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Explicit construction of decomposable Jacobians

Mesut Buğday, Mohammad Sadek

Abstract

In this note we give explicit constructions of decomposable hyperelliptic Jacobian varieties over fields of characteristic $0$. These include hyperelliptic Jacobian varieties that are isogenous to a product of two absolutely simple hyperelliptic Jacobian varieties, a square of a hyperelliptic Jacobian variety, and a product of four hyperelliptic Jacobian varieties three of which are of the same dimension. As an application, we produce families of hyperelliptic curves with infinitely many quadratic twists having at least two rational non-Weierstrass points; and families of quadruples of hyperelliptic curves together with infinitely many square-free $d$ such that the quadratic twists of each of the curves by $d$ possess at least one rational non-Weierstrass point.

Explicit construction of decomposable Jacobians

Abstract

In this note we give explicit constructions of decomposable hyperelliptic Jacobian varieties over fields of characteristic . These include hyperelliptic Jacobian varieties that are isogenous to a product of two absolutely simple hyperelliptic Jacobian varieties, a square of a hyperelliptic Jacobian variety, and a product of four hyperelliptic Jacobian varieties three of which are of the same dimension. As an application, we produce families of hyperelliptic curves with infinitely many quadratic twists having at least two rational non-Weierstrass points; and families of quadruples of hyperelliptic curves together with infinitely many square-free such that the quadratic twists of each of the curves by possess at least one rational non-Weierstrass point.

Paper Structure

This paper contains 5 sections, 16 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Decomposition into two abelian subvarieties
  4. Families of decomposable hyperelliptic Jacobian varieties
  5. Rational points on quadratic twists

Key Result

Theorem 2.1

Let $C$ be a hyperelliptic curve defined by the equation $y^2= f(x)$, where $f(x)$ is polynomial of degree $n$ without multiple roots in $K[x]$.

Theorems & Definitions (22)

  • Theorem 2.1
  • Proposition 2.2
  • Lemma 3.1
  • Proposition 3.2
  • Theorem 3.3
  • Proposition 3.4
  • Theorem 3.5
  • Proposition 3.6
  • Corollary 3.7
  • Definition 4.1
  • ...and 12 more