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$p$-Ordinary Part of Hyperbolic Cycles on Modular Curves

Hohto Bekki, Ryotaro Sakamoto

Abstract

In this paper, we study hyperbolic cycles in the first homology group with local coefficients of congruence subgroups of $\mathrm{SL}_2(\mathbb{Z})$. We prove that, for any prime number $p$, the $p$-ordinary part of the first homology group is generated by hyperbolic cycles.

$p$-Ordinary Part of Hyperbolic Cycles on Modular Curves

Abstract

In this paper, we study hyperbolic cycles in the first homology group with local coefficients of congruence subgroups of . We prove that, for any prime number , the -ordinary part of the first homology group is generated by hyperbolic cycles.
Paper Structure (19 sections, 19 theorems, 100 equations)

This paper contains 19 sections, 19 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $M_2^+(\mathbb{Z})$-modules $\mathcal{V}_k$
  4. Group homology
  5. $H_1(\overline{\Gamma},\mathcal{V})$
  6. Definition of hyperbolic cycles
  7. Hecke operators
  8. Definition of double coset operators
  9. $p$-th Hecke operator
  10. Hecke stability of $\mathfrak{Z}_{\Gamma, 2k}$
  11. Ordinary part
  12. Definition of the ordinary part
  13. Main result
  14. The case where $p \mid N$
  15. The case where $p \nmid N$
  16. ...and 4 more sections

Key Result

Theorem 1.1

For any prime number $p$ and any integer $N$ such that $p \mid N$ and $N \geq 4$, we have Here, $\mathcal{V}_{2k,\mathbb{Z}_p} := \mathcal{V}_{2k}\otimes \mathbb{Z}_p$, and $\mathfrak{Z}_{\Gamma_{1}(N), 2k}^{p\text{-}\mathrm{ord}}$ (resp. $H_1^\mathrm{ord}(\overline{\Gamma_1(N)}, \mathcal{V}_{2k, \mathbb{Z}_{p}})$) denotes the $p$-ordinary part of $\mathfrak{Z}_{\Gamma_{1}(N), 2k}$ (resp

Theorems & Definitions (44)

  • Theorem 1.1: BS25
  • Theorem 1.2: Theorem \ref{['thm:main']}
  • Remark 1.3
  • Theorem 1.4: Theorem \ref{['thm:quotient_finite_non-ordinary']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 3.1
  • Definition 3.2
  • ...and 34 more