On $\mathbb{Z}_p$-towers of graph coverings arising from a constant voltage assignment

Antonio Lei; Katharina M\"uller

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On $\mathbb{Z}_p$-towers of graph coverings arising from a constant voltage assignment

Antonio Lei, Katharina Müller

Number Theory · Submitted 7 Jan 2025

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We prove the existence and uniqueness (up to isomorphisms) of such towers.

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arXiv ID2501.03852 (version 2)

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CitationAntonio Lei, Katharina M\"uller. "On $\mathbb{Z}_p$-towers of graph coverings arising from a constant voltage assignment." arXiv:2501.03852 (2025). https://pith.science/paper/Y6AS5F3C

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We investigate properties of $\mathbb{Z}_p$-towers of graph coverings that arise from a constant voltage assignment. We prove the existence and uniqueness (up to isomorphisms) of such towers. Furthermore, we study the Iwasawa invariants of these towers, and apply our results to towers of isogeny graphs enhanced with level structures, as well as towers arising from volcano graphs.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We investigate properties of $\mathbb{Z}_p$-towers of graph coverings that arise from a constant voltage assignment.	contribution 0.82	0.80	0.07	asserted 0.77
s1	We prove the existence and uniqueness (up to isomorphisms) of such towers.	result 0.94	0.96	0.42	asserted 1.00
s2	Furthermore, we study the Iwasawa invariants of these towers, and apply our results to towers of isogeny graphs enhanced with level structures, as well as towers arising from volcano graphs.	contribution 0.99	0.94	0.25	asserted 1.00

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Original abstract

We investigate properties of $\mathbb{Z}_p$-towers of graph coverings that arise from a constant voltage assignment. We prove the existence and uniqueness (up to isomorphisms) of such towers. Furthermore, we study the Iwasawa invariants of these towers, and apply our results to towers of isogeny graphs enhanced with level structures, as well as towers arising from volcano graphs.

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1 arXiv paper cite this in 8 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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Compares with it 1 paper

While the edges of $(r,ℓ,e_p^n)$ are given by $ℓ$-isogenies up to automorphisms of their targets, the counterparts in [ref] , [ref] are given by $ℓ$-isogenies not up to automorphisms of their targets.

Distributions of Iwasawa $\lambda$-invariants of $\mathbf{Z}_p$-towers over supersingular isogeny graphs · Level structure aspects of constant $Z_p$-towers · 0.99

Cites it as background 1 paper

On the other hand, Lei and Müller [ref] , [ref] , [ref] , [ref] study $Z_p$-towers of certain directed graphs called isogeny graphs, in particular supersingular $ℓ$-isogeny graphs, abbreviated as $ℓ$-SIGs.

Distributions of Iwasawa $\lambda$-invariants of $\mathbf{Z}_p$-towers over supersingular isogeny graphs · Introduction · 0.94

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Builds on 1 paper

The following lemma is a slight generalization of [ref] , where the same result was proved under the assumption that $X_n$ is connected for all $n$.

On the non-commutative Iwasawa main conjecture for voltage covers of graphs · Derived graphs arising from a constant voltage assignment · 0.98

Cites as support 1 paper

This conjecture is of interest since it implies that the $K$-theoretic formulation of the non-commutative Iwasawa main conjecture is true for the tower $(Z_n)_n≥m_0$ (as demonstrated in [ref] ).

On the non-commutative Iwasawa main conjecture for voltage covers of graphs · Main results · 0.99

Cites as background 1 paper

These graphs are utilized to describe the isogeny graphs of ordinary elliptic curves in [ref] , where $l$ is a prime number; we refer the reader to [ref] for further details.

Ordinary isogeny graphs over $\mathbb{F}_p$: the inverse volcano problem · Constant $$-towers over volcano graphs · 0.93

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