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<p>In this paper, we will define analogues of multiple zeta values by replacing the differential forms defining multiple zeta values with some <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\mathbb {Q}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">Q</mi> </math></EquationSource> </InlineEquation>-rational differential forms on the Fermat curve <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(F_2\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>F</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> of degree 2, and discuss their arithmetic properties. We also investigate a motivic structure of the motivic periods corresponding to our periods. However, in order to study them, the current theory for motivic zeta elements is insufficient, and it leads us to study the base extension of the space of the motivic periods <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\mathcal {H}_4\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">H</mi> <mn>4</mn> </msub> </math></EquationSource> </InlineEquation> of level 4 and its Galois invariant part.</p>

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On arithmetic properties of periods for some rational differential forms over \(\mathbb {Q}\) on the Fermat curve \(F_2\) of degree 2

  • 期刊论文

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  • 发表日期：

    2026年3月14日

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  • Eisuke Otsuka

摘要

In this paper, we will define analogues of multiple zeta values by replacing the differential forms defining multiple zeta values with some \(\mathbb {Q}\) Q -rational differential forms on the Fermat curve \(F_2\) F 2 of degree 2, and discuss their arithmetic properties. We also investigate a motivic structure of the motivic periods corresponding to our periods. However, in order to study them, the current theory for motivic zeta elements is insufficient, and it leads us to study the base extension of the space of the motivic periods \(\mathcal {H}_4\) H 4 of level 4 and its Galois invariant part.