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Structure of fine Selmer groups in abelian p-adic Lie extensions

Debanjana Kundu, Filippo Alberto Edoardo Nuccio Mortarino Majno Di Capriglio, Sujatha Ramdorai

Abstract

This paper studies fine Selmer groups of elliptic curves in abelian $p$-adic Lie extensions. A class of elliptic curves are provided where both the Selmer group and the fine Selmer group are trivial in the cyclotomic $\mathbb{Z}_p$-extension. The fine Selmer groups of elliptic curves with complex multiplication are shown to be pseudonull over the trivializing extension in some new cases. Finally, a relationship between the structure of the fine Selmer group for some CM elliptic curves and the Generalized Greenberg's Conjecture is clarified.

Structure of fine Selmer groups in abelian p-adic Lie extensions

Abstract

This paper studies fine Selmer groups of elliptic curves in abelian -adic Lie extensions. A class of elliptic curves are provided where both the Selmer group and the fine Selmer group are trivial in the cyclotomic -extension. The fine Selmer groups of elliptic curves with complex multiplication are shown to be pseudonull over the trivializing extension in some new cases. Finally, a relationship between the structure of the fine Selmer group for some CM elliptic curves and the Generalized Greenberg's Conjecture is clarified.
Paper Structure (9 sections, 19 theorems, 53 equations, 1 figure)

This paper contains 9 sections, 19 theorems, 53 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fine Selmer Groups in the Cyclotomic Extension
  4. Trivial Fine Selmer Groups in the Cyclotomic Tower
  5. over -Rational Number Fields
  6. for Elliptic Curves with CM: Special Cases
  7. Reformulation of
  8. The noncommutative setting
  9. and the

Key Result

Theorem 2.2

Let $\mathsf{E}/F$ be an elliptic curve and suppose that $\mathop{\mathrm{Gal}}\nolimits(F_\infty/F)$ is pro-$p$. Then conj:A for $\mathsf{E}/F$ is equivalent to the conj:Iwasawa for $F$.

Figures (1)

  • Figure 1: The diagram of fields occurring in Theorem \ref{['Thm: conj B implies GGC in several']}

Theorems & Definitions (47)

  • Definition 2.1
  • Theorem 2.2: CS05
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Theorem 3.3
  • proof
  • Remark 3.4
  • Corollary 3.5
  • proof
  • ...and 37 more