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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| Accès en ligne: | https://arxiv.org/abs/2603.17286 |
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| _version_ | 1866915876146511872 |
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| author | Pereda, Aaron Blas Sulca, Diego |
| author_facet | Pereda, Aaron Blas Sulca, Diego |
| contents | Let $\mathfrak{o}$ be a compact discrete valuation ring and $n\geq 2$. We introduce a method to study the cotype zeta function of subalgebras of $\mathfrak{o}^n$. This multivariable series encodes the number of finite-index subalgebras $Λ$ of the $\mathfrak{o}$-algebra $\mathfrak{o}^n$ of a given elementary divisor type. We express this zeta function as a finite sum of $\mathfrak{o}$-adic integrals and compute these integrals in many cases.
As a first application, we recover known results in a natural way from our approach. For instance, we obtain a lower bound for the abscissa of convergence of the subalgebra zeta function of $\mathfrak{o}^n$ by exhibiting an explicit pole. We also determine the number of irreducible subrings of $\mathfrak{o}^n$ of small index.
As a second application, we give an explicit formula for the cotype zeta function of subalgebras of $\mathfrak{o}^4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_17286 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Counting subalgebras of $\mathfrak{o}^n$ Pereda, Aaron Blas Sulca, Diego Number Theory 11M41, 11S40 Let $\mathfrak{o}$ be a compact discrete valuation ring and $n\geq 2$. We introduce a method to study the cotype zeta function of subalgebras of $\mathfrak{o}^n$. This multivariable series encodes the number of finite-index subalgebras $Λ$ of the $\mathfrak{o}$-algebra $\mathfrak{o}^n$ of a given elementary divisor type. We express this zeta function as a finite sum of $\mathfrak{o}$-adic integrals and compute these integrals in many cases. As a first application, we recover known results in a natural way from our approach. For instance, we obtain a lower bound for the abscissa of convergence of the subalgebra zeta function of $\mathfrak{o}^n$ by exhibiting an explicit pole. We also determine the number of irreducible subrings of $\mathfrak{o}^n$ of small index. As a second application, we give an explicit formula for the cotype zeta function of subalgebras of $\mathfrak{o}^4$. |
| title | Counting subalgebras of $\mathfrak{o}^n$ |
| topic | Number Theory 11M41, 11S40 |
| url | https://arxiv.org/abs/2603.17286 |