Let $p$ be an odd prime number and let $K$ be an imaginary quadratic field in which $p$ is split. Let $f$ be a modular form with good reduction at $p$. We study the variation of the Bloch--Kato Selmer groups and the Bloch--Kato--Shafarevich--Tate groups of $f$ over the anticyclotomic $mathbf{Z}_p$-extension $K_infty$ of $K$. In particular, we show that under the generalized Heegner hypothesis, if the $p$-localization of the generalized Heegner cycle attached to $f$ is primitive and certain local conditions hold, then the Pontryagin dual of the Selmer group of $f$ over $K_infty$ is free over the Iwasawa algebra. Consequently, the Bloch--Kato--Shafarevich--Tate groups of $f$ vanish. This generalizes earlier works of Matar and Matar--Nekov'av{r} on elliptic curves. Furthermore, our proof applies uniformly to the ordinary and non-ordinary settings.
{"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}
We establish a Galois-theoretic trichotomy governing Diophantine stability for genus $0$ curves. We use it to prove that the curve associated to the Hilbert symbol is Diophantine stable with probability $1$. Our asymptotic formula for the second order term exhibits strong bias towards instability.
{"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}
We show that the systems of Hecke eigenvalues that appear in the coherent cohomology with coefficients in automorphic line bundles of any mod $p$ abelian type compact Shimura variety at hyperspecial level are the same as those appearing in any Hecke-equivariant closed subscheme. We also prove analogous results for noncompact Shimura varieties or nonclosed subschemes, such as Ekedahl-Oort strata, length strata and central leaves.
{"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}
Given a supersingular elliptic curve, the supersingular endomorphism ring problem is to compute all of its endomorphisms. We use the correspondence between maximal orders in quaternion algebra $B_{p,infty}$ and positive ternary quadratic forms with discriminant $p$ to solve the supersingular endomorphism ring problem. Let $c<3p/16$ be a prime or $c=1$. Let $E$ be a $mathbb{Z}[sqrt{-cp}]$-oriented supersingular elliptic curve defined over $mathbb{F}_{p^2}$. There exists a subgroup $G$ of order $c$, and $text{End}(E,G)$ is isomorphic to an Eichler order in $B_{p,infty}$ of level $c$. If the endomorphism ring $text{End}(E,G)$ is known, then we can compute $text{End}(E)$ by solving two square roots in $mathbb{F}_c$. In particular, let $D
{"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}
Let $k$ and $n$ be natural numbers. Let $omega_k(n)$ denote the number of distinct prime factors of $n$ with multiplicity $k$ as studied by Elma and the third author. We obtain asymptotic estimates for the first and the second moments of $omega_k(n)$ when restricted to the set of $h$-free and $h$-full numbers. We prove that $omega_1(n)$ has normal order $log log n$ over $h$-free numbers, $omega_h(n)$ has normal order $log log n$ over $h$-full numbers, and both of them satisfy the ErdH{o}s-Kac Theorem. Finally, we prove that the functions $omega_k(n)$ with $1 < k < h$ do not have normal order over $h$-free numbers and $omega_k(n)$ with $k > h$ do not have normal order over $h$-full numbers.
{"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}
Greither and Kurihara proved a theorem about the commutativity of projective limits and Fitting ideals for modules over the classical equivariant Iwasawa algebra $Lambda_G=mathbb{Z}_p[[T]][G]$, where $G$ is a finite, abelian group and $Bbb Z_p$ is the ring of $p$--adic integers, for some prime $p$. In this paper, we generalize their result first to the Noetherian Iwasawa algebra $mathbb{Z}_p[[T_1, T_2, cdots, T_n]][G]$ and, most importantly, to the non-Noetherian algebra $mathbb{Z}_p[[T_1, T_2, cdots, T_n, cdots]][G]$ of countably many generators. The latter generalization is motivated by the recent work of Bley-Popescu on the geometric Equivariant Iwasawa Conjecture for function fields, where the Iwasawa algebra is not Noetherian, of the type described above. Applications of these results to the emerging field of non-Noetherian Iwasawa Theory will be given in an upcoming paper.
{"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}
Caroline Cashman, Steven J. Miller, Jenna Shuffleton, Daeyoung Son
Zeckendorf proved a remarkable fact that every positive integer can be written as a decomposition of non-adjacent Fibonacci numbers. Baird-Smith, Epstein, Flint, and Miller converted the process of decomposing a positive integer into its Zeckendorf decomposition into a game, using the moves of $F_i + F_{i-1} = F_{i+1}$ and $2F_i = F_{i+1} + F_{i-2}$, where $F_i$ is the $i$thFibonacci number. Players take turns applying these moves, beginning with $n$ pieces in the $F_1$ column. They showed that for $n neq 2$, Player 2 has a winning strategy, though the proof is non-constructive, and a constructive solution is unknown. We expand on this by investigating "black hole'' variants of this game. The Black Hole Zeckendorf game on $F_m$ is played with any $n$ but solely in columns $F_i$ for $i < m$. Gameplay is similar to the original Zeckendorf game, except any piece that would be placed on $F_i$ for $i geq m$ is locked out in a ``black hole'' and removed from play. With these constraints, we analyze the games with black holes on $F_3$ and $F_4$ and construct a solution for specific configurations, using a parity-stealing based non-constructive proof to lead to a constructive one. We also examine a pre-game in which players take turns placing down $n$ pieces in the outermost columns before the decomposition phase, and find constructive solutions for any $n$.
{"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}
Assuming an integral quadratic polynomial with nonsingular quadratic part has a nontrivial zero on an integer lattice outside of a union of finite-index sublattices, we prove that there exists such a zero of bounded norm and provide an explicit bound. This is a contribution related to the celebrated theorem of Cassels on small-height zeros of quadratic forms, which builds on some previous work in this area. We also demonstrate an application of these results to the problem of effective distribution of angles between vectors in the integer lattice.
{"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}
Nicolas Daans, Vitezslav Kala, Siu Hang Man, Martin Widmer, Pavlo Yatsyna
We show that, in the space of all totally real fields equipped with the constructible topology, the set of fields that admit a universal quadratic form, or have the Northcott property, is meager. The main tool is a new theorem on the number of square classes of totally positive units represented by a quadratic lattice of a given rank.
{"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}
Chaudhry and Qadir obtained new identities for the gamma function by using a distributional representation for it. Here we obtain new identities for the Riemann zeta function and its family by using that representation for them. This also leads to new identities involving the Dirichlet eta and Lambda functions.
{"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}
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