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<p>This paper is a sequel to our previous work, where we proved the <i>“modularity theorem” for algebraic Witt vectors</i> over imaginary quadratic fields. This theorem states that, in the case of imaginary quadratic fields <i>K</i>, the algebraic Witt vectors over <i>K</i> are precisely those generated by the <i>modular vectors</i> whose components are given by special values of deformation family of Fricke modular functions; arithmetically, this theorem implies certain congruences between special values of modular functions that are not necessarily Galois conjugate. In order to take a closer look at this modularity theorem, the current paper extends it to the case of CM fields. The main results include (i) a construction of algebraic Witt vectors from special values of deformation family of Siegel modular functions on Siegel upper-half space <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3772_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathfrak {H}_g\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="fraktur">H</mi> <mi>g</mi> </msub> </math></EquationSource> </InlineEquation> given by ratios of theta functions, and (ii) a Galois-theoretic characterization of which algebraic Witt vectors arise in this modular way, intending to exemplify a general Galois-correspondence result which is also proved in this paper.</p>

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Semi-galois categories IV: a deformed reciprocity law for Siegel modular functions

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  • 发表日期：

    2025年6月3日

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  • Takeo Uramoto

摘要

This paper is a sequel to our previous work, where we proved the “modularity theorem” for algebraic Witt vectors over imaginary quadratic fields. This theorem states that, in the case of imaginary quadratic fields K, the algebraic Witt vectors over K are precisely those generated by the modular vectors whose components are given by special values of deformation family of Fricke modular functions; arithmetically, this theorem implies certain congruences between special values of modular functions that are not necessarily Galois conjugate. In order to take a closer look at this modularity theorem, the current paper extends it to the case of CM fields. The main results include (i) a construction of algebraic Witt vectors from special values of deformation family of Siegel modular functions on Siegel upper-half space \(\mathfrak {H}_g\) H g given by ratios of theta functions, and (ii) a Galois-theoretic characterization of which algebraic Witt vectors arise in this modular way, intending to exemplify a general Galois-correspondence result which is also proved in this paper.