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Explicit bounds for prime K-tuplets 素数 K 元组的明确界限
arXiv - MATH - Number Theory Pub Date : 2024-09-06 DOI: arxiv-2409.04261
Thomas Dubbe
{"title":"Explicit bounds for prime K-tuplets","authors":"Thomas Dubbe","doi":"arxiv-2409.04261","DOIUrl":"https://doi.org/arxiv-2409.04261","url":null,"abstract":"Let $Kgeq 2$ be a natural number and $a_i,b_iinmathbb{Z}$ for\u0000$i=1,ldots,K-1$. We use the large sieve to derive explicit upper bounds for\u0000the number of prime $k$-tuplets, i.e., for the number of primes $pleq x$ for\u0000which all $a_ip+b_i$ are also prime.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203788","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}
引用次数: 0
The automaticity of the set of primes 素数集的自动性
arXiv - MATH - Number Theory Pub Date : 2024-09-06 DOI: arxiv-2409.04314
Thomas Dubbe
{"title":"The automaticity of the set of primes","authors":"Thomas Dubbe","doi":"arxiv-2409.04314","DOIUrl":"https://doi.org/arxiv-2409.04314","url":null,"abstract":"The automaticity $A(x)$ of a set $mathcal{X}$ is the size of the smallest\u0000automaton that recognizes $mathcal{X}$ on all words of length $leq x$. We\u0000show that the automaticity of the set of primes is at least\u0000$xexpleft(-c(loglog x)^2logloglog xright)$, which is fairly close to\u0000the maximal automaticity.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203784","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}
引用次数: 0
Explicit desingularisation of Kummer surfaces in characteristic two via specialisation 通过特殊化实现库默尔曲面在特征二中的去奇点化
arXiv - MATH - Number Theory Pub Date : 2024-09-06 DOI: arxiv-2409.04532
Alvaro Gonzalez-Hernandez
{"title":"Explicit desingularisation of Kummer surfaces in characteristic two via specialisation","authors":"Alvaro Gonzalez-Hernandez","doi":"arxiv-2409.04532","DOIUrl":"https://doi.org/arxiv-2409.04532","url":null,"abstract":"We study the birational geometry of the Kummer surfaces associated to the\u0000Jacobian varieties of genus two curves, with a particular focus on fields of\u0000characteristic two. In order to do so, we explicitly compute a projective\u0000embedding of the Jacobian of a general genus two curve and, from this, we\u0000construct its associated Kummer surface. This explicit construction produces a\u0000model for desingularised Kummer surfaces over any field of characteristic not\u0000two, and specialising these equations to characteristic two provides a model of\u0000a partial desingularisation. Adapting the classic description of the Picard\u0000lattice in terms of tropes, we also describe how to explicitly find completely\u0000desingularised models of Kummer surfaces whenever the $p$-rank is not zero. In\u0000the final section of this paper, we compute an example of a Kummer surface with\u0000everywhere good reduction over a quadratic number field, and draw connections\u0000between the models we computed and a criterion that determines when a Kummer\u0000surface has good reduction at two.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"65 4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203783","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}
引用次数: 0
Siegel operators for holomorphic differential forms 全微分形式的西格尔算子
arXiv - MATH - Number Theory Pub Date : 2024-09-06 DOI: arxiv-2409.04315
Shouhei Ma
{"title":"Siegel operators for holomorphic differential forms","authors":"Shouhei Ma","doi":"arxiv-2409.04315","DOIUrl":"https://doi.org/arxiv-2409.04315","url":null,"abstract":"We give a geometric interpretation of the Siegel operators for holomorphic\u0000differential forms on Siegel modular varieties. This involves extension of the\u0000differential forms over a toroidal compactification, and we show that the\u0000Siegel operator essentially describes the restriction and descent to the\u0000boundary Kuga variety via holomorphic Leray filtration. As a consequence, we\u0000obtain equivalence of various notions of \"vanishing at boundary'' for\u0000holomorphic forms. We also study the case of orthogonal modular varieties.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"58 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203789","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}
引用次数: 0
Cyclotomic fields are generated by cyclotomic Hecke {it L}-values of totally real fields, II 循环域由完全实域的循环 Hecke {it L} 值生成，II
arXiv - MATH - Number Theory Pub Date : 2024-09-06 DOI: arxiv-2409.04661
Jaesung kwon, Hae-Sang Sun
{"title":"Cyclotomic fields are generated by cyclotomic Hecke {it L}-values of totally real fields, II","authors":"Jaesung kwon, Hae-Sang Sun","doi":"arxiv-2409.04661","DOIUrl":"https://doi.org/arxiv-2409.04661","url":null,"abstract":"Jun-Lee-Sun posed the question of whether the cyclotomic Hecke field can be\u0000generated by a single critical $L$-value of a cyclotomic Hecke character over a\u0000totally real field. They provided an answer to this question in the case where\u0000the tame Hecke character is trivial. In this paper, we extend their work to\u0000address the case of non-trivial Hecke characters over solvable totally real\u0000number fields. Our approach builds upon the primary estimation obtained by\u0000Jun-Lee-Sun, supplemented with new inputs, including global class field theory,\u0000duality principles, the analytic behavior of partial Hecke $L$-functions, and\u0000the non-vanishing of twisted Gauss sums and Hyper Kloosterman sums.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"60 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203779","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}
引用次数: 0
Weil-Barsotti formula for $mathbf{T}$-modules $mathbf{T}$ 模块的 Weil-Barsotti 公式
arXiv - MATH - Number Theory Pub Date : 2024-09-06 DOI: arxiv-2409.04029
Dawid E. Kędzierski, Piotr Krasoń
{"title":"Weil-Barsotti formula for $mathbf{T}$-modules","authors":"Dawid E. Kędzierski, Piotr Krasoń","doi":"arxiv-2409.04029","DOIUrl":"https://doi.org/arxiv-2409.04029","url":null,"abstract":"In the work of M. A. Papanikolas and N. Ramachandran [A Weil-Barsotti formula\u0000for Drinfeld modules, Journal of Number Theory 98, (2003), 407-431] the\u0000Weil-Barsotti formula for the function field case concerning\u0000$Ext_{tau}^1(E,C)$ where $E$ is a Drinfeld module and $C$ is the Carlitz\u0000module was proved. We generalize this formula to the case where $E$ is a\u0000strictly pure tm module $Phi$ with the zero nilpotent matrix $N_Phi.$ For\u0000such a tm module $Phi$ we explicitly compute its dual tm module\u0000${Phi}^{vee}$ as well as its double dual ${Phi}^{{vee}{vee}}.$ This\u0000computation is done in a a subtle way by combination of the tm reduction\u0000algorithm developed by F. G{l}och, D.E. K{k e}dzierski, P. Kraso{'n} [\u0000Algorithms for determination of tm module structures on some extension groups\u0000, arXiv:2408.08207] and the methods of the work of D.E. K{k e}dzierski and P.\u0000Kraso{'n} [On $Ext^1$ for Drinfeld modules, Journal of Number Theory 256\u0000(2024) 97-135]. We also give a counterexample to the Weil-Barsotti formula if\u0000the nilpotent matrix $N_{Phi}$ is non-zero.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203786","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}
引用次数: 0
Integer Factorization via Continued Fractions and Quadratic Forms 通过连续分数和二次型进行整数因式分解
arXiv - MATH - Number Theory Pub Date : 2024-09-05 DOI: arxiv-2409.03486
Nadir Murru, Giulia Salvatori
{"title":"Integer Factorization via Continued Fractions and Quadratic Forms","authors":"Nadir Murru, Giulia Salvatori","doi":"arxiv-2409.03486","DOIUrl":"https://doi.org/arxiv-2409.03486","url":null,"abstract":"We propose a novel factorization algorithm that leverages the theory\u0000underlying the SQUFOF method, including reduced quadratic forms,\u0000infrastructural distance, and Gauss composition. We also present an analysis of\u0000our method, which has a computational complexity of $O left( exp left(\u0000frac{3}{sqrt{8}} sqrt{ln N ln ln N} right) right)$, making it more\u0000efficient than the classical SQUFOF and CFRAC algorithms. Additionally, our\u0000algorithm is polynomial-time, provided knowledge of a (not too large) multiple\u0000of the regulator of $mathbb{Q}(N)$.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203791","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}
引用次数: 0
Integral models of Shimura varieties with parahoric level structure, II 具有准水平结构的志村变种积分模型，II
arXiv - MATH - Number Theory Pub Date : 2024-09-05 DOI: arxiv-2409.03689
Mark Kisin, Georgios Pappas, Rong Zhou
{"title":"Integral models of Shimura varieties with parahoric level structure, II","authors":"Mark Kisin, Georgios Pappas, Rong Zhou","doi":"arxiv-2409.03689","DOIUrl":"https://doi.org/arxiv-2409.03689","url":null,"abstract":"We construct integral models of Shimura varieties of abelian type with\u0000parahoric level structure over odd primes. These models are 'etale locally\u0000isomorphic to corresponding local models.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203790","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}
引用次数: 0
A Deceptively Simple Quadratic Recurrence 看似简单的二次递归
arXiv - MATH - Number Theory Pub Date : 2024-09-05 DOI: arxiv-2409.03510
Steven Finch
{"title":"A Deceptively Simple Quadratic Recurrence","authors":"Steven Finch","doi":"arxiv-2409.03510","DOIUrl":"https://doi.org/arxiv-2409.03510","url":null,"abstract":"Standard techniques for treating linear recurrences no longer apply for\u0000quadratic recurrences. It is not hard to determine asymptotics for a specific\u0000parametrized model over a wide domain of values (all $p neq 1/2$ here). The\u0000gap between theory and experimentation seems insurmountable, however, at a\u0000single outlier ($p = 1/2$).","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"40 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203793","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}
引用次数: 0
On pseudo-nullity of fine Mordell-Weil group 论精细莫德尔-魏尔群的伪无效性
arXiv - MATH - Number Theory Pub Date : 2024-09-05 DOI: arxiv-2409.03546
Meng Fai Lim, Chao Qin, Jun Wang
{"title":"On pseudo-nullity of fine Mordell-Weil group","authors":"Meng Fai Lim, Chao Qin, Jun Wang","doi":"arxiv-2409.03546","DOIUrl":"https://doi.org/arxiv-2409.03546","url":null,"abstract":"Let $E$ be an elliptic curve defined over $mathbb{Q}$ with good ordinary\u0000reduction at a prime $pgeq 5$, and let $F$ be an imaginary quadratic field.\u0000Under appropriate assumptions, we show that the Pontryagin dual of the fine\u0000Mordell-Weil group of $E$ over the $mathbb{Z}_p^2$-extension of $F$ is\u0000pseudo-null as a module over the Iwasawa algebra of the group $mathbb{Z}_p^2$.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203792","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}
引用次数: 0
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