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An Asymptotic formula for Tate-Shafarevich groups of CM elliptic curves at supersingular primes

Katharina Müller

Abstract

Let $K$ be an imaginary quadratic field and $E/\mathbb{Q}$ an elliptic curves with complex multiplication by $\mathcal{O}_K$. Let $K_\infty/K$ be the anticyclotomic $\mathbb{Z}_p$-extension of $K$ and $K_n$ the intermediate layers. Under additional assumptions on Kobayashi's signed Selmer groups we prove an asymptotic formula for the Tate-Shafarevich group over $K_n$.

An Asymptotic formula for Tate-Shafarevich groups of CM elliptic curves at supersingular primes

Abstract

Let be an imaginary quadratic field and an elliptic curves with complex multiplication by . Let be the anticyclotomic -extension of and the intermediate layers. Under additional assumptions on Kobayashi's signed Selmer groups we prove an asymptotic formula for the Tate-Shafarevich group over .

Paper Structure

This paper contains 7 sections, 26 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. plus/minus Selmer groups
  3. Plus/Minus Tate-Shafarevich groups
  4. Estimating $\kappa^0_{n,n-1}$
  5. Estimating the kernels
  6. Estimating $\kappa^{-\varepsilon}_{n,n-1}$
  7. Estimating $\sha_{n,n-1}$

Key Result

Theorem A

Assume that $\sha(E/K_n)$ and the fine Selmer group $\mathop{\mathrm{Sel}}\nolimits^0(E/K_n)[\omega_n^{-\varepsilon}]$ are finite for all $n$. Then for all $n$ large enough and such that $(-1)^n=\varepsilon$ one has The integers $\mu$ and $\lambda$ are the Iwasawa invariants of the fine Tate-Shafarevich groups.

Theorems & Definitions (49)

  • Theorem A
  • Theorem B
  • Remark 1.1
  • Theorem C
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 39 more