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An explicit version of Shimura's reciprocity law for Siegel modular functions

Marco Streng

Abstract

We give an explicit version of Shimura's reciprocity law for singular values of Siegel modular functions. We use this to construct the first examples of class invariants of quartic CM fields that are smaller than Igusa invariants. Our statement also enables a new proof of Shimura's reciprocity law by Tonghai Yang.

An explicit version of Shimura's reciprocity law for Siegel modular functions

Abstract

We give an explicit version of Shimura's reciprocity law for singular values of Siegel modular functions. We use this to construct the first examples of class invariants of quartic CM fields that are smaller than Igusa invariants. Our statement also enables a new proof of Shimura's reciprocity law by Tonghai Yang.

Paper Structure

This paper contains 36 sections, 19 theorems, 63 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Summary of results
  3. Overview of content
  4. Definitions and statement of the main results
  5. The upper half space
  6. The algebraic groups
  7. Modular forms and group actions
  8. Computing the group action
  9. Complex multiplication
  10. Primitive CM points
  11. Computing the primitive CM points
  12. The type norm
  13. The first main theorem
  14. The class fields generated by complex multiplication
  15. Class invariants and a special case of the main theorem
  16. ...and 21 more sections

Key Result

Proposition \oldthetheorem

There is a right action of $\mathrm{GSp}_{2g}(\mathbf{Z}/N\mathbf{Z})$ on $\mathcal{F}_N$ given as follows. For $A\in \mathrm{GSp}_{2g}(\mathbf{Z}/N\mathbf{Z})$, let $t = \nu(A)$ and $B = i(t)^{-1} A$. Then where we have:

Theorems & Definitions (53)

  • Proposition \oldthetheorem
  • Remark \oldthetheorem
  • Example \oldthetheorem
  • Theorem \oldthetheorem: General reciprocity law
  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • proof : Proof of Algorithm \ref{['alg:mu']}
  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • ...and 43 more