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Construction of graph coverings with prescribed Iwasawa invariants

Takenori Kataoka

Abstract

For a $\mathbb{Z}_p$-covering of connected graphs, an analogue of Iwasawa's class number formula describes the growth of the number of spanning trees in terms of Iwasawa $λ$- and $μ$-invariants. In this paper, we show that any pair $(λ, μ)$ can be realized as the Iwasawa invariants of an unramified $\mathbb{Z}_p$-covering of a bouquet, provided that the necessary condition that $λ$ is odd is satisfied. We further show that any pair $(λ, μ)$, without a parity condition, can be realized if we allow ramified $\mathbb{Z}_p$-coverings.

Construction of graph coverings with prescribed Iwasawa invariants

Abstract

For a -covering of connected graphs, an analogue of Iwasawa's class number formula describes the growth of the number of spanning trees in terms of Iwasawa - and -invariants. In this paper, we show that any pair can be realized as the Iwasawa invariants of an unramified -covering of a bouquet, provided that the necessary condition that is odd is satisfied. We further show that any pair , without a parity condition, can be realized if we allow ramified -coverings.
Paper Structure (23 sections, 18 theorems, 71 equations, 2 tables)

This paper contains 23 sections, 18 theorems, 71 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Iwasawa invariants
  4. The analogue of Iwasawa's class number formula
  5. Graphs
  6. Unramified $\mathbb{Z}_p$-coverings
  7. The formula
  8. The parity condition
  9. Statement and a reduction step
  10. $\iota$-invariant power series
  11. Proof of Theorem \ref{['thm:main']}
  12. $\mathbb{Z}_p$-coverings of bouquets
  13. Study of $h + \iota(h)$
  14. Construction
  15. Remarks
  16. ...and 8 more sections

Key Result

Theorem 1.1

For any odd $\lambda \geq 1$ and any $\mu \geq 0$, there exists an unramified $\mathbb{Z}_p$-covering $X_{\infty}/X$ of connected graphs satisfying $\lambda(X_{\infty}/X) = \lambda$ and $\mu(X_{\infty}/X) = \mu$. Indeed, we construct such a covering with $X$ a bouquet (i.e., a graph with a single ve

Theorems & Definitions (40)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • Proposition 3.1
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • ...and 30 more