Higher a-numbers in $\mathbf{Z}_p$-towers via Counting Lattice Points
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arXiv
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| Autori principali: | , , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911521275117568 |
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| author | Booher, Jeremy Hsieh, Jack Rivera, Rakesh Tran, Vincent Upton, James Wu, Carol |
| author_facet | Booher, Jeremy Hsieh, Jack Rivera, Rakesh Tran, Vincent Upton, James Wu, Carol |
| contents | Booher, Cais, Kramer-Miller and Upton study a class of $\mathbf{Z}_p$-tower of curves in characteristic $p$ with ramification controlled by an integer $d$. In the special case that $d$ divides $p-1$, they prove a formula for the higher $a$-numbers of these curves involving the number of lattice points in a complicated region of the plane. Booher and Cais had previously conjectured that for $n$ sufficiently large the higher $a$-numbers of the $n$th curve are given by formulae of the form $α(n) p^{2n} + β(n) p^n + λ_r(n) n + ν(n) $, where $α,β,ν,λ_r$ are periodic functions of $n$. This is an example of a new kind of Iwasawa theory. We establish this conjecture by carefully studying these lattice points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_13969 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Higher a-numbers in $\mathbf{Z}_p$-towers via Counting Lattice Points Booher, Jeremy Hsieh, Jack Rivera, Rakesh Tran, Vincent Upton, James Wu, Carol Number Theory Algebraic Geometry Booher, Cais, Kramer-Miller and Upton study a class of $\mathbf{Z}_p$-tower of curves in characteristic $p$ with ramification controlled by an integer $d$. In the special case that $d$ divides $p-1$, they prove a formula for the higher $a$-numbers of these curves involving the number of lattice points in a complicated region of the plane. Booher and Cais had previously conjectured that for $n$ sufficiently large the higher $a$-numbers of the $n$th curve are given by formulae of the form $α(n) p^{2n} + β(n) p^n + λ_r(n) n + ν(n) $, where $α,β,ν,λ_r$ are periodic functions of $n$. This is an example of a new kind of Iwasawa theory. We establish this conjecture by carefully studying these lattice points. |
| title | Higher a-numbers in $\mathbf{Z}_p$-towers via Counting Lattice Points |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2407.13969 |