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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2305.13265 |
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| _version_ | 1866929290081206272 |
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| author | Uramoto, Takeo |
| author_facet | Uramoto, Takeo |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_13265 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Semi-galois Categories IV: A deformed reciprocity law for Siegel modular functions Uramoto, Takeo Number Theory 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. |
| title | Semi-galois Categories IV: A deformed reciprocity law for Siegel modular functions |
| topic | Number Theory |
| url | https://arxiv.org/abs/2305.13265 |