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<p>Triple symbols are arithmetic analogues of the mod <i>n</i> triple linking number in topology, where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(n &gt; 1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>&gt;</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> is an integer. In this paper, we introduce a cohomological formulation of a mod <i>n</i> triple symbol for characters over a number field containing a primitive <i>n</i>-th root of unity. Our definition is motivated by the arithmetic Chern–Simons theory and in this respect it differs from earlier approaches to triple symbols. We show that our symbol agrees with that of Rédei when <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(n=2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> and of Amano–Mizusawa–Morishita when <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(n=3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

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Triple symbols in arithmetic

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年9月5日

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  • Dohyeong Kim,
  • Masanori Morishita

摘要

Triple symbols are arithmetic analogues of the mod n triple linking number in topology, where \(n > 1\) n > 1 is an integer. In this paper, we introduce a cohomological formulation of a mod n triple symbol for characters over a number field containing a primitive n-th root of unity. Our definition is motivated by the arithmetic Chern–Simons theory and in this respect it differs from earlier approaches to triple symbols. We show that our symbol agrees with that of Rédei when \(n=2\) n = 2 and of Amano–Mizusawa–Morishita when \(n=3\) n = 3 .