representations. It develops a two-variable Iwasawa-theoretic framework to compare Selmer groups attached to elliptic curves and to modular forms congruent mod $p$, and proves that under explicit hypotheses (including the two-variable Iwasawa main conjecture and Heegner-type conditions) the Selmer group over the anticyclotomic/cyclotomic sum is cotorsion with $oldsymbol{}=0$ and $ lambda=1$, yielding Mazur's conjecture for $(E,K,p)$. It further extends these congruence-propagation results to Greenberg Selmer groups attached to $p$-ordinary Hecke eigenforms with residual isomorphism to $E[p]$, and shows that non-anticyclotomic specializations have finite corank, giving bounded Mordell–Weil ranks in these towers; an application to Hida families demonstrates weight-variable congruences preserve the finiteness properties. The results connect Iwasawa invariants, Heegner classes, and $p$-adic $L$-functions to derive explicit, testable criteria for rank stability in families of elliptic curves and modular forms. Overall, the paper advances the understanding of how $p$-congruences influence Selmer growth and rank behavior in rich Iwasawa-theoretic settings.

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.

Mazur's growth number conjecture and congruences

TL;DR

This work investigates the persistence of Mazur's Growth Number Conjecture for -Selmer groups in -extensions of imaginary quadratic fields under -congruences of Galois representations. It develops a two-variable Iwasawa-theoretic framework to compare Selmer groups attached to elliptic curves and to modular forms congruent mod , and proves that under explicit hypotheses (including the two-variable Iwasawa main conjecture and Heegner-type conditions) the Selmer group over the anticyclotomic/cyclotomic sum is cotorsion with and , yielding Mazur's conjecture for . It further extends these congruence-propagation results to Greenberg Selmer groups attached to -ordinary Hecke eigenforms with residual isomorphism to , and shows that non-anticyclotomic specializations have finite corank, giving bounded Mordell–Weil ranks in these towers; an application to Hida families demonstrates weight-variable congruences preserve the finiteness properties. The results connect Iwasawa invariants, Heegner classes, and -adic -functions to derive explicit, testable criteria for rank stability in families of elliptic curves and modular forms. Overall, the paper advances the understanding of how -congruences influence Selmer growth and rank behavior in rich Iwasawa-theoretic settings.

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 -extensions of an imaginary quadratic field persists under -congruences between Galois representations. As a first step, I establish Mazur's conjecture for certain triples under explicit hypotheses. Building on this, I prove analogous results for Greenberg Selmer groups attached to modular forms that are congruent mod to , including all specializations arising from Hida families of fixed tame level. In particular, I show that the Mordell-Weil ranks in non-anticyclotomic -extensions of remain bounded for elliptic curves such that and are isomorphic as Galois modules.

Paper Structure

This paper contains 12 sections, 13 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Prior results
  4. Main results
  5. Iwasawa theory of Selmer groups and congruences
  6. Selmer groups over $\mathbb{Z}_p$-extensions
  7. Iwasawa invariants of $\mathbb{Z}_p\llbracket T\rrbracket$-modules
  8. Greenberg Selmer groups and $p$-congruences
  9. Propagating the growth number conjecture via congruences
  10. Prior results
  11. Main results
  12. An application involving Hida families

Key Result

Theorem A

Let $E$ be an elliptic curve over $\mathbb{Q}$ with good ordinary reduction at a prime $p\geq 5$ and $K$ be an imaginary quadratic number field. Let $N$ be the conductor of $E$. Assume that the following conditions are satisfied for $(E,K, p)$: Let $\mathcal{K}$ be a $\mathbb{Z}_p$-extension of $K$ in which the primes of $K$ that lie above $p$ are ramified. Then, the following assertions hold.

Theorems & Definitions (30)

  • Conjecture 1.1: Mazur's Growth Number Conjecture
  • Theorem A: Proposition \ref{['main long propn']}
  • Theorem B: Theorem \ref{['main theorem of paper']}
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6
  • ...and 20 more