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Harder's denominator problem for $\mathrm{SL}_2(\mathbb{Z})$ and its applications

Hohto Bekki, Ryotaro Sakamoto

Abstract

The aim of this paper is to give a full detail of the proof given by Harder of a theorem on the denominator of the Eisenstein class for $\mathrm{SL}_2(\mathbb{Z})$ and to show that the theorem has some interesting applications including the proof of a recent conjecture by Duke on the integrality of the higher Rademacher symbols. We also present a sharp universal upper bound for the denominators of the values of partial zeta functions associated with narrow ideal classes of real quadratic fields in terms of the denominator of the values of the Riemann zeta function.

Harder's denominator problem for $\mathrm{SL}_2(\mathbb{Z})$ and its applications

Abstract

The aim of this paper is to give a full detail of the proof given by Harder of a theorem on the denominator of the Eisenstein class for and to show that the theorem has some interesting applications including the proof of a recent conjecture by Duke on the integrality of the higher Rademacher symbols. We also present a sharp universal upper bound for the denominators of the values of partial zeta functions associated with narrow ideal classes of real quadratic fields in terms of the denominator of the values of the Riemann zeta function.
Paper Structure (50 sections, 88 theorems, 379 equations)

This paper contains 50 sections, 88 theorems, 379 equations.

Table of Contents

  1. Introduction
  2. The denominator of Eisenstein classes for $\mathrm{SL}_2(\mathbb{Z})$
  3. Reformulation in terms of the holomorphic Eisenstein series
  4. Strategy of the proof of Theorem \ref{['intro:reformulated theorem']}
  5. Applications to Duke's conjecture and to the special values of the partial zeta functions of real quadratic fields
  6. Duke's conjecture
  7. The denominators of the partial zeta functions of real quadratic fields
  8. Preliminaries and the Eisenstein class
  9. Definitions of Modular curve and Borel--Serre compactification
  10. Modular symbols and (co)homology
  11. $M_2^+(\mathbb{Z})$-modules $\mathcal{M}_n$ and $\mathcal{M}_n^{\flat}$
  12. Hecke operators
  13. Formal duality
  14. Eichler--Shimura homomorphism
  15. Definition of the Eisenstein class
  16. ...and 35 more sections

Key Result

Theorem 1.1

For any even integer $n \geq 2$, we have where $\zeta(s)$ denotes the Riemann zeta functionIn the present paper, the numerator and the denominator of a rational number are always defined to be positive integers..

Theorems & Definitions (190)

  • Theorem 1.1: Harder CAG
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4: Lemma \ref{['lem:eisenstein-pairing-eigen-rational']}, Proposition \ref{['prop:eisenstein-rational']}, and Theorem \ref{['thm:main result']}
  • Remark 1.5
  • Theorem 1.6: Corollary \ref{["cor:duke's conjecture"]}
  • Remark 1.7
  • Proposition 1.8: Corollary \ref{['cor:partial zeta denominator and J_n']}
  • Theorem 1.9: Corollary \ref{['cor:sherpness of the universal bound of partial zeta functions']}
  • Remark 1.10
  • ...and 180 more