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arXiv · 2508.07916

Composition laws of binary quadratic forms and isolations of quadratic forms

Jangwon Ju·Daejun Kim·Kyoungmin Kim·Mingyu Kim·Byeong-Kweon Oh
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Abstract

A positive definite and integral quadratic form $f$ is called irrecoverable if there is a quadratic form $F$ such that it represents all proper subforms of $f$, whereas it does not represent $f$ itself. In this case, $F$ is called an isolation of $f$. In this article, we prove that there does not exist a binary isolation of any unary quadratic form. We also prove that there does not exist a ternary isolation of any binary quadratic form. Furthermore, if the form class group of a primitive binary quadratic form has no element of order $4$, then the discriminant of any quaternary isolation of it, if exists, is a square of an integer. The composition laws of primitive binary quadratic forms play an essential role in the proofs of the results.

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Jangwon Ju, Daejun Kim, Kyoungmin Kim, Mingyu Kim, Byeong-Kweon Oh. 2025-08-11. Composition laws of binary quadratic forms and isolations of quadratic forms. https://arxiv.org/abs/2508.07916

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