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| Format: | Preprint |
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2022
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| Online Access: | https://arxiv.org/abs/2209.04890 |
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| _version_ | 1866909660323250176 |
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| author | Ray, Anwesh Vallières, Daniel |
| author_facet | Ray, Anwesh Vallières, Daniel |
| contents | Let $\ell$ be a rational prime and let $p:Y\rightarrow X$ be a Galois cover of finite graphs whose Galois group is a finite $\ell$-group. Consider a $\mathbb{Z}_{\ell}$-tower above $X$ and its pullback along $p$. Assuming that all the graphs in the pullback are connected, one obtains a $\mathbb{Z}_{\ell}$-tower above $Y$. Under the assumption that the Iwasawa $μ$-invariant of the tower above $X$ vanishes, we prove a formula relating the Iwasawa $λ$-invariant of the $\mathbb{Z}_{\ell}$-tower above $X$ to the Iwasawa $λ$-invariant of the pullback. This formula is analogous to Kida's formula in classical Iwasawa theory. We present an application to the study of structural properties of certain noncommutative pro-$\ell$ towers of graphs, based on an analogy with classical results of Cuoco on the growth of Iwasawa invariants in $\mathbb{Z}_\ell^2$-extensions of number fields. Our investigations are illustrated by explicit examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_04890 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | An analogue of Kida's formula in graph theory Ray, Anwesh Vallières, Daniel Number Theory Combinatorics 11R23, 05C25 Let $\ell$ be a rational prime and let $p:Y\rightarrow X$ be a Galois cover of finite graphs whose Galois group is a finite $\ell$-group. Consider a $\mathbb{Z}_{\ell}$-tower above $X$ and its pullback along $p$. Assuming that all the graphs in the pullback are connected, one obtains a $\mathbb{Z}_{\ell}$-tower above $Y$. Under the assumption that the Iwasawa $μ$-invariant of the tower above $X$ vanishes, we prove a formula relating the Iwasawa $λ$-invariant of the $\mathbb{Z}_{\ell}$-tower above $X$ to the Iwasawa $λ$-invariant of the pullback. This formula is analogous to Kida's formula in classical Iwasawa theory. We present an application to the study of structural properties of certain noncommutative pro-$\ell$ towers of graphs, based on an analogy with classical results of Cuoco on the growth of Iwasawa invariants in $\mathbb{Z}_\ell^2$-extensions of number fields. Our investigations are illustrated by explicit examples. |
| title | An analogue of Kida's formula in graph theory |
| topic | Number Theory Combinatorics 11R23, 05C25 |
| url | https://arxiv.org/abs/2209.04890 |