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On the structure of the Bloch--Kato Selmer groups of modular forms over anticyclotomic $mathbf{Z}_p$-towers 论反双环$mathbf{Z}_p$塔上模态形式的布洛赫--加藤塞尔默群的结构
arXiv - MATH - Number Theory Pub Date : 2024-09-18 DOI: arxiv-2409.11966
Antonio Lei, Luca Mastella, Luochen Zhao
{"title":"On the structure of the Bloch--Kato Selmer groups of modular forms over anticyclotomic $mathbf{Z}_p$-towers","authors":"Antonio Lei, Luca Mastella, Luochen Zhao","doi":"arxiv-2409.11966","DOIUrl":"https://doi.org/arxiv-2409.11966","url":null,"abstract":"Let $p$ be an odd prime number and let $K$ be an imaginary quadratic field in\u0000which $p$ is split. Let $f$ be a modular form with good reduction at $p$. We\u0000study the variation of the Bloch--Kato Selmer groups and the\u0000Bloch--Kato--Shafarevich--Tate groups of $f$ over the anticyclotomic\u0000$mathbf{Z}_p$-extension $K_infty$ of $K$. In particular, we show that under\u0000the generalized Heegner hypothesis, if the $p$-localization of the generalized\u0000Heegner cycle attached to $f$ is primitive and certain local conditions hold,\u0000then the Pontryagin dual of the Selmer group of $f$ over $K_infty$ is free\u0000over the Iwasawa algebra. Consequently, the Bloch--Kato--Shafarevich--Tate\u0000groups of $f$ vanish. This generalizes earlier works of Matar and\u0000Matar--Nekov'av{r} on elliptic curves. Furthermore, our proof applies\u0000uniformly to the ordinary and non-ordinary settings.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259845","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
Diophantine stability and second order terms Diophantine 稳定性和二阶项
arXiv - MATH - Number Theory Pub Date : 2024-09-18 DOI: arxiv-2409.12144
Carlo Pagano, Efthymios Sofos
{"title":"Diophantine stability and second order terms","authors":"Carlo Pagano, Efthymios Sofos","doi":"arxiv-2409.12144","DOIUrl":"https://doi.org/arxiv-2409.12144","url":null,"abstract":"We establish a Galois-theoretic trichotomy governing Diophantine stability\u0000for genus $0$ curves. We use it to prove that the curve associated to the\u0000Hilbert symbol is Diophantine stable with probability $1$. Our asymptotic\u0000formula for the second order term exhibits strong bias towards instability.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259844","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
Systems of Hecke eigenvalues on subschemes of Shimura varieties 志村变子图上的赫克特征值系统
arXiv - MATH - Number Theory Pub Date : 2024-09-18 DOI: arxiv-2409.11720
Stefan Reppen
{"title":"Systems of Hecke eigenvalues on subschemes of Shimura varieties","authors":"Stefan Reppen","doi":"arxiv-2409.11720","DOIUrl":"https://doi.org/arxiv-2409.11720","url":null,"abstract":"We show that the systems of Hecke eigenvalues that appear in the coherent\u0000cohomology with coefficients in automorphic line bundles of any mod $p$ abelian\u0000type compact Shimura variety at hyperspecial level are the same as those\u0000appearing in any Hecke-equivariant closed subscheme. We also prove analogous\u0000results for noncompact Shimura varieties or nonclosed subschemes, such as\u0000Ekedahl-Oort strata, length strata and central leaves.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259847","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
Endomorphism Rings of Supersingular Elliptic Curves and Quadratic Forms 超奇异椭圆曲线和二次型的同构环
arXiv - MATH - Number Theory Pub Date : 2024-09-17 DOI: arxiv-2409.11025
Guanju Xiao, Zijian Zhou, Longjiang Qu
{"title":"Endomorphism Rings of Supersingular Elliptic Curves and Quadratic Forms","authors":"Guanju Xiao, Zijian Zhou, Longjiang Qu","doi":"arxiv-2409.11025","DOIUrl":"https://doi.org/arxiv-2409.11025","url":null,"abstract":"Given a supersingular elliptic curve, the supersingular endomorphism ring\u0000problem is to compute all of its endomorphisms. We use the correspondence\u0000between maximal orders in quaternion algebra $B_{p,infty}$ and positive\u0000ternary quadratic forms with discriminant $p$ to solve the supersingular\u0000endomorphism ring problem. Let $c<3p/16$ be a prime or $c=1$. Let $E$ be a\u0000$mathbb{Z}[sqrt{-cp}]$-oriented supersingular elliptic curve defined over\u0000$mathbb{F}_{p^2}$. There exists a subgroup $G$ of order $c$, and\u0000$text{End}(E,G)$ is isomorphic to an Eichler order in $B_{p,infty}$ of level\u0000$c$. If the endomorphism ring $text{End}(E,G)$ is known, then we can compute\u0000$text{End}(E)$ by solving two square roots in $mathbb{F}_c$. In particular,\u0000let $D<p$ be a prime. If an imaginary quadratic order with discriminant $-D$ or\u0000$-4D$ can be embedded into $text{End}(E)$, then we can compute $text{End}(E)$\u0000by solving one square root in $mathbb{F}_D$ and two square roots in\u0000$mathbb{F}_c$. As we know, isogenies between supersingular elliptic curves can be translated\u0000to kernel ideals of endomorphism rings. We study the action of these kernel\u0000ideals and express right orders of them by ternary quadratic forms.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259903","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 the number of prime factors with a given multiplicity over h-free and h-full numbers 关于在无h和满h数中具有给定倍数的质因数个数
arXiv - MATH - Number Theory Pub Date : 2024-09-17 DOI: arxiv-2409.11275
Sourabhashis Das, Wentang Kuo, Yu-Ru Liu
{"title":"On the number of prime factors with a given multiplicity over h-free and h-full numbers","authors":"Sourabhashis Das, Wentang Kuo, Yu-Ru Liu","doi":"arxiv-2409.11275","DOIUrl":"https://doi.org/arxiv-2409.11275","url":null,"abstract":"Let $k$ and $n$ be natural numbers. Let $omega_k(n)$ denote the number of\u0000distinct prime factors of $n$ with multiplicity $k$ as studied by Elma and the\u0000third author. We obtain asymptotic estimates for the first and the second\u0000moments of $omega_k(n)$ when restricted to the set of $h$-free and $h$-full\u0000numbers. We prove that $omega_1(n)$ has normal order $log log n$ over\u0000$h$-free numbers, $omega_h(n)$ has normal order $log log n$ over $h$-full\u0000numbers, and both of them satisfy the ErdH{o}s-Kac Theorem. Finally, we prove\u0000that the functions $omega_k(n)$ with $1 < k < h$ do not have normal order over\u0000$h$-free numbers and $omega_k(n)$ with $k > h$ do not have normal order over\u0000$h$-full numbers.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142269310","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
Fitting Ideals of Projective Limits of Modules over Non-Noetherian Iwasawa Algebras 非etherian岩泽代数上模块的投影极限的拟合顶点
arXiv - MATH - Number Theory Pub Date : 2024-09-17 DOI: arxiv-2409.11562
Cristian D. Popescu, Wei Yin
{"title":"Fitting Ideals of Projective Limits of Modules over Non-Noetherian Iwasawa Algebras","authors":"Cristian D. Popescu, Wei Yin","doi":"arxiv-2409.11562","DOIUrl":"https://doi.org/arxiv-2409.11562","url":null,"abstract":"Greither and Kurihara proved a theorem about the commutativity of projective\u0000limits and Fitting ideals for modules over the classical equivariant Iwasawa\u0000algebra $Lambda_G=mathbb{Z}_p[[T]][G]$, where $G$ is a finite, abelian group\u0000and $Bbb Z_p$ is the ring of $p$--adic integers, for some prime $p$. In this\u0000paper, we generalize their result first to the Noetherian Iwasawa algebra\u0000$mathbb{Z}_p[[T_1, T_2, cdots, T_n]][G]$ and, most importantly, to the\u0000non-Noetherian algebra $mathbb{Z}_p[[T_1, T_2, cdots, T_n, cdots]][G]$ of\u0000countably many generators. The latter generalization is motivated by the recent\u0000work of Bley-Popescu on the geometric Equivariant Iwasawa Conjecture for\u0000function fields, where the Iwasawa algebra is not Noetherian, of the type\u0000described above. Applications of these results to the emerging field of\u0000non-Noetherian Iwasawa Theory will be given in an upcoming paper.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"39 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259846","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
Black Hole Zeckendorf Games 黑洞泽肯多夫游戏
arXiv - MATH - Number Theory Pub Date : 2024-09-17 DOI: arxiv-2409.10981
Caroline Cashman, Steven J. Miller, Jenna Shuffleton, Daeyoung Son
{"title":"Black Hole Zeckendorf Games","authors":"Caroline Cashman, Steven J. Miller, Jenna Shuffleton, Daeyoung Son","doi":"arxiv-2409.10981","DOIUrl":"https://doi.org/arxiv-2409.10981","url":null,"abstract":"Zeckendorf proved a remarkable fact that every positive integer can be\u0000written as a decomposition of non-adjacent Fibonacci numbers. Baird-Smith,\u0000Epstein, Flint, and Miller converted the process of decomposing a positive\u0000integer into its Zeckendorf decomposition into a game, using the moves of $F_i\u0000+ F_{i-1} = F_{i+1}$ and $2F_i = F_{i+1} + F_{i-2}$, where $F_i$ is the\u0000$i$thFibonacci number. Players take turns applying these moves, beginning with\u0000$n$ pieces in the $F_1$ column. They showed that for $n neq 2$, Player 2 has a\u0000winning strategy, though the proof is non-constructive, and a constructive\u0000solution is unknown. We expand on this by investigating \"black hole'' variants of this game. The\u0000Black Hole Zeckendorf game on $F_m$ is played with any $n$ but solely in\u0000columns $F_i$ for $i < m$. Gameplay is similar to the original Zeckendorf game,\u0000except any piece that would be placed on $F_i$ for $i geq m$ is locked out in\u0000a ``black hole'' and removed from play. With these constraints, we analyze the\u0000games with black holes on $F_3$ and $F_4$ and construct a solution for specific\u0000configurations, using a parity-stealing based non-constructive proof to lead to\u0000a constructive one. We also examine a pre-game in which players take turns\u0000placing down $n$ pieces in the outermost columns before the decomposition\u0000phase, and find constructive solutions for any $n$.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259904","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 zeros of quadratic polynomials avoiding sublattices 避开子网格的二次多项式积分零点
arXiv - MATH - Number Theory Pub Date : 2024-09-17 DOI: arxiv-2409.10867
Lenny Fukshansky, Sehun Jeong
{"title":"Integral zeros of quadratic polynomials avoiding sublattices","authors":"Lenny Fukshansky, Sehun Jeong","doi":"arxiv-2409.10867","DOIUrl":"https://doi.org/arxiv-2409.10867","url":null,"abstract":"Assuming an integral quadratic polynomial with nonsingular quadratic part has\u0000a nontrivial zero on an integer lattice outside of a union of finite-index\u0000sublattices, we prove that there exists such a zero of bounded norm and provide\u0000an explicit bound. This is a contribution related to the celebrated theorem of\u0000Cassels on small-height zeros of quadratic forms, which builds on some previous\u0000work in this area. We also demonstrate an application of these results to the\u0000problem of effective distribution of angles between vectors in the integer\u0000lattice.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259905","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
Most totally real fields do not have universal forms or Northcott property 大多数完全真实的领域没有通用形式或诺斯考特属性
arXiv - MATH - Number Theory Pub Date : 2024-09-17 DOI: arxiv-2409.11082
Nicolas Daans, Vitezslav Kala, Siu Hang Man, Martin Widmer, Pavlo Yatsyna
{"title":"Most totally real fields do not have universal forms or Northcott property","authors":"Nicolas Daans, Vitezslav Kala, Siu Hang Man, Martin Widmer, Pavlo Yatsyna","doi":"arxiv-2409.11082","DOIUrl":"https://doi.org/arxiv-2409.11082","url":null,"abstract":"We show that, in the space of all totally real fields equipped with the\u0000constructible topology, the set of fields that admit a universal quadratic\u0000form, or have the Northcott property, is meager. The main tool is a new theorem\u0000on the number of square classes of totally positive units represented by a\u0000quadratic lattice of a given rank.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"40 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259849","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
New identities for the family of Zeta function by using distributional representations 利用分布表示的 Zeta 函数族新特性
arXiv - MATH - Number Theory Pub Date : 2024-09-17 DOI: arxiv-2409.11029
Asghar Qadir, Aamina Jamshaid
{"title":"New identities for the family of Zeta function by using distributional representations","authors":"Asghar Qadir, Aamina Jamshaid","doi":"arxiv-2409.11029","DOIUrl":"https://doi.org/arxiv-2409.11029","url":null,"abstract":"Chaudhry and Qadir obtained new identities for the gamma function by using a\u0000distributional representation for it. Here we obtain new identities for the\u0000Riemann zeta function and its family by using that representation for them.\u0000This also leads to new identities involving the Dirichlet eta and Lambda\u0000functions.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259850","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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