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<p>In this paper, we obtain some sufficient and necessary conditions for indecomposable positive definite integral lattices with discriminants 2, 3, 4 and 5 over <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10114_2025_2562_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\({\mathbb Z}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </math></EquationSource> </InlineEquation> being additively indecomposable lattices. Using these results, we prove that there exist additively indecomposable positive integral quadratic lattices with discriminants 2, 3, 4 and 5 and rank greater than or equal to 2 but for 35 exceptions. In the exceptions there are no lattices with the desired properties. We also give a lifting theorem of additively indecomposable positive definite integral lattices.</p>

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Additively Indecomposable Positive Definite Integral Lattices

  • 期刊论文

  • 发表日期：

    2025年1月24日

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  • Ruiqing Wang

摘要

In this paper, we obtain some sufficient and necessary conditions for indecomposable positive definite integral lattices with discriminants 2, 3, 4 and 5 over \({\mathbb Z}\) Z being additively indecomposable lattices. Using these results, we prove that there exist additively indecomposable positive integral quadratic lattices with discriminants 2, 3, 4 and 5 and rank greater than or equal to 2 but for 35 exceptions. In the exceptions there are no lattices with the desired properties. We also give a lifting theorem of additively indecomposable positive definite integral lattices.