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Semi-galois Categories IV: A deformed reciprocity law for Siegel modular functions

Takeo Uramoto

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.

Semi-galois Categories IV: A deformed reciprocity law for Siegel modular functions

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 , the algebraic Witt vectors over 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.
Paper Structure (17 sections, 18 theorems, 38 equations)

This paper contains 17 sections, 18 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Related works
  3. Acknowledgements
  4. Preliminaries
  5. Modular vectors
  6. Witt vectors as functions on $DR_K$
  7. Modular vectors
  8. Basic facts on CM fields
  9. CM fields and Riemann forms
  10. Projective embedding
  11. Complex multiplication
  12. Modular vectors over CM fields
  13. Construction of modular vectors
  14. Basic properties
  15. Galois correspondence
  16. ...and 2 more sections

Key Result

Theorem 2.1.2

We have a canonical isomorphism of $\Lambda$-rings over $K$:

Theorems & Definitions (45)

  • Definition 2.1.1: Deligne-Ribet monoid $DR_K$
  • Theorem 2.1.2: Corollary 3.1.6, Uramoto20
  • Proposition 2.1.3: Proposition 8.2, Yalkinoglu Yalkinoglu13
  • Definition 2.1.4: modular vectors $\widehat{f}_a$
  • Proposition 2.1.5
  • proof
  • Definition 2.2.1: polarization on complex tori
  • Remark 2.2.2: symplectic basis and type
  • Definition 2.2.3: theta function $\theta^k$
  • Definition 2.2.4: projective embedding
  • ...and 35 more