{"title":"模素数半群轨道的大小","authors":"Wade Hindes, Joseph H. Silverman","doi":"10.2140/pjm.2023.325.281","DOIUrl":null,"url":null,"abstract":"Let $V$ be a projective variety defined over a number field $K$, let $S$ be a polarized set of endomorphisms of $V$ all defined over $K$, and let $P\\in V(K)$. For each prime $\\mathfrak{p}$ of $K$, let $m_{\\mathfrak{p}}(S,P)$ denote the number of points in the orbit of $P\\bmod\\mathfrak{p}$ for the semigroup of maps generated by $S$. Under suitable hypotheses on $S$ and $P$, we prove an analytic estimate for $m_{\\mathfrak{p}}(S,P)$ and use it to show that the set of primes for which $m_{\\mathfrak{p}}(S,P)$ grows subexponentially as a function of $\\operatorname{\\mathsf{N}}_{K/\\mathbb{Q}}\\mathfrak{p}$ is a set of density zero. For $V=\\mathbb{P}^1$ we show that this holds for a generic set of maps $S$ provided that at least two of the maps in $S$ have degree at least four.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":"27 11","pages":"0"},"PeriodicalIF":0.7000,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"The size of semigroup orbits modulo primes\",\"authors\":\"Wade Hindes, Joseph H. Silverman\",\"doi\":\"10.2140/pjm.2023.325.281\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let $V$ be a projective variety defined over a number field $K$, let $S$ be a polarized set of endomorphisms of $V$ all defined over $K$, and let $P\\\\in V(K)$. For each prime $\\\\mathfrak{p}$ of $K$, let $m_{\\\\mathfrak{p}}(S,P)$ denote the number of points in the orbit of $P\\\\bmod\\\\mathfrak{p}$ for the semigroup of maps generated by $S$. Under suitable hypotheses on $S$ and $P$, we prove an analytic estimate for $m_{\\\\mathfrak{p}}(S,P)$ and use it to show that the set of primes for which $m_{\\\\mathfrak{p}}(S,P)$ grows subexponentially as a function of $\\\\operatorname{\\\\mathsf{N}}_{K/\\\\mathbb{Q}}\\\\mathfrak{p}$ is a set of density zero. For $V=\\\\mathbb{P}^1$ we show that this holds for a generic set of maps $S$ provided that at least two of the maps in $S$ have degree at least four.\",\"PeriodicalId\":54651,\"journal\":{\"name\":\"Pacific Journal of Mathematics\",\"volume\":\"27 11\",\"pages\":\"0\"},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2023-11-03\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Pacific Journal of Mathematics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.2140/pjm.2023.325.281\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Pacific Journal of Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.2140/pjm.2023.325.281","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
Let $V$ be a projective variety defined over a number field $K$, let $S$ be a polarized set of endomorphisms of $V$ all defined over $K$, and let $P\in V(K)$. For each prime $\mathfrak{p}$ of $K$, let $m_{\mathfrak{p}}(S,P)$ denote the number of points in the orbit of $P\bmod\mathfrak{p}$ for the semigroup of maps generated by $S$. Under suitable hypotheses on $S$ and $P$, we prove an analytic estimate for $m_{\mathfrak{p}}(S,P)$ and use it to show that the set of primes for which $m_{\mathfrak{p}}(S,P)$ grows subexponentially as a function of $\operatorname{\mathsf{N}}_{K/\mathbb{Q}}\mathfrak{p}$ is a set of density zero. For $V=\mathbb{P}^1$ we show that this holds for a generic set of maps $S$ provided that at least two of the maps in $S$ have degree at least four.
期刊介绍:
Founded in 1951, PJM has published mathematics research for more than 60 years. PJM is run by mathematicians from the Pacific Rim. PJM aims to publish high-quality articles in all branches of mathematics, at low cost to libraries and individuals. The Pacific Journal of Mathematics is incorporated as a 501(c)(3) California nonprofit.