{"title":"An infinitesimal variant of the Guo-Jacquet trace formula. I: The case of $(mathrm{GL}_{2n, D}, mathrm{GL}_{n, D}times mathrm{GL}_{n, D})$","authors":"Huajie Li","doi":"10.4171/dm/872","DOIUrl":"https://doi.org/10.4171/dm/872","url":null,"abstract":"","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":"68 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77149909","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Okubo quasigroups","authors":"Jonathan D. H. Smith, P. Vojtěchovský","doi":"10.4171/dm/877","DOIUrl":"https://doi.org/10.4171/dm/877","url":null,"abstract":"","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82993412","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In a recent paper, the author introduced a rich class NC(R) of “noncommutative C” functions R → C whose operator functional calculus is k-times differentiable and has derivatives expressible in terms of multiple operator integrals (MOIs). In the present paper, we explore a connection between free stochastic calculus and the theory of MOIs by proving an Itô formula for noncommutative C functions of self-adjoint free Itô processes. To do this, we first extend P. Biane and R. Speicher’s theory of free stochastic calculus – including their free Itô formula for polynomials – to allow free Itô processes driven by multiple freely independent semicircular Brownian motions. Then, in the self-adjoint case, we reinterpret the objects appearing in the free Itô formula for polynomials in terms of MOIs. This allows us to enlarge the class of functions for which one can formulate and prove a free Itô formula from the space originally considered by Biane and Speicher (Fourier transforms of complex measures with two finite moments) to the strictly larger space NC(R). Along the way, we also obtain a useful “traced” Itô formula for arbitrary C scalar functions of self-adjoint free Itô processes. Finally, as motivation, we study an Itô formula for C scalar functions of N ×N Hermitian matrix Itô processes. Keyphrases: free probability, free stochastic calculus, matrix stochastic calculus, Itô formula, functional calculus, multiple operator integral Mathematics Subject Classification: 46L54, 47A60, 60H05
{"title":"Itô's formula for noncommutative $C^2$ functions of free Itô processes","authors":"Evangelos A. Nikitopoulos","doi":"10.4171/dm/902","DOIUrl":"https://doi.org/10.4171/dm/902","url":null,"abstract":"In a recent paper, the author introduced a rich class NC(R) of “noncommutative C” functions R → C whose operator functional calculus is k-times differentiable and has derivatives expressible in terms of multiple operator integrals (MOIs). In the present paper, we explore a connection between free stochastic calculus and the theory of MOIs by proving an Itô formula for noncommutative C functions of self-adjoint free Itô processes. To do this, we first extend P. Biane and R. Speicher’s theory of free stochastic calculus – including their free Itô formula for polynomials – to allow free Itô processes driven by multiple freely independent semicircular Brownian motions. Then, in the self-adjoint case, we reinterpret the objects appearing in the free Itô formula for polynomials in terms of MOIs. This allows us to enlarge the class of functions for which one can formulate and prove a free Itô formula from the space originally considered by Biane and Speicher (Fourier transforms of complex measures with two finite moments) to the strictly larger space NC(R). Along the way, we also obtain a useful “traced” Itô formula for arbitrary C scalar functions of self-adjoint free Itô processes. Finally, as motivation, we study an Itô formula for C scalar functions of N ×N Hermitian matrix Itô processes. Keyphrases: free probability, free stochastic calculus, matrix stochastic calculus, Itô formula, functional calculus, multiple operator integral Mathematics Subject Classification: 46L54, 47A60, 60H05","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":"103 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80339081","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-22DOI: 10.25537/dm.2022v27.699-717
L. Barbieri-Viale, B. Kahn
We prove that any rigid additive symmetric monoidal category can be mapped to a rigid abelian symmetric monoidal category in a universal way. This yields a novel approach to Grothendieck’s standard conjecture D and Voevodsky’s smash nilpotence conjecture.
{"title":"A universal rigid abelian tensor category","authors":"L. Barbieri-Viale, B. Kahn","doi":"10.25537/dm.2022v27.699-717","DOIUrl":"https://doi.org/10.25537/dm.2022v27.699-717","url":null,"abstract":"We prove that any rigid additive symmetric monoidal category can be mapped to a rigid abelian symmetric monoidal category in a universal way. This yields a novel approach to Grothendieck’s standard conjecture D and Voevodsky’s smash nilpotence conjecture.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":"24 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78783808","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We investigate the structure of the Chow ring of the classifying stacks BT of algebraic tori, as it has been defined by B. Totaro. Some previous work of N. Karpenko, A. Merkurjev, S. Blinstein and F. Scavia has shed some light on the structure of such rings. In particular Karpenko showed the absence of torsion classes in the case of permutation tori, while Merkurjev and Blinstein described in a very effective way the second Chow group A2(BT ) in the general case. Building on this work, Scavia exhibited an example where A2(BT )tors 6= 0. Here, by making use of a very elementary approach, we extend the result of Karpenko to special tori and we completely determine the Chow ring A∗(BT ) when T is an algebraic torus admitting a resolution with special tori 0 → T → Q → P . In particular we show that there can be torsion in the Chow ring of such tori.
我们研究了B. Totaro定义的代数环面分类堆BT的Chow环的结构。N. Karpenko, A. Merkurjev, S. Blinstein和F. Scavia先前的一些研究已经揭示了这种环的结构。特别是Karpenko证明了在置换环面情况下不存在扭转类，而Merkurjev和Blinstein在一般情况下以一种非常有效的方式描述了第二个Chow群A2(BT)。在这项工作的基础上，Scavia展示了一个A2(BT)tors 6= 0的例子。本文利用一种非常初等的方法，将Karpenko的结果推广到特殊环面，并完全确定了当T是具有特殊环面0→T→Q→P的代数环面时的Chow环a * (BT)。特别地，我们证明了这种环面的周氏环可能存在扭转。
{"title":"On the Chow ring of the classifying stack of algebraic tori","authors":"Francesco Sala","doi":"10.4171/dm/888","DOIUrl":"https://doi.org/10.4171/dm/888","url":null,"abstract":"We investigate the structure of the Chow ring of the classifying stacks BT of algebraic tori, as it has been defined by B. Totaro. Some previous work of N. Karpenko, A. Merkurjev, S. Blinstein and F. Scavia has shed some light on the structure of such rings. In particular Karpenko showed the absence of torsion classes in the case of permutation tori, while Merkurjev and Blinstein described in a very effective way the second Chow group A2(BT ) in the general case. Building on this work, Scavia exhibited an example where A2(BT )tors 6= 0. Here, by making use of a very elementary approach, we extend the result of Karpenko to special tori and we completely determine the Chow ring A∗(BT ) when T is an algebraic torus admitting a resolution with special tori 0 → T → Q → P . In particular we show that there can be torsion in the Chow ring of such tori.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":"35 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81091856","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove asymptotic lower bounds on the variance of the number of vertices and missed area of random disc-polygons in convex discs whose boundary is $C_+^2$ smooth. The established lower bounds are of the same order as the upper bounds proved previously by Fodor and V'{i}gh (2018).
{"title":"Variance bounds for disc-polygons","authors":"F. Fodor, B. Grunfelder, V. V'igh","doi":"10.4171/dm/891","DOIUrl":"https://doi.org/10.4171/dm/891","url":null,"abstract":"We prove asymptotic lower bounds on the variance of the number of vertices and missed area of random disc-polygons in convex discs whose boundary is $C_+^2$ smooth. The established lower bounds are of the same order as the upper bounds proved previously by Fodor and V'{i}gh (2018).","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":"20 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87926544","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $p$ be an odd prime. Associated to a pair $(E, mathcal{F}_infty)$ consisting of a rational elliptic curve $E$ and a $p$-adic Lie extension $mathcal{F}_infty$ of $mathbb{Q}$, is the $p$-primary Selmer group $Sel_{p^infty}(E/mathcal{F}_infty)$ of $E$ over $mathcal{F}_infty$. In this paper, we study the arithmetic statistics for the algebraic structure of this Selmer group. The results provide insights into the asymptotics for the growth of Mordell--Weil ranks of elliptic curves in noncommutative towers.
设$p$为奇素数。与$mathbb{Q}$的有理椭圆曲线$E$和$p$ -adic Lie扩展$mathcal{F}_infty$组成的$(E, mathcal{F}_infty)$对相关联的是$E$的$Sel_{p^infty}(E/mathcal{F}_infty)$ -primary Selmer群$p$ over $mathcal{F}_infty$。本文研究了这类Selmer群的代数结构的算术统计。结果提供了非交换塔中椭圆曲线的莫德尔-韦尔秩增长的渐近性的见解。
{"title":"Arithmetic statistics and noncommutative Iwasawa theory","authors":"Debanjana Kundu, Antonio Lei, Anwesh Ray","doi":"10.25537/dm.2022v27","DOIUrl":"https://doi.org/10.25537/dm.2022v27","url":null,"abstract":"Let $p$ be an odd prime. Associated to a pair $(E, mathcal{F}_infty)$ consisting of a rational elliptic curve $E$ and a $p$-adic Lie extension $mathcal{F}_infty$ of $mathbb{Q}$, is the $p$-primary Selmer group $Sel_{p^infty}(E/mathcal{F}_infty)$ of $E$ over $mathcal{F}_infty$. In this paper, we study the arithmetic statistics for the algebraic structure of this Selmer group. The results provide insights into the asymptotics for the growth of Mordell--Weil ranks of elliptic curves in noncommutative towers.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":"29 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87169502","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study twisted vector bundles of infinite rank on gerbes, giving a new spin on Grothendieck's famous problem on the equality of the Brauer group and cohomological Brauer group. We show that the relaxed version of the question has an affirmative answer in many, but not all, cases, including for any algebraic space with the resolution property and any algebraic space obtained by pinching two closed subschemes of a projective scheme. We also discuss some possible theories of infinite rank Azumaya algebras, consider a new class of"very positive"infinite rank vector bundles on projective varieties, and show that an infinite rank vector bundle on a curve in a surface can be lifted to the surface away from finitely many points.
{"title":"Locally free twisted sheaves of infinite rank","authors":"A. Jong, Max Lieblich, Minseon Shin","doi":"10.4171/dm/909","DOIUrl":"https://doi.org/10.4171/dm/909","url":null,"abstract":"We study twisted vector bundles of infinite rank on gerbes, giving a new spin on Grothendieck's famous problem on the equality of the Brauer group and cohomological Brauer group. We show that the relaxed version of the question has an affirmative answer in many, but not all, cases, including for any algebraic space with the resolution property and any algebraic space obtained by pinching two closed subschemes of a projective scheme. We also discuss some possible theories of infinite rank Azumaya algebras, consider a new class of\"very positive\"infinite rank vector bundles on projective varieties, and show that an infinite rank vector bundle on a curve in a surface can be lifted to the surface away from finitely many points.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":"8 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90277336","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Multiple zeta values are the convergent iterated integrals from 0 to 1 of the differential forms ω0 = dt/t and ω1 = dt/(1− t). They form an algebra over Q, which has many interesting connections with different domains, including knot theory and perturbative quantum field theory [18, 11]. This algebra is expected to be graded by the weight, and a famous conjecture of Zagier [19] states that the dimensions of homogeneous components are given by the Padovan numbers. The algebra AMZV of motivic multiple zeta values is a more subtle construction, in the setting of periods and mixed motives [5, 6, 11]. It can be defined as the quotient of the commutative algebra A1,0, whose elements are seen as formal iterated integrals of ω0 and ω1, by the non-explicit ideal of all relations that can be proved using algebraic geometry. This algebra is known to be graded by the weight and its dimensions are given by the Padovan sequence, by results of Brown [5]. There is a surjective morphism, called the period map, from the motivic algebra AMZV to the usual algebra of multiple zeta values, defined by taking the numerical value of a formal iterated integral. This period map is expected to be injective, hence an isomorphism.
{"title":"Zinbiel algebras and multiple zeta values","authors":"F. Chapoton","doi":"10.4171/dm/876","DOIUrl":"https://doi.org/10.4171/dm/876","url":null,"abstract":"Multiple zeta values are the convergent iterated integrals from 0 to 1 of the differential forms ω0 = dt/t and ω1 = dt/(1− t). They form an algebra over Q, which has many interesting connections with different domains, including knot theory and perturbative quantum field theory [18, 11]. This algebra is expected to be graded by the weight, and a famous conjecture of Zagier [19] states that the dimensions of homogeneous components are given by the Padovan numbers. The algebra AMZV of motivic multiple zeta values is a more subtle construction, in the setting of periods and mixed motives [5, 6, 11]. It can be defined as the quotient of the commutative algebra A1,0, whose elements are seen as formal iterated integrals of ω0 and ω1, by the non-explicit ideal of all relations that can be proved using algebraic geometry. This algebra is known to be graded by the weight and its dimensions are given by the Padovan sequence, by results of Brown [5]. There is a surjective morphism, called the period map, from the motivic algebra AMZV to the usual algebra of multiple zeta values, defined by taking the numerical value of a formal iterated integral. This period map is expected to be injective, hence an isomorphism.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81721254","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-08-20DOI: 10.25537/dm.2022v27.1275-1297
Benjamin W. Passer
. Evert and Helton proved that real free spectrahedra are the matrix convex hulls of their absolute extreme points. However, this result does not extend to complex free spectrahedra, and we examine multiple ways in which the analogous result can fail. We also develop some local techniques to determine when matrix convex sets are not (duals of) free spectrahedra, as part of a continued study of minimal and maximal matrix convex sets and operator systems. These results apply to both the real and complex cases.
{"title":"Complex free spectrahedra, absolute extreme points, and dilations","authors":"Benjamin W. Passer","doi":"10.25537/dm.2022v27.1275-1297","DOIUrl":"https://doi.org/10.25537/dm.2022v27.1275-1297","url":null,"abstract":". Evert and Helton proved that real free spectrahedra are the matrix convex hulls of their absolute extreme points. However, this result does not extend to complex free spectrahedra, and we examine multiple ways in which the analogous result can fail. We also develop some local techniques to determine when matrix convex sets are not (duals of) free spectrahedra, as part of a continued study of minimal and maximal matrix convex sets and operator systems. These results apply to both the real and complex cases.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":"48 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76752620","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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