We formulate and prove a version of the arithmetic Siegel–Weil formula for (zero dimensional) Shimura varieties attached to tori, equipped with some additional data. More precisely, we define a family of “special” divisors in terms of Green functions at archimedean and non-archimedean places and prove that their degrees coincide with the Fourier coefficients of the central derivative of an Eisenstein series. The proof relies on the usual Siegel–Weil formula to provide a direct link between both sides of the identity, and in some sense, offers a more conceptual point of view on prior results in the literature.
{"title":"Zero-Dimensional Shimura Varieties and Central Derivatives of Eisenstein Series","authors":"Siddarth Sankaran","doi":"10.1093/imrn/rnae179","DOIUrl":"https://doi.org/10.1093/imrn/rnae179","url":null,"abstract":"We formulate and prove a version of the arithmetic Siegel–Weil formula for (zero dimensional) Shimura varieties attached to tori, equipped with some additional data. More precisely, we define a family of “special” divisors in terms of Green functions at archimedean and non-archimedean places and prove that their degrees coincide with the Fourier coefficients of the central derivative of an Eisenstein series. The proof relies on the usual Siegel–Weil formula to provide a direct link between both sides of the identity, and in some sense, offers a more conceptual point of view on prior results in the literature.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203306","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $p$ and $l$ be two distinct odd primes, and let $ngeq 2$ be a positive integer. Let $E$ be a finite Galois extension of degree $l$ of a $p$-adic field $F$. Let $q$ be the cardinality of the residue field of $F$. Let $pi _{F}$ be an integral $l$-adic generic representation of $mathrm{GL}_{n}(F)$, and let $pi _{E}$ be the base change of $pi _{F}$. Let $J_{l}(pi _{F})$ (resp. $J_{l}(pi _{E})$) be the unique generic component of the mod-$l$ reduction $r_{l}(pi _{F})$ (resp. $r_{l}(pi _{E})$). Assuming that $l$ does not divide $|mathrm{GL}_{n-1}(mathbb{F}_{q})|$, we prove that the Frobenius twist of $J_{l}(pi _{F})$ is the unique generic subquotient of the Tate cohomology group $widehat{H}^{0}(mathrm{Gal}(E/F), J_{l}(pi _{E}))$—considered as a representation of $mathrm{GL}_{n}(F)$.
{"title":"Tate Cohomology of Whittaker Lattices and Base Change of Generic Representations of GLn","authors":"Santosh Nadimpalli, Sabyasachi Dhar","doi":"10.1093/imrn/rnae183","DOIUrl":"https://doi.org/10.1093/imrn/rnae183","url":null,"abstract":"Let $p$ and $l$ be two distinct odd primes, and let $ngeq 2$ be a positive integer. Let $E$ be a finite Galois extension of degree $l$ of a $p$-adic field $F$. Let $q$ be the cardinality of the residue field of $F$. Let $pi _{F}$ be an integral $l$-adic generic representation of $mathrm{GL}_{n}(F)$, and let $pi _{E}$ be the base change of $pi _{F}$. Let $J_{l}(pi _{F})$ (resp. $J_{l}(pi _{E})$) be the unique generic component of the mod-$l$ reduction $r_{l}(pi _{F})$ (resp. $r_{l}(pi _{E})$). Assuming that $l$ does not divide $|mathrm{GL}_{n-1}(mathbb{F}_{q})|$, we prove that the Frobenius twist of $J_{l}(pi _{F})$ is the unique generic subquotient of the Tate cohomology group $widehat{H}^{0}(mathrm{Gal}(E/F), J_{l}(pi _{E}))$—considered as a representation of $mathrm{GL}_{n}(F)$.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203308","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Anna Marie Bohmann, Teena Gerhardt, Cary Malkiewich, Mona Merling, Inna Zakharevich
Cut-and-paste $K$-theory has recently emerged as an important variant of higher algebraic $K$-theory. However, many of the powerful tools used to study classical higher algebraic $K$-theory do not yet have analogues in the cut-and-paste setting. In particular, there does not yet exist a sensible notion of the Dennis trace for cut-and-paste $K$-theory. In this paper we address the particular case of the $K$-theory of polyhedra, also called scissors congruence $K$-theory. We introduce an explicit, computable trace map from the higher scissors congruence groups to group homology, and use this trace to prove the existence of some nonzero classes in the higher scissors congruence groups. We also show that the $K$-theory of polyhedra is a homotopy orbit spectrum. This fits into Thomason’s general framework of $K$-theory commuting with homotopy colimits, but we give a self-contained proof. We then use this result to re-interpret the trace map as a partial inverse to the map that commutes homotopy orbits with algebraic $K$-theory.
{"title":"A Trace Map on Higher Scissors Congruence Groups","authors":"Anna Marie Bohmann, Teena Gerhardt, Cary Malkiewich, Mona Merling, Inna Zakharevich","doi":"10.1093/imrn/rnae153","DOIUrl":"https://doi.org/10.1093/imrn/rnae153","url":null,"abstract":"Cut-and-paste $K$-theory has recently emerged as an important variant of higher algebraic $K$-theory. However, many of the powerful tools used to study classical higher algebraic $K$-theory do not yet have analogues in the cut-and-paste setting. In particular, there does not yet exist a sensible notion of the Dennis trace for cut-and-paste $K$-theory. In this paper we address the particular case of the $K$-theory of polyhedra, also called scissors congruence $K$-theory. We introduce an explicit, computable trace map from the higher scissors congruence groups to group homology, and use this trace to prove the existence of some nonzero classes in the higher scissors congruence groups. We also show that the $K$-theory of polyhedra is a homotopy orbit spectrum. This fits into Thomason’s general framework of $K$-theory commuting with homotopy colimits, but we give a self-contained proof. We then use this result to re-interpret the trace map as a partial inverse to the map that commutes homotopy orbits with algebraic $K$-theory.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203310","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We develop a theory of adjunctions in semigroup categories, that is, monoidal categories without a unit object. We show that a rigid semigroup category is promonoidal, and thus one can naturally adjoin a unit object to it. This extends the previous results of Houston in the symmetric case, and addresses a question of his. It also extends the results in the non-symmetric case with additional finiteness assumptions, obtained by Benson–Etingof–Ostrik, Coulembier, and Ko–Mazorchuk–Zhang. We give an interpretation of these results using comonad cohomology, and, in the absence of finiteness conditions, using enriched traces of monoidal categories. As an application of our results, we give a characterization of finite tensor categories in terms of the finitary $2$-representation theory of Mazorchuk–Miemietz.
{"title":"Identity in the Presence of Adjunction","authors":"Mateusz Stroiński","doi":"10.1093/imrn/rnae166","DOIUrl":"https://doi.org/10.1093/imrn/rnae166","url":null,"abstract":"We develop a theory of adjunctions in semigroup categories, that is, monoidal categories without a unit object. We show that a rigid semigroup category is promonoidal, and thus one can naturally adjoin a unit object to it. This extends the previous results of Houston in the symmetric case, and addresses a question of his. It also extends the results in the non-symmetric case with additional finiteness assumptions, obtained by Benson–Etingof–Ostrik, Coulembier, and Ko–Mazorchuk–Zhang. We give an interpretation of these results using comonad cohomology, and, in the absence of finiteness conditions, using enriched traces of monoidal categories. As an application of our results, we give a characterization of finite tensor categories in terms of the finitary $2$-representation theory of Mazorchuk–Miemietz.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203312","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Gompf showed that for $K$ in a certain family of double-twist knots, the swallow-follow operation makes $1/n$-surgery on $K # -K$ into a cork boundary. We derive a general Floer-theoretic condition on $K$ under which this is the case. Our formalism allows us to produce many further examples of corks, partially answering a question of Gompf. Unlike Gompf’s method, our proof does not rely on any closed 4-manifold invariants or effective embeddings, and also generalizes to other diffeomorphisms.
{"title":"Gompf’s Cork and Heegaard Floer Homology","authors":"Irving Dai, Abhishek Mallick, Ian Zemke","doi":"10.1093/imrn/rnae180","DOIUrl":"https://doi.org/10.1093/imrn/rnae180","url":null,"abstract":"Gompf showed that for $K$ in a certain family of double-twist knots, the swallow-follow operation makes $1/n$-surgery on $K # -K$ into a cork boundary. We derive a general Floer-theoretic condition on $K$ under which this is the case. Our formalism allows us to produce many further examples of corks, partially answering a question of Gompf. Unlike Gompf’s method, our proof does not rely on any closed 4-manifold invariants or effective embeddings, and also generalizes to other diffeomorphisms.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203311","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The generating function of the Segre integrals on Hilbert schemes of points on a surface $X$ can be determined by five universal series $A_{0}(z)$, $A_{1}(z)$, $A_{2}(z)$, $A_{3}(z)$, $A_{4}(z)$, due to the result of Ellingsrud–Göttsche–Lehn. These five series do not depend on the surface $X$ and depend on the element $alpha in K(X)$, to which the Segre integrals are associated, only through the rank. Marian–Oprea–Pandharipande have determined $A_{0}(z),A_{1}(z),A_{2}(z)$ for all ranks. For rank 0, it is easy to see $A_{4}(z)=1$. Marian–Oprea–Pandharipande also conjectured that $A_{3}(z)=A_{0}(z)A_{1}(z)$ for rank 0. We prove this conjecture by showing that when $X$ is the projective plan, the Segre integrals associated to the structure sheaf of a curve in the anti-canoncial class are all zero. Hence, the rank zero Segre integrals on the Hilbert schemes of points for all surfaces are determined.
{"title":"Rank Zero Segre Integrals on Hilbert Schemes of Points on Surfaces","authors":"Yao Yuan","doi":"10.1093/imrn/rnae173","DOIUrl":"https://doi.org/10.1093/imrn/rnae173","url":null,"abstract":"The generating function of the Segre integrals on Hilbert schemes of points on a surface $X$ can be determined by five universal series $A_{0}(z)$, $A_{1}(z)$, $A_{2}(z)$, $A_{3}(z)$, $A_{4}(z)$, due to the result of Ellingsrud–Göttsche–Lehn. These five series do not depend on the surface $X$ and depend on the element $alpha in K(X)$, to which the Segre integrals are associated, only through the rank. Marian–Oprea–Pandharipande have determined $A_{0}(z),A_{1}(z),A_{2}(z)$ for all ranks. For rank 0, it is easy to see $A_{4}(z)=1$. Marian–Oprea–Pandharipande also conjectured that $A_{3}(z)=A_{0}(z)A_{1}(z)$ for rank 0. We prove this conjecture by showing that when $X$ is the projective plan, the Segre integrals associated to the structure sheaf of a curve in the anti-canoncial class are all zero. Hence, the rank zero Segre integrals on the Hilbert schemes of points for all surfaces are determined.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203314","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Suppose that $Sigma ^{n}subset mathbb{S}^{n+1}$ is a closed embedded minimal hypersurface. We prove that the first non-zero eigenvalue $lambda _{1}$ of the induced Laplace–Beltrami operator on $Sigma $ satisfies $lambda _{1} geq frac{n}{2}+ a_{n}(Lambda ^{6} + b_{n})^{-1}$, where $a_{n}$ and $b_{n}$ are explicit dimensional constants and $Lambda $ is an upper bound for the length of the second fundamental form of $Sigma $. This provides the first explicitly computable improvement on Choi and Wang’s lower bound $lambda _{1} geq frac{n}{2}$ without any further assumptions on $Sigma $.
{"title":"An Improved Eigenvalue Estimate for Embedded Minimal Hypersurfaces in the Sphere","authors":"Jonah A J Duncan, Yannick Sire, Joel Spruck","doi":"10.1093/imrn/rnae154","DOIUrl":"https://doi.org/10.1093/imrn/rnae154","url":null,"abstract":"Suppose that $Sigma ^{n}subset mathbb{S}^{n+1}$ is a closed embedded minimal hypersurface. We prove that the first non-zero eigenvalue $lambda _{1}$ of the induced Laplace–Beltrami operator on $Sigma $ satisfies $lambda _{1} geq frac{n}{2}+ a_{n}(Lambda ^{6} + b_{n})^{-1}$, where $a_{n}$ and $b_{n}$ are explicit dimensional constants and $Lambda $ is an upper bound for the length of the second fundamental form of $Sigma $. This provides the first explicitly computable improvement on Choi and Wang’s lower bound $lambda _{1} geq frac{n}{2}$ without any further assumptions on $Sigma $.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203315","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove the existence of singular del Pezzo surfaces that are neither K-semistable nor contain any anticanonical polar cylinder.
我们证明了既不可 K-semistable 又不包含任何反谐极圆柱的奇异 del Pezzo 曲面的存在性。
{"title":"K-Unstable Singular del Pezzo Surfaces Without Anticanonical Polar Cylinder","authors":"In-Kyun Kim, Jaehyun Kim, Joonyeong Won","doi":"10.1093/imrn/rnae175","DOIUrl":"https://doi.org/10.1093/imrn/rnae175","url":null,"abstract":"We prove the existence of singular del Pezzo surfaces that are neither K-semistable nor contain any anticanonical polar cylinder.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203313","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
It is proven that, in any given base, there are infinitely many palindromic numbers having at most six prime divisors, each relatively large. The work involves equidistribution estimates for the palindromes in residue classes to large moduli, offering upper bounds for moments and averages of certain products closely related to exponential sums over palindromes.
{"title":"Infinitude of Palindromic Almost-Prime Numbers","authors":"Aleksandr Tuxanidy, Daniel Panario","doi":"10.1093/imrn/rnae174","DOIUrl":"https://doi.org/10.1093/imrn/rnae174","url":null,"abstract":"It is proven that, in any given base, there are infinitely many palindromic numbers having at most six prime divisors, each relatively large. The work involves equidistribution estimates for the palindromes in residue classes to large moduli, offering upper bounds for moments and averages of certain products closely related to exponential sums over palindromes.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141943845","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We analyze an algorithmic question about immersion theory: for which $m$, $n$, and $CAT=textbf{Diff}$ or $textbf{PL}$ is the question of whether an $m$-dimensional $CAT$-manifold is immersible in $mathbb{R}^{n}$ decidable? We show that PL immersibility is decidable in all cases except for codimension 2, whereas smooth immersibility is decidable in all odd codimensions and undecidable in many even codimensions. As a corollary, we show that the smooth embeddability of an $m$-manifold with boundary in $mathbb{R}^{n}$ is undecidable when $n-m$ is even and $11m geq 10n+1$.
{"title":"Algorithmic Aspects of Immersibility and Embeddability","authors":"Fedor Manin, Shmuel Weinberger","doi":"10.1093/imrn/rnae170","DOIUrl":"https://doi.org/10.1093/imrn/rnae170","url":null,"abstract":"We analyze an algorithmic question about immersion theory: for which $m$, $n$, and $CAT=textbf{Diff}$ or $textbf{PL}$ is the question of whether an $m$-dimensional $CAT$-manifold is immersible in $mathbb{R}^{n}$ decidable? We show that PL immersibility is decidable in all cases except for codimension 2, whereas smooth immersibility is decidable in all odd codimensions and undecidable in many even codimensions. As a corollary, we show that the smooth embeddability of an $m$-manifold with boundary in $mathbb{R}^{n}$ is undecidable when $n-m$ is even and $11m geq 10n+1$.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141943844","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：