Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Class groups of imaginary biquadratic fields

Kalyan Banerjee, Kalyan Chakraborty, Arkabrata Ghosh

TL;DR

The paper addresses constructing infinite families of imaginary biquadratic fields with large class groups by combining Soleng's elliptic-curve–to–class-group technique with a hybrid elliptic–hyperelliptic approach, and further employing Banerjee–Hoque results. It propagates torsion from elliptic curves to imaginary quadratic fields and then to biquadratic fields via norm/transfer maps and going-up/going-down, yielding class-group elements of order $n$ or $2n$ (with $n\le 16$). It also proves a second result by linking elements of order $l$ from ${\bf Q}(\sqrt{k^2-p^l})$ to a biquadratic field with an element of order $ln$ in its class group, under disjointness conditions for infinitely many primes. Together, these yield two infinite families of imaginary biquadratic fields with large class groups, highlighting a deep connection between elliptic/hyperelliptic curves and higher-degree arithmetic.

Abstract

We present two distinct families of imaginary biquadratic fields, each of which contains infinitely many members, with each member having large class groups. Construction of the first family involves elliptic curves and their quadratic twists, whereas to find the other family, we use a combination of elliptic and hyperelliptic curves. Two main results are used, one from Soleng and the other from Banerjee and Hoque.

Class groups of imaginary biquadratic fields

TL;DR

The paper addresses constructing infinite families of imaginary biquadratic fields with large class groups by combining Soleng's elliptic-curve–to–class-group technique with a hybrid elliptic–hyperelliptic approach, and further employing Banerjee–Hoque results. It propagates torsion from elliptic curves to imaginary quadratic fields and then to biquadratic fields via norm/transfer maps and going-up/going-down, yielding class-group elements of order or (with ). It also proves a second result by linking elements of order from to a biquadratic field with an element of order in its class group, under disjointness conditions for infinitely many primes. Together, these yield two infinite families of imaginary biquadratic fields with large class groups, highlighting a deep connection between elliptic/hyperelliptic curves and higher-degree arithmetic.

Abstract

We present two distinct families of imaginary biquadratic fields, each of which contains infinitely many members, with each member having large class groups. Construction of the first family involves elliptic curves and their quadratic twists, whereas to find the other family, we use a combination of elliptic and hyperelliptic curves. Two main results are used, one from Soleng and the other from Banerjee and Hoque.

Paper Structure

This paper contains 4 sections, 7 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Soleng's technique
  3. The connection between the class groups of biquadratic fields and elliptic curves
  4. Proof of Theorem Thm:1.2

Key Result

Theorem 1.1

Let $d$ be a square-free integer and let $E_d$ be the quadratic twist of an elliptic curve $E$, defined over ${\mathbb Q}(\sqrt{-d})$ such that $d^3 b$ is not a square. Let $E_d({\mathbb Q})$ has a primitive element of order $n\leq 16$. Then there are infinitely many biquadratic fields ${\mathbb Q}(

Theorems & Definitions (13)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Theorem 2.2: Theorem 2.1 S
  • Lemma 3.1
  • proof
  • proof
  • Lemma 3.2
  • proof
  • Example 3.3
  • ...and 3 more