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A short remark on the $\ell$-torsion part of class groups

Martin Widmer

Abstract

In a 2008 paper Ellenberg suggested a strategy to improve the known upper bounds for the $\ell$-torsion part of class groups of number fields of fixed degree $d$. Motivated by this he proposed a question about the number of primitive elements of small height in a number field. Here we answer Ellenberg's question. We also improve Heath-Brown's bound for the $\ell$-torsion part of class groups of purely cubic number fields, and we generalize our improvement to pure fields of arbitrary odd degree $d$.

A short remark on the $\ell$-torsion part of class groups

Abstract

In a 2008 paper Ellenberg suggested a strategy to improve the known upper bounds for the -torsion part of class groups of number fields of fixed degree . Motivated by this he proposed a question about the number of primitive elements of small height in a number field. Here we answer Ellenberg's question. We also improve Heath-Brown's bound for the -torsion part of class groups of purely cubic number fields, and we generalize our improvement to pure fields of arbitrary odd degree .
Paper Structure (6 sections, 8 theorems, 43 equations)

This paper contains 6 sections, 8 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Background and definitions
  3. A lower bound for $M_{K,\ell}$
  4. proof of Proposition \ref{['prop:fdl']}
  5. Constructing suitable primes
  6. A height lower bound for generators of pure fields and proof of Proposition \ref{['prop:GB']}

Key Result

Proposition 1

Let $d>1$ and $\ell\geq d/2$ be integers. Then $f(\ell,d)=1/(2\ell(d-1))$.

Theorems & Definitions (12)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Lemma 1: Ellenberg and Venkatesh
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • ...and 2 more