Pub Date : 2023-01-01DOI: 10.4310/pamq.2023.v19.n2.a9
Weidong Wang, Yanping Zhou
{"title":"Some inequalities for the dual $p$-quermassintegrals","authors":"Weidong Wang, Yanping Zhou","doi":"10.4310/pamq.2023.v19.n2.a9","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n2.a9","url":null,"abstract":"","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70526842","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.4310/pamq.2023.v19.n1.a9
V. Golyshev, Duco van Straten
{"title":"Congruences via fibered motives","authors":"V. Golyshev, Duco van Straten","doi":"10.4310/pamq.2023.v19.n1.a9","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n1.a9","url":null,"abstract":"","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70526725","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.4310/pamq.2023.v19.n1.a1
B. Gross
{"title":"Working with Don","authors":"B. Gross","doi":"10.4310/pamq.2023.v19.n1.a1","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n1.a1","url":null,"abstract":"","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70527007","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-22DOI: 10.4310/pamq.2023.v19.n1.a4
G. Bogo, Yingkun Li
In this paper, we study the diagonal restrictions of certain Hilbert theta series for a totally real field $F$, and prove that they span the corresponding space of elliptic modular forms when the $F$ is quadratic or cubic. Furthermore, we give evidence of this phenomenon when $F$ is quartic, quintic and sextic.
{"title":"Span of restriction of Hilbert theta functions","authors":"G. Bogo, Yingkun Li","doi":"10.4310/pamq.2023.v19.n1.a4","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n1.a4","url":null,"abstract":"In this paper, we study the diagonal restrictions of certain Hilbert theta series for a totally real field $F$, and prove that they span the corresponding space of elliptic modular forms when the $F$ is quadratic or cubic. Furthermore, we give evidence of this phenomenon when $F$ is quartic, quintic and sextic.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48288818","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-21DOI: 10.4310/pamq.2023.v19.n2.a6
Andrew Yu
Let $S_n$ be the symmetric group of all permutations of ${1, cdots, n}$ with two generators: the transposition switching $1$ with $2$ and the cyclic permutation sending $k$ to $k+1$ for $1leq kleq n-1$ and $n$ to $1$ (denoted by $sigma$ and $tau$). In this article, we study quantum complexity of permutations in $S_n$ using ${sigma, tau, tau^{-1}}$ as logic gates. We give an explicit construction of permutations in $S_n$ with quadratic quantum complexity lower bound $frac{n^2-2n-7}{4}$. We also prove that all permutations in $S_n$ have quadratic quantum complexity upper bound $3(n-1)^2$. Finally, we show that almost all permutations in $S_n$ have quadratic quantum complexity lower bound when $nrightarrow infty$.
{"title":"Quantum complexity of permutations","authors":"Andrew Yu","doi":"10.4310/pamq.2023.v19.n2.a6","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n2.a6","url":null,"abstract":"Let $S_n$ be the symmetric group of all permutations of ${1, cdots, n}$ with two generators: the transposition switching $1$ with $2$ and the cyclic permutation sending $k$ to $k+1$ for $1leq kleq n-1$ and $n$ to $1$ (denoted by $sigma$ and $tau$). In this article, we study quantum complexity of permutations in $S_n$ using ${sigma, tau, tau^{-1}}$ as logic gates. We give an explicit construction of permutations in $S_n$ with quadratic quantum complexity lower bound $frac{n^2-2n-7}{4}$. We also prove that all permutations in $S_n$ have quadratic quantum complexity upper bound $3(n-1)^2$. Finally, we show that almost all permutations in $S_n$ have quadratic quantum complexity lower bound when $nrightarrow infty$.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44878114","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-11DOI: 10.4310/pamq.2022.v18.n5.a5
J. Thorne
Let $rho$ be the $p$-adic Galois representation attached to a cuspidal, regular algebraic, polarizable automorphic representation of $GL_n$. Assuming only that $rho$ satisfies an irreducibility condition, we prove the vanishing of the adjoint Bloch--Kato Selmer group attached to $rho$. This generalizes previous work of the author and James Newton.
{"title":"On the vanishing of adjoint Bloch–Kato Selmer groups of irreducible automorphic Galois representations","authors":"J. Thorne","doi":"10.4310/pamq.2022.v18.n5.a5","DOIUrl":"https://doi.org/10.4310/pamq.2022.v18.n5.a5","url":null,"abstract":"Let $rho$ be the $p$-adic Galois representation attached to a cuspidal, regular algebraic, polarizable automorphic representation of $GL_n$. Assuming only that $rho$ satisfies an irreducibility condition, we prove the vanishing of the adjoint Bloch--Kato Selmer group attached to $rho$. This generalizes previous work of the author and James Newton.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46129152","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-04-27DOI: 10.4310/pamq.2022.v18.n4.a10
S. Katz, W. Taylor
We describe the dimensional reduction of the IIB B-fields in F-theory using a conjectured description of normalizable B-fields in terms of perverse sheaves. Computations are facilitated using the Decomposition Theorem. Many of our descriptions are new, and all our results are all consistent with known results in physics. We also conjecture a physical framework for normalizable B-fields and show consistency with mathematics. We dedicate this paper to Herb Clemens, in admiration for his myriad fundamental contributions to complex algebraic geometry, together with his more recent interest in F-theory in physics. This paper deals with three of Herb’s interests: Hodge theory, topology of algebraic varieties, and F-theory, and so is a fitting way for us to express our appreciation for his contributions over a period of more than five decades.
{"title":"Dimensional reduction of B-fields in F-theory","authors":"S. Katz, W. Taylor","doi":"10.4310/pamq.2022.v18.n4.a10","DOIUrl":"https://doi.org/10.4310/pamq.2022.v18.n4.a10","url":null,"abstract":"We describe the dimensional reduction of the IIB B-fields in F-theory using a conjectured description of normalizable B-fields in terms of perverse sheaves. Computations are facilitated using the Decomposition Theorem. Many of our descriptions are new, and all our results are all consistent with known results in physics. We also conjecture a physical framework for normalizable B-fields and show consistency with mathematics. We dedicate this paper to Herb Clemens, in admiration for his myriad fundamental contributions to complex algebraic geometry, together with his more recent interest in F-theory in physics. This paper deals with three of Herb’s interests: Hodge theory, topology of algebraic varieties, and F-theory, and so is a fitting way for us to express our appreciation for his contributions over a period of more than five decades.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43239355","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-04-23DOI: 10.4310/pamq.2022.v18.n5.a6
Zhiwei Yun
. In this paper we give a different proof of a theorem of Paul Breutmann: for a Bruhat-Tits group scheme H over a smooth projective curve X and a closed embedding into another smooth affine group scheme G , the induced map on the moduli of Shtukas Sht r H → Sht r G is schematic, finite and unramified. This result enables one to define special cycles on the moduli stack of Shtukas.
在本文中，我们给出了Paul Breutmann定理的不同证明：对于光滑投影曲线X上的Bruhat-Tits群方案H和另一个光滑群方案G中的闭嵌入，Shtukas Sht r H的模上的诱导映射→ Sht r G是示意性的、有限的和不可分割的。这一结果使人们能够在Shtukas的模量堆栈上定义特殊循环。
{"title":"Special cycles for Shtukas are closed","authors":"Zhiwei Yun","doi":"10.4310/pamq.2022.v18.n5.a6","DOIUrl":"https://doi.org/10.4310/pamq.2022.v18.n5.a6","url":null,"abstract":". In this paper we give a different proof of a theorem of Paul Breutmann: for a Bruhat-Tits group scheme H over a smooth projective curve X and a closed embedding into another smooth affine group scheme G , the induced map on the moduli of Shtukas Sht r H → Sht r G is schematic, finite and unramified. This result enables one to define special cycles on the moduli stack of Shtukas.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45757624","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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