Pub Date : 2024-10-15DOI: 10.1016/j.aim.2024.109970
Ben Castor
We compare several different methods involving Hodge-theoretic spectra of singularities which produce constraints on the number and type of isolated singularities on a projective hypersurface of fixed degree. In particular, we introduce a method based on the spectrum of the nonisolated singularity at the origin of the affine cone on such a hypersurface, and relate the resulting explicit formula to Varchenko's bound.
{"title":"Bounding projective hypersurface singularities","authors":"Ben Castor","doi":"10.1016/j.aim.2024.109970","DOIUrl":"10.1016/j.aim.2024.109970","url":null,"abstract":"<div><div>We compare several different methods involving Hodge-theoretic spectra of singularities which produce constraints on the number and type of isolated singularities on a projective hypersurface of fixed degree. In particular, we introduce a method based on the spectrum of the nonisolated singularity at the origin of the affine cone on such a hypersurface, and relate the resulting explicit formula to Varchenko's bound.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142441604","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-15DOI: 10.1016/j.aim.2024.109972
Kees Kok , Lin Zhou
In this paper we show that Bloch's higher cycle class map with finite coefficients for quasi-projective equi-dimensional schemes over a field fits naturally in a long exact sequence involving Schreieder's refined unramified cohomology. We also show that the refined unramified cohomology satisfies the localization sequence. Using this we conjecture in the end that refined unramified cohomology is a motivic homology theory and explain how this is related to the aforementioned results.
{"title":"Higher Chow groups with finite coefficients and refined unramified cohomology","authors":"Kees Kok , Lin Zhou","doi":"10.1016/j.aim.2024.109972","DOIUrl":"10.1016/j.aim.2024.109972","url":null,"abstract":"<div><div>In this paper we show that Bloch's higher cycle class map with finite coefficients for quasi-projective equi-dimensional schemes over a field fits naturally in a long exact sequence involving Schreieder's refined unramified cohomology. We also show that the refined unramified cohomology satisfies the localization sequence. Using this we conjecture in the end that refined unramified cohomology is a motivic homology theory and explain how this is related to the aforementioned results.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142441602","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-14DOI: 10.1016/j.aim.2024.109967
Junnosuke Koizumi
In this paper, we study cohomology theories of -modulus pairs, which are pairs consisting of a scheme X and a -divisor D. Our main theorem provides a sufficient condition for such a cohomology theory to be invariant under blow-ups with centers contained in the divisor. This yields a short proof of the blow-up invariance of the Hodge cohomology with modulus proved by Kelly-Miyazaki. We also define the Witt vector cohomology with modulus using the Brylinski-Kato filtration and prove its blow-up invariance.
本文研究 Q 模对的同调理论，即由方案 X 和 Q 分因子 D 组成的对 (X,D)。我们的主要定理提供了一个充分条件，使这种同调理论在中心包含在分因子中的吹胀下保持不变。这就产生了凯利-宫崎（Kelly-Miyazaki）所证明的带模霍奇同调的炸毁不变性的简短证明。我们还利用布赖林斯基-加藤滤波定义了带模的维特向量同调，并证明了它的炸毁不变性。
{"title":"Blow-up invariance of cohomology theories with modulus","authors":"Junnosuke Koizumi","doi":"10.1016/j.aim.2024.109967","DOIUrl":"10.1016/j.aim.2024.109967","url":null,"abstract":"<div><div>In this paper, we study cohomology theories of <span><math><mi>Q</mi></math></span>-modulus pairs, which are pairs <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>D</mi><mo>)</mo></math></span> consisting of a scheme <em>X</em> and a <span><math><mi>Q</mi></math></span>-divisor <em>D</em>. Our main theorem provides a sufficient condition for such a cohomology theory to be invariant under blow-ups with centers contained in the divisor. This yields a short proof of the blow-up invariance of the Hodge cohomology with modulus proved by Kelly-Miyazaki. We also define the Witt vector cohomology with modulus using the Brylinski-Kato filtration and prove its blow-up invariance.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142434434","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-11DOI: 10.1016/j.aim.2024.109968
W. Duke , Ö. Imamoḡlu , Á. Tóth
We give class number formulas for binary cubic and n-ary quadratic forms using a method of Hurwitz. We also show how the same method can be applied to give identities for certain multiple zeta values attached to symmetric cones.
我们用赫尔维茨的方法给出了二元三次方和 n 元二次方的类数公式。我们还展示了如何运用同样的方法给出对称锥的某些多重zeta值的等价性。
{"title":"On a method of Hurwitz and its applications","authors":"W. Duke , Ö. Imamoḡlu , Á. Tóth","doi":"10.1016/j.aim.2024.109968","DOIUrl":"10.1016/j.aim.2024.109968","url":null,"abstract":"<div><div>We give class number formulas for binary cubic and <em>n</em>-ary quadratic forms using a method of Hurwitz. We also show how the same method can be applied to give identities for certain multiple zeta values attached to symmetric cones.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142424644","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-08DOI: 10.1016/j.aim.2024.109966
Daniel C. Isaksen , Hana Jia Kong , Guchuan Li , Yangyang Ruan , Heyi Zhu
We analyze the -motivic (and classical) Adams-Novikov spectral sequence for the -motivic modular forms spectrum mmf (and for the classical topological modular forms spectrum tmf). We primarily use purely algebraic techniques, with a few exceptions. Along the way, we settle a previously unresolved detail about the multiplicative structure of the homotopy groups of tmf.
{"title":"The C-motivic Adams-Novikov spectral sequence for topological modular forms","authors":"Daniel C. Isaksen , Hana Jia Kong , Guchuan Li , Yangyang Ruan , Heyi Zhu","doi":"10.1016/j.aim.2024.109966","DOIUrl":"10.1016/j.aim.2024.109966","url":null,"abstract":"<div><div>We analyze the <span><math><mi>C</mi></math></span>-motivic (and classical) Adams-Novikov spectral sequence for the <span><math><mi>C</mi></math></span>-motivic modular forms spectrum <em>mmf</em> (and for the classical topological modular forms spectrum <em>tmf</em>). We primarily use purely algebraic techniques, with a few exceptions. Along the way, we settle a previously unresolved detail about the multiplicative structure of the homotopy groups of <em>tmf</em>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142424643","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-07DOI: 10.1016/j.aim.2024.109964
Federico Scavia , Fumiaki Suzuki
We give the first examples of smooth projective varieties X over a finite field admitting a non-algebraic torsion ℓ-adic cohomology class of degree 4 which vanishes over . We use them to show that two versions of the integral Tate conjecture over are not equivalent to one another and that a fundamental exact sequence of Colliot-Thélène and Kahn does not necessarily split. Some of our examples have dimension 4, and are the first known examples of fourfolds with non-vanishing .
我们给出了有限域 F 上的光滑投影变项 X 的第一个例子，该投影变项承认一个在 F‾ 上消失的 4 度非代数扭转 ℓ-adic 同调类。我们用它们来证明在 F 上的积分泰特猜想的两个版本并不等同，而且科利奥-泰莱与卡恩的基本精确序列并不一定分裂。我们的一些例子维数为 4，是已知的第一个 Hnr3(X,Q2/Z2(2))不求和的四折的例子。
{"title":"Non-algebraic geometrically trivial cohomology classes over finite fields","authors":"Federico Scavia , Fumiaki Suzuki","doi":"10.1016/j.aim.2024.109964","DOIUrl":"10.1016/j.aim.2024.109964","url":null,"abstract":"<div><div>We give the first examples of smooth projective varieties <em>X</em> over a finite field <span><math><mi>F</mi></math></span> admitting a non-algebraic torsion <em>ℓ</em>-adic cohomology class of degree 4 which vanishes over <span><math><mover><mrow><mi>F</mi></mrow><mo>‾</mo></mover></math></span>. We use them to show that two versions of the integral Tate conjecture over <span><math><mi>F</mi></math></span> are not equivalent to one another and that a fundamental exact sequence of Colliot-Thélène and Kahn does not necessarily split. Some of our examples have dimension 4, and are the first known examples of fourfolds with non-vanishing <span><math><msubsup><mrow><mi>H</mi></mrow><mrow><mi>nr</mi></mrow><mrow><mn>3</mn></mrow></msubsup><mo>(</mo><mi>X</mi><mo>,</mo><msub><mrow><mi>Q</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>/</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mn>2</mn><mo>)</mo><mo>)</mo></math></span>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142425599","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-04DOI: 10.1016/j.aim.2024.109965
Jianhao Shen
In this paper, we define edge zeta functions for spherical buildings associated with finite general linear groups. We derive elegant formulas for these zeta functions and reveal patterns of eigenvalues of these buildings, by introducing and applying insightful tools including digraphs and , cyclic n-partite graphs, partite-transitive group actions, and Springer's theorem on Hecke algebras.
在本文中，我们定义了与有限一般线性群相关的球形建筑的边缘zeta函数。我们通过引入和应用具有洞察力的工具，包括数字图 X0 和 X2、循环 n 部分图、部分传递群作用和关于赫克代数的 Springer 定理，推导出这些 zeta 函数的优雅公式，并揭示了这些建筑物的特征值模式。
{"title":"Zeta functions for spherical tits buildings of finite general linear groups","authors":"Jianhao Shen","doi":"10.1016/j.aim.2024.109965","DOIUrl":"10.1016/j.aim.2024.109965","url":null,"abstract":"<div><div>In this paper, we define edge zeta functions for spherical buildings associated with finite general linear groups. We derive elegant formulas for these zeta functions and reveal patterns of eigenvalues of these buildings, by introducing and applying insightful tools including digraphs <span><math><msub><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, cyclic <em>n</em>-partite graphs, partite-transitive group actions, and Springer's theorem on Hecke algebras.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142425600","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-03DOI: 10.1016/j.aim.2024.109963
Guram Bezhanishvili , Andre Kornell
We prove that the category of topological spaces and open maps does not have binary products, thus resolving the Esakia problem in the negative. We also prove that the category of Kripke frames does not have binary products and that the category of complete Heyting algebras does not have binary coproducts.
{"title":"The category of topological spaces and open maps does not have products","authors":"Guram Bezhanishvili , Andre Kornell","doi":"10.1016/j.aim.2024.109963","DOIUrl":"10.1016/j.aim.2024.109963","url":null,"abstract":"<div><div>We prove that the category of topological spaces and open maps does not have binary products, thus resolving the Esakia problem in the negative. We also prove that the category of Kripke frames does not have binary products and that the category of complete Heyting algebras does not have binary coproducts.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142425598","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-01DOI: 10.1016/j.aim.2024.109960
Vincenzo Morinelli, Karl-Hermann Neeb
Various aspects of the geometric setting of Algebraic Quantum Field Theory (AQFT) models related to representations of the Poincaré group can be studied for general Lie groups, whose Lie algebra contains an Euler element, i.e., ad h is diagonalizable with eigenvalues in . This has been explored by the authors and their collaborators during recent years. A key property in this construction is the Bisognano–Wichmann property (thermal property for wedge region algebras) concerning the geometric implementation of modular groups of local algebras.
In the present paper we prove that under a natural regularity condition, geometrically implemented modular groups arising from the Bisognano–Wichmann property are always generated by Euler elements. We also show the converse, namely that in presence of Euler elements and the Bisognano–Wichmann property, regularity and localizability hold in a quite general setting. Lastly we show that, in this generalized AQFT, in the vacuum representation, under analogous assumptions (regularity and Bisognano–Wichmann), the von Neumann algebras associated to wedge regions are type III1 factors, a property that is well-known in the AQFT context.
{"title":"From local nets to Euler elements","authors":"Vincenzo Morinelli, Karl-Hermann Neeb","doi":"10.1016/j.aim.2024.109960","DOIUrl":"10.1016/j.aim.2024.109960","url":null,"abstract":"<div><div>Various aspects of the geometric setting of Algebraic Quantum Field Theory (AQFT) models related to representations of the Poincaré group can be studied for general Lie groups, whose Lie algebra contains an Euler element, i.e., ad <em>h</em> is diagonalizable with eigenvalues in <span><math><mo>{</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></math></span>. This has been explored by the authors and their collaborators during recent years. A key property in this construction is the Bisognano–Wichmann property (thermal property for wedge region algebras) concerning the geometric implementation of modular groups of local algebras.</div><div>In the present paper we prove that under a natural regularity condition, geometrically implemented modular groups arising from the Bisognano–Wichmann property are always generated by Euler elements. We also show the converse, namely that in presence of Euler elements and the Bisognano–Wichmann property, regularity and localizability hold in a quite general setting. Lastly we show that, in this generalized AQFT, in the vacuum representation, under analogous assumptions (regularity and Bisognano–Wichmann), the von Neumann algebras associated to wedge regions are type III<sub>1</sub> factors, a property that is well-known in the AQFT context.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142425596","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-01DOI: 10.1016/j.aim.2024.109962
Nathan Green
We define two pairings relating the A-motive with the dual A-motive of an abelian Anderson A-module. We show that specializations of these pairings give the exponential and logarithm functions of this Anderson A-module, and we use these specializations to give precise formulas for the coefficients of the exponential and logarithm functions. We then use these pairings to express the exponential and logarithm functions as evaluations of certain infinite products. As an application of these ideas, we prove an analogue of the Mellin transform formula for the Riemann zeta function in the case of Carlitz zeta values. We also give an example showing how our results apply to Carlitz multiple zeta values.
我们定义了两种配对关系，它们分别涉及无性安德森 A 模块的 A 动量和对偶 A 动量。我们证明这些配对的特殊化给出了这个安德森 A 模块的指数函数和对数函数，并利用这些特殊化给出了指数函数和对数函数系数的精确公式。然后，我们利用这些配对将指数函数和对数函数表示为某些无穷积的求值。作为这些思想的应用，我们证明了黎曼zeta函数在卡利茨zeta值情况下的梅林变换公式。我们还举例说明了我们的结果如何适用于卡利茨多重zeta值。
{"title":"A motivic pairing and the Mellin transform in function fields","authors":"Nathan Green","doi":"10.1016/j.aim.2024.109962","DOIUrl":"10.1016/j.aim.2024.109962","url":null,"abstract":"<div><div>We define two pairings relating the <em>A</em>-motive with the dual <em>A</em>-motive of an abelian Anderson <em>A</em>-module. We show that specializations of these pairings give the exponential and logarithm functions of this Anderson <em>A</em>-module, and we use these specializations to give precise formulas for the coefficients of the exponential and logarithm functions. We then use these pairings to express the exponential and logarithm functions as evaluations of certain infinite products. As an application of these ideas, we prove an analogue of the Mellin transform formula for the Riemann zeta function in the case of Carlitz zeta values. We also give an example showing how our results apply to Carlitz multiple zeta values.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142425597","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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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