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On the distribution of Iwasawa invariants associated to multigraphs

Cédric Dion, Antonio Lei, Anwesh Ray, Daniel Vallières

Abstract

Let $\ell$ be a prime number. The Iwasawa theory of multigraphs is the systematic study of growth patterns in the number of spanning trees in abelian $\ell$-towers of multigraphs. In this context, growth patterns are realized by certain analogues of Iwasawa invariants, which depend on the prime $\ell$ and the abelian $\ell$-tower of multigraphs. We formulate and study statistical questions about the behaviour of the Iwasawa $μ$ and $λ$ invariants.

On the distribution of Iwasawa invariants associated to multigraphs

Abstract

Let be a prime number. The Iwasawa theory of multigraphs is the systematic study of growth patterns in the number of spanning trees in abelian -towers of multigraphs. In this context, growth patterns are realized by certain analogues of Iwasawa invariants, which depend on the prime and the abelian -tower of multigraphs. We formulate and study statistical questions about the behaviour of the Iwasawa and invariants.
Paper Structure (16 sections, 38 theorems, 205 equations, 1 figure)

This paper contains 16 sections, 38 theorems, 205 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic properties of multigraphs
  4. Iwasawa theory of multigraphs
  5. Connectedness of multigraphs in abelian $\ell$-towers
  6. Basic properties of Iwasawa invariants
  7. Equations over finite fields
  8. Systems of equations
  9. Quadratic forms and their solutions
  10. Statistics for the Iwasawa invariants of bouquets
  11. The vanishing of the $\mu$-invariant
  12. The distribution of Iwasawa invariants of a fixed $X_t$ varying $\alpha$
  13. Results for small values of $t$
  14. Vary $t$ and $\alpha$
  15. Statistics for multigraphs with two vertices
  16. ...and 1 more sections

Key Result

Lemma 2.7

Let $X$ be a multigraph, $\ell$ be a prime and $\alpha:S\rightarrow \mathbb{Z}_\ell$ a function. The constant term of the characteristic series $f(T)$ (see char series def) is equal to $0$.

Figures (1)

  • Figure :

Theorems & Definitions (92)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.7
  • proof
  • Definition 2.9
  • Lemma 2.10
  • proof
  • ...and 82 more