Let $Kgeq 2$ be a natural number and $a_i,b_iinmathbb{Z}$ for $i=1,ldots,K-1$. We use the large sieve to derive explicit upper bounds for the number of prime $k$-tuplets, i.e., for the number of primes $pleq x$ for which all $a_ip+b_i$ are also prime.
{"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}
The automaticity $A(x)$ of a set $mathcal{X}$ is the size of the smallest automaton that recognizes $mathcal{X}$ on all words of length $leq x$. We show that the automaticity of the set of primes is at least $xexpleft(-c(loglog x)^2logloglog xright)$, which is fairly close to the maximal automaticity.
{"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}
We study the birational geometry of the Kummer surfaces associated to the Jacobian varieties of genus two curves, with a particular focus on fields of characteristic two. In order to do so, we explicitly compute a projective embedding of the Jacobian of a general genus two curve and, from this, we construct its associated Kummer surface. This explicit construction produces a model for desingularised Kummer surfaces over any field of characteristic not two, and specialising these equations to characteristic two provides a model of a partial desingularisation. Adapting the classic description of the Picard lattice in terms of tropes, we also describe how to explicitly find completely desingularised models of Kummer surfaces whenever the $p$-rank is not zero. In the final section of this paper, we compute an example of a Kummer surface with everywhere good reduction over a quadratic number field, and draw connections between the models we computed and a criterion that determines when a Kummer surface has good reduction at two.
{"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}
We give a geometric interpretation of the Siegel operators for holomorphic differential forms on Siegel modular varieties. This involves extension of the differential forms over a toroidal compactification, and we show that the Siegel operator essentially describes the restriction and descent to the boundary Kuga variety via holomorphic Leray filtration. As a consequence, we obtain equivalence of various notions of "vanishing at boundary'' for holomorphic forms. We also study the case of orthogonal modular varieties.
{"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}
Jun-Lee-Sun posed the question of whether the cyclotomic Hecke field can be generated by a single critical $L$-value of a cyclotomic Hecke character over a totally real field. They provided an answer to this question in the case where the tame Hecke character is trivial. In this paper, we extend their work to address the case of non-trivial Hecke characters over solvable totally real number fields. Our approach builds upon the primary estimation obtained by Jun-Lee-Sun, supplemented with new inputs, including global class field theory, duality principles, the analytic behavior of partial Hecke $L$-functions, and the non-vanishing of twisted Gauss sums and Hyper Kloosterman sums.
{"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}
In the work of M. A. Papanikolas and N. Ramachandran [A Weil-Barsotti formula for Drinfeld modules, Journal of Number Theory 98, (2003), 407-431] the Weil-Barsotti formula for the function field case concerning $Ext_{tau}^1(E,C)$ where $E$ is a Drinfeld module and $C$ is the Carlitz module was proved. We generalize this formula to the case where $E$ is a strictly pure tm module $Phi$ with the zero nilpotent matrix $N_Phi.$ For such a tm module $Phi$ we explicitly compute its dual tm module ${Phi}^{vee}$ as well as its double dual ${Phi}^{{vee}{vee}}.$ This computation is done in a a subtle way by combination of the tm reduction algorithm developed by F. G{l}och, D.E. K{k e}dzierski, P. Kraso{'n} [ Algorithms for determination of tm module structures on some extension groups , arXiv:2408.08207] and the methods of the work of D.E. K{k e}dzierski and P. Kraso{'n} [On $Ext^1$ for Drinfeld modules, Journal of Number Theory 256 (2024) 97-135]. We also give a counterexample to the Weil-Barsotti formula if the nilpotent matrix $N_{Phi}$ is non-zero.
在 M. A. Papanikolas 和 N. Ramachandran 的工作[A Weil-Barsotti formulafor Drinfeld modules, Journal of Number Theory 98, (2003), 407-431]中，证明了关于$Ext_{tau}^1(E,C)$（其中$E$是德林菲尔德模块，$C$是卡利茨模块）的函数场情况的魏尔-巴索提公式。我们把这个公式推广到 $E$ 是严格纯粹的 tm 模块 $Phi$ 与零零势矩阵 $N_Phi 的情况。对于这样的 tm 模块 $Phi$ 我们明确地计算它的对偶 tm 模块 ${Phi}^{vee}$ 以及它的双重对偶 ${Phi}^{vee}{vee}}.这种计算是通过结合 F. G{l}och, D.E. K{k e}dzierski, P. Kraso{'n} 开发的 tm 还原算法以一种微妙的方式完成的。[一些扩展群上的tm 模块结构的确定算法，arXiv:2408.08207] 以及 D. E. K{k e}dzierski 和 P. Kraso{'n} 的工作方法[On $Ext^1$ for Drinfeld modules, Journal of Number Theory 256(2024) 97-135].如果无穷矩阵 $N_{Phi}$ 非零，我们还给出了 Weil-Barsotti 公式的一个反例。
{"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}
We propose a novel factorization algorithm that leverages the theory underlying the SQUFOF method, including reduced quadratic forms, infrastructural distance, and Gauss composition. We also present an analysis of our method, which has a computational complexity of $O left( exp left( frac{3}{sqrt{8}} sqrt{ln N ln ln N} right) right)$, making it more efficient than the classical SQUFOF and CFRAC algorithms. Additionally, our algorithm is polynomial-time, provided knowledge of a (not too large) multiple of the regulator of $mathbb{Q}(N)$.
{"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}
We construct integral models of Shimura varieties of abelian type with parahoric level structure over odd primes. These models are 'etale locally isomorphic to corresponding local models.
{"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}
Standard techniques for treating linear recurrences no longer apply for quadratic recurrences. It is not hard to determine asymptotics for a specific parametrized model over a wide domain of values (all $p neq 1/2$ here). The gap between theory and experimentation seems insurmountable, however, at a single outlier ($p = 1/2$).
{"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}
Let $E$ be an elliptic curve defined over $mathbb{Q}$ with good ordinary reduction at a prime $pgeq 5$, and let $F$ be an imaginary quadratic field. Under appropriate assumptions, we show that the Pontryagin dual of the fine Mordell-Weil group of $E$ over the $mathbb{Z}_p^2$-extension of $F$ is pseudo-null as a module over the Iwasawa algebra of the group $mathbb{Z}_p^2$.
{"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}
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