Let $f(x_1,ldots,x_s)$ be a translation invariant indefinite quadratic form of integer coefficients with $sge 10$. Let $mathcal{A}subseteq mathcal{P}cap {1,2,ldots,X}$. Let $X$ be sufficiently large. Subject to a rank condition, we prove that there exist distinct primes $p_1,ldots,p_sin mathcal{A}$ such that $f(p_1,ldots,p_s)=0$ as soon as $|mathcal{A}|ge frac{X}{log X} (loglog X)^{-frac{1}{80}}.$
{"title":"Translation invariant quadratic forms and dense sets of primes","authors":"Lilu Zhao","doi":"10.1090/tran/8530","DOIUrl":"https://doi.org/10.1090/tran/8530","url":null,"abstract":"Let $f(x_1,ldots,x_s)$ be a translation invariant indefinite quadratic form of integer coefficients with $sge 10$. Let $mathcal{A}subseteq mathcal{P}cap {1,2,ldots,X}$. Let $X$ be sufficiently large. Subject to a rank condition, we prove that there exist distinct primes $p_1,ldots,p_sin mathcal{A}$ such that $f(p_1,ldots,p_s)=0$ as soon as $|mathcal{A}|ge frac{X}{log X} (loglog X)^{-frac{1}{80}}.$","PeriodicalId":312337,"journal":{"name":"arXiv: Number Theory","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114333919","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we determine a constant occurring in a local analogue of the Siegel-Weil formula, and describe the behavior of the formal degrees under the local theta correspondence for quaternionic dual pairs of almost equal rank over a non-Archimedean local field of characteristic $0$. As an application, we prove the formal degree conjecture of Hiraga-Ichino-Ikeda for the non-split inner forms of ${rm Sp}_4$ and ${rm GSp}_4$.
{"title":"FORMAL DEGREES AND LOCAL THETA CORRESPONDENCE: QUATERNIONIC CASE","authors":"Hirotala Kakuhama","doi":"10.14989/DOCTOR.K22968","DOIUrl":"https://doi.org/10.14989/DOCTOR.K22968","url":null,"abstract":"In this paper, we determine a constant occurring in a local analogue of the Siegel-Weil formula, and describe the behavior of the formal degrees under the local theta correspondence for quaternionic dual pairs of almost equal rank over a non-Archimedean local field of characteristic $0$. As an application, we prove the formal degree conjecture of Hiraga-Ichino-Ikeda for the non-split inner forms of ${rm Sp}_4$ and ${rm GSp}_4$.","PeriodicalId":312337,"journal":{"name":"arXiv: Number Theory","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134159891","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $K$ be an imaginary quadratic field, with associated quadratic character $alpha$. We construct an analytic $p$-adic $L$-function interpolating the twisted adjoint $L$-values $L(1, mathrm{ad}(f) otimes alpha)$ as $f$ varies in a Hida family; these special values are non-critical in the sense of Deligne. Our approach is based on Greenberg--Stevens' idea of $Lambda$-adic modular symbols, which considers cohomology with values in a space of $p$-adic measures.
{"title":"p-adic L-functions for non-critical adjoint L-values","authors":"P. Lee","doi":"10.7916/D8-RVN9-R814","DOIUrl":"https://doi.org/10.7916/D8-RVN9-R814","url":null,"abstract":"Let $K$ be an imaginary quadratic field, with associated quadratic character $alpha$. We construct an analytic $p$-adic $L$-function interpolating the twisted adjoint $L$-values $L(1, mathrm{ad}(f) otimes alpha)$ as $f$ varies in a Hida family; these special values are non-critical in the sense of Deligne. Our approach is based on Greenberg--Stevens' idea of $Lambda$-adic modular symbols, which considers cohomology with values in a space of $p$-adic measures.","PeriodicalId":312337,"journal":{"name":"arXiv: Number Theory","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114166114","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-03-01DOI: 10.13140/RG.2.2.19894.42568
Georges Gras
Let k be a number field and let p ≥ 2 be a prime number. We call K/k a (pro)-cyclic p-tower if Gal(K/k)~ Z/p^N Z or Z_p. We give an elementary proof of a stability theorem about generalized p-class groups in the tower (Main Theorem 3.1). This result, using generalizations of Chevalley's formula (Gras, J. Math. Soc. Japan 46(3) (1994), Proc. Math. Sci. 127(1) (2017)), finds again or generalizes results by Fukuda (1994), Li--Ouyang--Xu--Zhang (2020), Mizusawa--Yamamoto (2020) and many others proving particular cases from Iwasawa's theory, pro-p-group theory or specific methods.
{"title":"ON THE p-CLASS GROUP STABILITY ALONG CYCLIC p-TOWERS OF A NUMBER FIELD","authors":"Georges Gras","doi":"10.13140/RG.2.2.19894.42568","DOIUrl":"https://doi.org/10.13140/RG.2.2.19894.42568","url":null,"abstract":"Let k be a number field and let p ≥ 2 be a prime number. We call K/k a (pro)-cyclic p-tower if Gal(K/k)~ Z/p^N Z or Z_p. We give an elementary proof of a stability theorem about generalized p-class groups in the tower (Main Theorem 3.1). This result, using generalizations of Chevalley's formula (Gras, J. Math. Soc. Japan 46(3) (1994), Proc. Math. Sci. 127(1) (2017)), finds again or generalizes results by Fukuda (1994), Li--Ouyang--Xu--Zhang (2020), Mizusawa--Yamamoto (2020) and many others proving particular cases from Iwasawa's theory, pro-p-group theory or specific methods.","PeriodicalId":312337,"journal":{"name":"arXiv: Number Theory","volume":"50 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128787728","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-02-23DOI: 10.4310/cjm.2021.v9.n1.a1
Yifeng Liu
In this article, we develop an arithmetic analogue of Fourier--Jacobi period integrals for a pair of unitary groups of equal rank. We construct the so-called Fourier--Jacobi cycles, which are algebraic cycles on the product of unitary Shimura varieties and abelian varieties. We propose the arithmetic Gan--Gross--Prasad conjecture for these cycles, which is related to central derivatives of certain Rankin--Selberg $L$-functions, and develop a relative trace formula approach toward this conjecture. As a necessary ingredient, we propose the conjecture of the corresponding arithmetic fundamental lemma, and confirm it for unitary groups of rank at most two and for the minuscule case.
{"title":"Fourier–Jacobi cycles and arithmetic relative trace formula (with an appendix by Chao Li and Yihang Zhu)","authors":"Yifeng Liu","doi":"10.4310/cjm.2021.v9.n1.a1","DOIUrl":"https://doi.org/10.4310/cjm.2021.v9.n1.a1","url":null,"abstract":"In this article, we develop an arithmetic analogue of Fourier--Jacobi period integrals for a pair of unitary groups of equal rank. We construct the so-called Fourier--Jacobi cycles, which are algebraic cycles on the product of unitary Shimura varieties and abelian varieties. We propose the arithmetic Gan--Gross--Prasad conjecture for these cycles, which is related to central derivatives of certain Rankin--Selberg $L$-functions, and develop a relative trace formula approach toward this conjecture. As a necessary ingredient, we propose the conjecture of the corresponding arithmetic fundamental lemma, and confirm it for unitary groups of rank at most two and for the minuscule case.","PeriodicalId":312337,"journal":{"name":"arXiv: Number Theory","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133313481","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-02-12DOI: 10.21203/RS.3.RS-503616/V1
Ahmed Diab
We will make two algorithms that generate all the prime numbers up to a given limit, they are a development of sieve of Eratosthenes algorithm, we will use two formulas of mine to achieve this development, where all the multiples of the prime number 2 are pre-eliminated in the first formula, and all the multiples of the prime numbers 2 and 3 are pre-eliminated in the second formula. We will proof sieve of Sundaram's algorithm by using the first algorithm we will make, then we will make some improvement to make it an efficient prime generator algorithm. We will show the difference in performance between all the algorithms we will make and sieve of Eratosthenes algorithm in terms of speed and space usage.
{"title":"Development of sieve of Eratosthenes and sieve of Sundaram's proof","authors":"Ahmed Diab","doi":"10.21203/RS.3.RS-503616/V1","DOIUrl":"https://doi.org/10.21203/RS.3.RS-503616/V1","url":null,"abstract":"We will make two algorithms that generate all the prime numbers up to a given limit, they are a development of sieve of Eratosthenes algorithm, we will use two formulas of mine to achieve this development, where all the multiples of the prime number 2 are pre-eliminated in the first formula, and all the multiples of the prime numbers 2 and 3 are pre-eliminated in the second formula. \u0000We will proof sieve of Sundaram's algorithm by using the first algorithm we will make, then we will make some improvement to make it an efficient prime generator algorithm. \u0000We will show the difference in performance between all the algorithms we will make and sieve of Eratosthenes algorithm in terms of speed and space usage.","PeriodicalId":312337,"journal":{"name":"arXiv: Number Theory","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127794757","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We extend Faltings' finiteness criteria to determine the equivalence of two $ell$-adic, semisimple representations of the absolute Galois group of a global field, to the context of potential equivalence. We observe that potential equivalence of a representation $rho$ is determined by an equality of an $m$-power character $gmapsto Tr(rho(g^m))$ for some natural number $m$. We also discuss finiteness results for twist unramified representations.
{"title":"Finiteness theorems for potentially equivalent Galois representations: Extension of Faltings’ finiteness criteria","authors":"Plawan Das, C. Rajan","doi":"10.1090/proc/15856","DOIUrl":"https://doi.org/10.1090/proc/15856","url":null,"abstract":"We extend Faltings' finiteness criteria to determine the equivalence of two $ell$-adic, semisimple representations of the absolute Galois group of a global field, to the context of potential equivalence. We observe that potential equivalence of a representation $rho$ is determined by an equality of an $m$-power character $gmapsto Tr(rho(g^m))$ for some natural number $m$. \u0000We also discuss finiteness results for twist unramified representations.","PeriodicalId":312337,"journal":{"name":"arXiv: Number Theory","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124716804","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-16DOI: 10.31801/CFSUASMAS.831024
Irem Kucukoglu
The main purpose of this paper is to present various identities and computation formulas for certain classes of Apostol-type numbers and polynomials. The results of this paper contain not only the $lambda$-Apostol-Daehee numbers and polynomials, but also Simsek numbers and polynomials, the Stirling numbers of the first kind, the Daehee numbers, and the Chu-Vandermonde identity. Furthermore, we derive an infinite series representation for the $lambda$-Apostol-Daehee polynomials. By using functional equations containing the generating functions for the Cauchy numbers and the Riemann integrals of the generating functions for the $lambda$-Apostol-Daehee numbers and polynomials, we also derive some identities and formulas for these numbers and polynomials. Moreover, we give implementation of a computation formula for the $lambda$-Apostol-Daehee polynomials in Mathematica by Wolfram language. By this implementation, we also present some plots of these polynomials in order to investigate their behaviour some randomly selected special cases of its parameters. Finally, we conclude the paper with some comments and observations on our results.
{"title":"Implementation of computation formulas for certain classes of Apostol-type polynomials and some properties associated with these polynomials","authors":"Irem Kucukoglu","doi":"10.31801/CFSUASMAS.831024","DOIUrl":"https://doi.org/10.31801/CFSUASMAS.831024","url":null,"abstract":"The main purpose of this paper is to present various identities and computation formulas for certain classes of Apostol-type numbers and polynomials. The results of this paper contain not only the $lambda$-Apostol-Daehee numbers and polynomials, but also Simsek numbers and polynomials, the Stirling numbers of the first kind, the Daehee numbers, and the Chu-Vandermonde identity. Furthermore, we derive an infinite series representation for the $lambda$-Apostol-Daehee polynomials. By using functional equations containing the generating functions for the Cauchy numbers and the Riemann integrals of the generating functions for the $lambda$-Apostol-Daehee numbers and polynomials, we also derive some identities and formulas for these numbers and polynomials. Moreover, we give implementation of a computation formula for the $lambda$-Apostol-Daehee polynomials in Mathematica by Wolfram language. By this implementation, we also present some plots of these polynomials in order to investigate their behaviour some randomly selected special cases of its parameters. Finally, we conclude the paper with some comments and observations on our results.","PeriodicalId":312337,"journal":{"name":"arXiv: Number Theory","volume":"63 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116984793","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce a new $p$-adic analogue of the incomplete gamma function. We also introduce a closely related family of combinatorial sequences counting derangements and arrangements in certain wreath products.
{"title":"Derangements and the $p$-adic incomplete gamma function.","authors":"Andrew O'Desky, David Harry Richman","doi":"10.1090/tran/8716","DOIUrl":"https://doi.org/10.1090/tran/8716","url":null,"abstract":"We introduce a new $p$-adic analogue of the incomplete gamma function. We also introduce a closely related family of combinatorial sequences counting derangements and arrangements in certain wreath products.","PeriodicalId":312337,"journal":{"name":"arXiv: Number Theory","volume":"91 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124707736","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Suppose $f in K[x]$ is a polynomial. The absolute Galois group of $K$ acts on the preimage tree $mathrm{T}$ of $0$ under $f$. The resulting homomorphism $rho_f: mathrm{Gal}_K to mathrm{Aut} mathrm{T}$ is called the arboreal Galois representation. Odoni conjectured that for all Hilbertian fields $K$ there exists a polynomial $f$ for which $rho_f$ is surjective. We show that this conjecture is false.
假设f in K[x]$是一个多项式。K$的绝对伽罗瓦群作用于$f$下$0$的原象树$ mathm {T}$。由此产生的同态$rho_f: mathrm{Gal}_K $到mathrm{Aut} mathrm{T}$称为树状伽罗瓦表示。Odoni推测，对于所有Hilbertian域，存在一个多项式f，对于该多项式f是满射。我们证明这个猜想是错误的。
{"title":"Odoni’s conjecture an arboreal Galois representations is false","authors":"Philip Dittmann, Borys Kadets","doi":"10.1090/proc/15920","DOIUrl":"https://doi.org/10.1090/proc/15920","url":null,"abstract":"Suppose $f in K[x]$ is a polynomial. The absolute Galois group of $K$ acts on the preimage tree $mathrm{T}$ of $0$ under $f$. The resulting homomorphism $rho_f: mathrm{Gal}_K to mathrm{Aut} mathrm{T}$ is called the arboreal Galois representation. Odoni conjectured that for all Hilbertian fields $K$ there exists a polynomial $f$ for which $rho_f$ is surjective. We show that this conjecture is false.","PeriodicalId":312337,"journal":{"name":"arXiv: Number Theory","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115661583","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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