Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Rank Jumps and Growth of Shafarevich--Tate Groups for Elliptic Curves in $\mathbb{Z}/p\mathbb{Z}$-Extensions

Lea Beneish, Debanjana Kundu, Anwesh Ray

Abstract

In this paper, we use techniques from Iwasawa theory to study questions about rank jump of elliptic curves in cyclic extensions of prime degree. We also study growth of the $p$-primary Selmer group and the Shafarevich--Tate group in cyclic degree-$p$ extensions and improve upon previously known results in this direction.

Rank Jumps and Growth of Shafarevich--Tate Groups for Elliptic Curves in $\mathbb{Z}/p\mathbb{Z}$-Extensions

Abstract

In this paper, we use techniques from Iwasawa theory to study questions about rank jump of elliptic curves in cyclic extensions of prime degree. We also study growth of the -primary Selmer group and the Shafarevich--Tate group in cyclic degree- extensions and improve upon previously known results in this direction.

Paper Structure

This paper contains 11 sections, 32 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Recollections from Iwasawa theory
  4. Results, conjectures, and heuristics on ranks of elliptic curves
  5. Example: CM case
  6. Growth of Shafarevich--Tate groups in cyclic extensions
  7. Large 2-rank of the Shafarevich--Tate group in quadratic extensions
  8. Reviewing the proof of Matsuno's construction
  9. Effective version conditional on Goldfeld's conjecture
  10. An unconditional effective version
  11. Tables

Key Result

Theorem A

Let $E_{/{\mathbb Q}}$ be a non-CM elliptic curve and $p$ be a fixed prime number $\geq 5$ such that the residual representation at $p$ is surjective. Further suppose that $\mu_p(E/{\mathbb Q}) = \lambda_p(E/{\mathbb Q}) =0$. Then, there are infinitely many ${\mathbb Z}/p{\mathbb Z}$-extensions of $

Theorems & Definitions (69)

  • Theorem A
  • Corollary A
  • Theorem B
  • Theorem C
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Remark
  • ...and 59 more