Quadratic forms with a strong regularity property on the representations of squares

TL;DR

This paper investigates strongly s-regular quadratic forms, proving finiteness of their classes for fixed rank and minimal represented square, highlighting a significant regularity property in quadratic form representations.

Contribution

It establishes a finiteness result for strongly s-regular quadratic forms of fixed rank with a given minimal represented square.

Findings

Finiteness of isometry classes for fixed rank and minimal square

Strong regularity property on representations of squares

Results applicable to positive definite quadratic forms

Abstract

A (positive definite and non-classic integral) quadratic form is called strongly $s$-regular if it satisfies a strong regularity property on the number of representations of squares of integers. In this article, we prove that for any integer $k \ge 2$, there are only finitely many isometry classes of strongly $s$-regular quadratic forms with rank $k$ if the minimum of the nonzero squares that are represented by them is fixed.