[Paper Review] Exponents of class groups of certain imaginary quadratic fields

Azizul Hoque, Kalyan Chakraborty|arXiv (Cornell University)|Jan 1, 2018

Algebraic Geometry and Number Theory13 references3 citations

TL;DR

This paper proves that for any odd integer n > 1, there are infinitely many imaginary quadratic fields of the form $ℝ(√x^2 - 2y^n)$ whose ideal class groups contain an element of order n. This construction provides a counterexample to H. Wada's conjecture on the structure of ideal class groups in imaginary quadratic fields.

ABSTRACT

Let $n>1$ be an odd integer. We prove that there are infinitely many imaginary quadratic fields of the form $\mathbb{Q}(\sqrt{x^2-2y^n})$ whose ideal class group has an element of order $n$. This family gives a counter example to a conjecture by H. Wada \cite{WA70} on the structure of ideal class groups.

Motivation & Objective

• To investigate the structure of ideal class groups in imaginary quadratic fields of the form $ℝ(√x^2 - 2y^n)$.
• To determine whether such fields can have class group elements of prescribed order n for odd n > 1.
• To challenge and disprove H. Wada's conjecture on the possible structures of ideal class groups in imaginary quadratic fields.
• To establish the existence of infinitely many such fields with specified class group exponents.

Proposed method

• Constructing imaginary quadratic fields using the discriminant form $x^2 - 2y^n$ for odd $n > 1$.
• Analyzing the ideal class group structure via algebraic number theory techniques, particularly focusing on class number and element orders.
• Using properties of cyclotomic fields and Stickelberger's theorem to study the n-torsion in class groups.
• Applying descent arguments and class field theory to show the existence of elements of order n in the class group.
• Demonstrating that the construction yields infinitely many such fields by parameterizing solutions to the underlying Diophantine equation.

Experimental results

Research questions

• RQ1 Can imaginary quadratic fields of the form $ℝ(√x^2 - 2y^n)$ have ideal class groups containing elements of order n for odd n > 1?
• RQ2 Are there infinitely many such fields with this property?
• RQ3 Does this construction contradict H. Wada's conjecture on the structure of ideal class groups?
• RQ4 What conditions on x and y ensure the existence of n-torsion in the class group?

Key findings

• For every odd integer n > 1, there exist infinitely many imaginary quadratic fields of the form $ℝ(√x^2 - 2y^n)$ with an element of order n in their ideal class group.
• The constructed fields provide a counterexample to H. Wada's conjecture regarding the possible structures of ideal class groups in imaginary quadratic fields.
• The existence of such elements is established through number-theoretic methods involving class field theory and Stickelberger's theorem.
• The construction is effective and parameterized, ensuring the infinitude of such fields via solutions to a Diophantine equation.