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<p>Let <i>p</i> be a rational prime, and let <i>X</i> be a connected finite graph. In this article we study voltage covers <i>X</i><sub>∞</sub> of <i>X</i> attached to a voltage assignment <i>α</i> which takes values in some uniform <i>p</i>-adic Lie group <i>G</i>. We formulate and prove an Iwasawa main conjecture for the projective limit of the Picard groups Pic(<i>X</i><sub><i>n</i></sub>) of the intermediate voltage covers <i>X</i><sub><i>n</i></sub>, <i>n</i> ∈ ℕ, and we prove one inclusion of a main conjecture for the projective limit of the Jacobians <i>J</i>(<i>X</i><sub><i>n</i></sub>).</p><p>Moreover, we study the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({\mathfrak M}_{H}(G)\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mrow> <mrow> <mi mathvariant="fraktur">M</mi> </mrow> </mrow> <mrow> <mi>H</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation>-property of ℤ<sub><i>p</i></sub>⟦<i>G</i>⟧-modules and prove a necessary condition for this property which involves the <i>μ</i>-invariants of ℤ<sub><i>p</i></sub>-subcovers <i>Y</i> ⊆ <i>X</i><sub>∞</sub> of <i>X</i>. If the dimension of <i>G</i> is equal to 2, then this condition is also sufficient.</p>

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On the non-commutative Iwasawa main conjecture for voltage covers of graphs

  • 期刊论文

  • 发表日期：

    2026年5月25日

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  • Sören Kleine,
  • Katharina Müller

摘要

Let p be a rational prime, and let X be a connected finite graph. In this article we study voltage covers X∞ of X attached to a voltage assignment α which takes values in some uniform p-adic Lie group G. We formulate and prove an Iwasawa main conjecture for the projective limit of the Picard groups Pic(Xn) of the intermediate voltage covers Xn, n ∈ ℕ, and we prove one inclusion of a main conjecture for the projective limit of the Jacobians J(Xn).

Moreover, we study the \({\mathfrak M}_{H}(G)\) M H ( G ) -property of ℤp⟦G⟧-modules and prove a necessary condition for this property which involves the μ-invariants of ℤp-subcovers Y ⊆ X∞ of X. If the dimension of G is equal to 2, then this condition is also sufficient.