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<p>Let <i>p</i> be a prime number and let <i>d</i> be a positive integer. In this paper, we generalize Iwasawa theory for graphs initiated by Gonet and Vallières to weighted graphs. In particular, we prove an analogue of Iwasawa’s class number formula and that of Kida’s formula for compatible systems of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\((\mathbb {Z}/p^n\mathbb {Z})^d\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">/</mo> <msup> <mi>p</mi> <mi>n</mi> </msup> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">)</mo> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation>-covers of weighted graphs. We also provide numerical examples of characteristic elements and Iwasawa invariants. At the end of this paper, we give an application of the ideas of Iwasawa theory to the theory of discrete-time quantum walks in graphs.</p>

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Iwasawa theory for weighted graphs

  • 期刊论文

  • 发表日期：

    2025年10月4日

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  • Taiga Adachi,
  • Kosuke Mizuno,
  • Sohei Tateno

摘要

Let p be a prime number and let d be a positive integer. In this paper, we generalize Iwasawa theory for graphs initiated by Gonet and Vallières to weighted graphs. In particular, we prove an analogue of Iwasawa’s class number formula and that of Kida’s formula for compatible systems of \((\mathbb {Z}/p^n\mathbb {Z})^d\) ( Z / p n Z ) d -covers of weighted graphs. We also provide numerical examples of characteristic elements and Iwasawa invariants. At the end of this paper, we give an application of the ideas of Iwasawa theory to the theory of discrete-time quantum walks in graphs.