Pub Date : 2023-08-04DOI: 10.1017/s0305004123000397
{"title":"PSP volume 175 issue 2 Cover and Back matter","authors":"","doi":"10.1017/s0305004123000397","DOIUrl":"https://doi.org/10.1017/s0305004123000397","url":null,"abstract":"","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89919205","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-04DOI: 10.1017/s0305004123000385
{"title":"PSP volume 175 issue 2 Cover and Front matter","authors":"","doi":"10.1017/s0305004123000385","DOIUrl":"https://doi.org/10.1017/s0305004123000385","url":null,"abstract":"","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91373969","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-07-10DOI: 10.1017/s030500412300035x
THOMAS MICHAEL KELLER, ALEXANDER MORETÓ
We prove that there exists a universal constant D such that if p is a prime divisor of the index of the Fitting subgroup of a finite group G, then the number of conjugacy classes of G is at least $Dp/log_2p$ . We conjecture that we can take $D=1$ and prove that for solvable groups, we can take $D=1/3$ .
{"title":"Prime divisors and the number of conjugacy classes of finite groups","authors":"THOMAS MICHAEL KELLER, ALEXANDER MORETÓ","doi":"10.1017/s030500412300035x","DOIUrl":"https://doi.org/10.1017/s030500412300035x","url":null,"abstract":"We prove that there exists a universal constant <jats:italic>D</jats:italic> such that if <jats:italic>p</jats:italic> is a prime divisor of the index of the Fitting subgroup of a finite group <jats:italic>G</jats:italic>, then the number of conjugacy classes of <jats:italic>G</jats:italic> is at least <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S030500412300035X_inline1.png\" /> <jats:tex-math> $Dp/log_2p$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. We conjecture that we can take <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S030500412300035X_inline2.png\" /> <jats:tex-math> $D=1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and prove that for solvable groups, we can take <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S030500412300035X_inline3.png\" /> <jats:tex-math> $D=1/3$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138532107","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-13DOI: 10.1017/s0305004123000312
{"title":"PSP volume 175 issue 1 Cover and Back matter","authors":"","doi":"10.1017/s0305004123000312","DOIUrl":"https://doi.org/10.1017/s0305004123000312","url":null,"abstract":"","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75771505","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-13DOI: 10.1017/s0305004123000300
{"title":"PSP volume 175 issue 1 Cover and Front matter","authors":"","doi":"10.1017/s0305004123000300","DOIUrl":"https://doi.org/10.1017/s0305004123000300","url":null,"abstract":"","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78896243","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-05-23DOI: 10.1017/S0305004123000294
Alexander P. Mangerel
Abstract Let f be a non-CM Hecke eigencusp form of level 1 and fixed weight, and let ${lambda_f(n)}_n$ be its sequence of normalised Fourier coefficients. We show that if $K/ mathbb{Q}$ is any number field, and $mathcal{N}_K$ denotes the collection of integers representable as norms of integral ideals of K, then a positive proportion of the positive integers $n in mathcal{N}_K$ yield a sign change for the sequence ${lambda_f(n)}_{n in mathcal{N}_K}$ . More precisely, for a positive proportion of $n in mathcal{N}_K cap [1,X]$ we have $lambda_f(n)lambda_f(n') < 0$ , where n ′ is the first element of $mathcal{N}_K$ greater than n for which $lambda_f(n') neq 0$ . For example, for $K = mathbb{Q}(i)$ and $mathcal{N}_K = {m^2+n^2 ;:; m,n in mathbb{Z}}$ the set of sums of two squares, we obtain $gg_f X/sqrt{log X}$ such sign changes, which is best possible (up to the implicit constant) and improves upon work of Banerjee and Pandey. Our proof relies on recent work of Matomäki and Radziwiłł on sparsely-supported multiplicative functions, together with some technical refinements of their results due to the author. In a related vein, we also consider the question of sign changes along shifted sums of two squares, for which multiplicative techniques do not directly apply. Using estimates for shifted convolution sums among other techniques, we establish that for any fixed $a neq 0$ there are $gg_{f,varepsilon} X^{1/2-varepsilon}$ sign changes for $lambda_f$ along the sequence of integers of the form $a + m^2 + n^2 leq X$ .
设f为一级定权的非cm Hecke特征形式，设${lambda_f(n)}_n$为其归一化傅里叶系数序列。我们证明，如果$K/ mathbb{Q}$是任意数字域，并且$mathcal{N}_K$表示可表示为K的积分理想的范数的整数集合，则正整数$n in mathcal{N}_K$的正比例产生序列${lambda_f(n)}_{n in mathcal{N}_K}$的符号变化。更准确地说，对于$n in mathcal{N}_K cap [1,X]$的正比例，我们有$lambda_f(n)lambda_f(n') < 0$，其中n '是$mathcal{N}_K$的第一个大于n的元素，对于$lambda_f(n') neq 0$。例如，对于$K = mathbb{Q}(i)$和$mathcal{N}_K = {m^2+n^2 ;:; m,n in mathbb{Z}}$两个平方和的集合，我们得到$gg_f X/sqrt{log X}$这样的符号变化，这是最好的可能(直到隐式常数)，并改进了Banerjee和Pandey的工作。我们的证明依赖于Matomäki和Radziwiłł最近关于稀疏支持的乘法函数的工作，以及作者对其结果的一些技术改进。在一个相关的静脉，我们也考虑沿移位的两个平方和的符号变化的问题，其中乘法技术不直接适用。在其他技术中使用移位卷积和的估计，我们建立了对于任何固定的$a neq 0$, $lambda_f$沿着形式为$a + m^2 + n^2 leq X$的整数序列有$gg_{f,varepsilon} X^{1/2-varepsilon}$符号变化。
{"title":"Sign changes of fourier coefficients of holomorphic cusp forms at norm form arguments","authors":"Alexander P. Mangerel","doi":"10.1017/S0305004123000294","DOIUrl":"https://doi.org/10.1017/S0305004123000294","url":null,"abstract":"Abstract Let f be a non-CM Hecke eigencusp form of level 1 and fixed weight, and let \u0000${lambda_f(n)}_n$\u0000 be its sequence of normalised Fourier coefficients. We show that if \u0000$K/ mathbb{Q}$\u0000 is any number field, and \u0000$mathcal{N}_K$\u0000 denotes the collection of integers representable as norms of integral ideals of K, then a positive proportion of the positive integers \u0000$n in mathcal{N}_K$\u0000 yield a sign change for the sequence \u0000${lambda_f(n)}_{n in mathcal{N}_K}$\u0000 . More precisely, for a positive proportion of \u0000$n in mathcal{N}_K cap [1,X]$\u0000 we have \u0000$lambda_f(n)lambda_f(n') < 0$\u0000 , where n ′ is the first element of \u0000$mathcal{N}_K$\u0000 greater than n for which \u0000$lambda_f(n') neq 0$\u0000 . For example, for \u0000$K = mathbb{Q}(i)$\u0000 and \u0000$mathcal{N}_K = {m^2+n^2 ;:; m,n in mathbb{Z}}$\u0000 the set of sums of two squares, we obtain \u0000$gg_f X/sqrt{log X}$\u0000 such sign changes, which is best possible (up to the implicit constant) and improves upon work of Banerjee and Pandey. Our proof relies on recent work of Matomäki and Radziwiłł on sparsely-supported multiplicative functions, together with some technical refinements of their results due to the author. In a related vein, we also consider the question of sign changes along shifted sums of two squares, for which multiplicative techniques do not directly apply. Using estimates for shifted convolution sums among other techniques, we establish that for any fixed \u0000$a neq 0$\u0000 there are \u0000$gg_{f,varepsilon} X^{1/2-varepsilon}$\u0000 sign changes for \u0000$lambda_f$\u0000 along the sequence of integers of the form \u0000$a + m^2 + n^2 leq X$\u0000 .","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81657536","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-05-08DOI: 10.1017/S0305004123000282
N. Yagita
Abstract Let BG be the classifying space of an algebraic group G over the field ${mathbb C}$ of complex numbers. There are smooth projective approximations X of $BGtimes {mathbb P}^{infty}$ , by Ekedahl. We compute a new stable birational invariant of X defined by the difference of two coniveau filtrations of X, by Benoist and Ottem. Hence we give many examples such that two coniveau filtrations are different.
{"title":"Coniveau filtrations and Milnor operation \u0000$Q_n$","authors":"N. Yagita","doi":"10.1017/S0305004123000282","DOIUrl":"https://doi.org/10.1017/S0305004123000282","url":null,"abstract":"Abstract Let BG be the classifying space of an algebraic group G over the field \u0000${mathbb C}$\u0000 of complex numbers. There are smooth projective approximations X of \u0000$BGtimes {mathbb P}^{infty}$\u0000 , by Ekedahl. We compute a new stable birational invariant of X defined by the difference of two coniveau filtrations of X, by Benoist and Ottem. Hence we give many examples such that two coniveau filtrations are different.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76210168","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-04-24DOI: 10.1017/s0305004123000257
Olli Järviniemi
Abstract Let $d ge 3$ be an integer and let $P in mathbb{Z}[x]$ be a polynomial of degree d whose Galois group is $S_d$ . Let $(a_n)$ be a non-degenerate linearly recursive sequence of integers which has P as its characteristic polynomial. We prove, under the generalised Riemann hypothesis, that the lower density of the set of primes which divide at least one non-zero element of the sequence $(a_n)$ is positive.
摘要设$d ge 3$是一个整数，设$P in mathbb{Z}[x]$是一个d次多项式，其伽罗瓦群为$S_d$。设$(a_n)$是一个非退化线性递归整数序列，其特征多项式为P。在广义黎曼假设下，证明了能除数列$(a_n)$中至少一个非零元素的素数集合的下密度是正的。
{"title":"Positive lower density for prime divisors of generic linear recurrences","authors":"Olli Järviniemi","doi":"10.1017/s0305004123000257","DOIUrl":"https://doi.org/10.1017/s0305004123000257","url":null,"abstract":"Abstract Let $d ge 3$ be an integer and let $P in mathbb{Z}[x]$ be a polynomial of degree d whose Galois group is $S_d$ . Let $(a_n)$ be a non-degenerate linearly recursive sequence of integers which has P as its characteristic polynomial. We prove, under the generalised Riemann hypothesis, that the lower density of the set of primes which divide at least one non-zero element of the sequence $(a_n)$ is positive.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135223521","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-04-13DOI: 10.1017/s030500412300018x
{"title":"PSP volume 174 issue 3 Cover and Front matter","authors":"","doi":"10.1017/s030500412300018x","DOIUrl":"https://doi.org/10.1017/s030500412300018x","url":null,"abstract":"","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87113622","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-04-13DOI: 10.1017/s0305004123000191
{"title":"PSP volume 174 issue 3 Cover and Back matter","authors":"","doi":"10.1017/s0305004123000191","DOIUrl":"https://doi.org/10.1017/s0305004123000191","url":null,"abstract":"","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86465633","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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