References & Citations
Mathematics > Number Theory
Title: Semi-galois Categories IV: A deformed reciprocity law for Siegel modular functions
(Submitted on 22 May 2023 (v1), last revised 27 Mar 2024 (this version, v5))
Abstract: 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 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.
Submission history
From: Takeo Uramoto [view email][v1] Mon, 22 May 2023 17:33:30 GMT (29kb)
[v2] Wed, 24 May 2023 13:55:15 GMT (29kb)
[v3] Thu, 25 May 2023 14:43:57 GMT (29kb)
[v4] Tue, 26 Mar 2024 09:32:57 GMT (29kb)
[v5] Wed, 27 Mar 2024 04:45:58 GMT (29kb)
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