Let 0 0, once T is large enough. This was refined by Bagchi who showed that the measure of such t∈[0,T] is (c(e)+o(1))T, for all but at most countably many e>0. Using a completely different approach, we obtain the first effective version of Voronin's Theorem, by showing that in the rate of convergence one can save a small power of the logarithm of T. Our method is flexible, and can be generalized to other L-functions in the t-aspect, as well as to families of L-functions in the conductor aspect.
{"title":"An effective universality theorem for the Riemann zeta function","authors":"Youness Lamzouri, S. Lester, Maksym Radziwill","doi":"10.4171/CMH/448","DOIUrl":"https://doi.org/10.4171/CMH/448","url":null,"abstract":"Let 0 0, once T is large enough. This was refined by Bagchi who showed that the measure of such t∈[0,T] is (c(e)+o(1))T, for all but at most countably many e>0. Using a completely different approach, we obtain the first effective version of Voronin's Theorem, by showing that in the rate of convergence one can save a small power of the logarithm of T. Our method is flexible, and can be generalized to other L-functions in the t-aspect, as well as to families of L-functions in the conductor aspect.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2016-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/CMH/448","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70841795","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 construct some non-arithmetic ball quotients as branched covers of a quotient of an Abelian surface by a finite group, and compare them with lattices that previously appear in the literature. This gives an alternative construction, which is independent of the computer, of some lattices constructed by the author with Parker and Paupert.
{"title":"Non-arithmetic ball quotients from a configuration of elliptic curves in an Abelian surface","authors":"M. Deraux","doi":"10.4171/CMH/443","DOIUrl":"https://doi.org/10.4171/CMH/443","url":null,"abstract":"We construct some non-arithmetic ball quotients as branched covers of a quotient of an Abelian surface by a finite group, and compare them with lattices that previously appear in the literature. This gives an alternative construction, which is independent of the computer, of some lattices constructed by the author with Parker and Paupert.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":"5 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2016-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/CMH/443","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70841834","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 two related invariants of Lagrangian submanifolds in symplectic manifolds. For a Lagrangian torus these invariants are functions on the first cohomology of the torus. The first invariant is of topological nature and is related to the study of Lagrangian isotopies with a given Lagrangian flux. More specifically, it measures the length of straight paths in the first cohomology that can be realized as the Lagrangian flux of a Lagrangian isotopy. The second invariant is of analytical nature and comes from symplectic function theory. It is defined for Lagrangian submanifolds admitting fibrations over a circle and has a dynamical interpretation. We partially compute these invariants for certain Lagrangian tori.
{"title":"Lagrangian isotopies and symplectic function theory","authors":"Michael Entov, Y. Ganor, Cedric Membrez","doi":"10.4171/CMH/451","DOIUrl":"https://doi.org/10.4171/CMH/451","url":null,"abstract":"We study two related invariants of Lagrangian submanifolds in symplectic manifolds. For a Lagrangian torus these invariants are functions on the first cohomology of the torus. The first invariant is of topological nature and is related to the study of Lagrangian isotopies with a given Lagrangian flux. More specifically, it measures the length of straight paths in the first cohomology that can be realized as the Lagrangian flux of a Lagrangian isotopy. The second invariant is of analytical nature and comes from symplectic function theory. It is defined for Lagrangian submanifolds admitting fibrations over a circle and has a dynamical interpretation. We partially compute these invariants for certain Lagrangian tori.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2016-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/CMH/451","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70841763","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 pairs of Engel structures on four-manifolds whose intersection has constant rank one and which define the same even contact structure, but induce different orientations on it. We establish a correspondence between such pairs of Engel structures and a class of weakly hyperbolic flows. This correspondence is analogous to the correspondence between bi-contact structures and projectively or conformally Anosov flows on three-manifolds found by Eliashberg--Thurston and by Mitsumatsu.
{"title":"Engel structures and weakly hyperbolic flows on four-manifolds","authors":"D. Kotschick, T. Vogel","doi":"10.4171/CMH/441","DOIUrl":"https://doi.org/10.4171/CMH/441","url":null,"abstract":"We study pairs of Engel structures on four-manifolds whose intersection has constant rank one and which define the same even contact structure, but induce different orientations on it. We establish a correspondence between such pairs of Engel structures and a class of weakly hyperbolic flows. This correspondence is analogous to the correspondence between bi-contact structures and projectively or conformally Anosov flows on three-manifolds found by Eliashberg--Thurston and by Mitsumatsu.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2016-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/CMH/441","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70841055","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 address the problem to characterise closed type I subgroups of the automorphism group of a tree. Even in the well-studied case of Burger-Mozes' universal groups, non-type I criteria were unknown. We prove that a huge class of groups acting properly on trees are not of type I. In the case of Burger-Mozes groups, this yields a complete classification of type I groups among them. Our key novelty is the use of von Neumann algebraic techniques to prove the stronger statement that the group von Neumann algebra of the groups under consideration is non-amenable.
{"title":"Locally compact groups acting on trees, the type I conjecture and non-amenable von Neumann algebras","authors":"Cyril Houdayer, Sven Raum","doi":"10.4171/CMH/458","DOIUrl":"https://doi.org/10.4171/CMH/458","url":null,"abstract":"We address the problem to characterise closed type I subgroups of the automorphism group of a tree. Even in the well-studied case of Burger-Mozes' universal groups, non-type I criteria were unknown. We prove that a huge class of groups acting properly on trees are not of type I. In the case of Burger-Mozes groups, this yields a complete classification of type I groups among them. Our key novelty is the use of von Neumann algebraic techniques to prove the stronger statement that the group von Neumann algebra of the groups under consideration is non-amenable.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":"21 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2016-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/CMH/458","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70842439","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}
Given a $1$-cocycle $b$ with coefficients in an orthogonal representation, we show that any finite dimensional summand of $b$ is cohomologically trivial if and only if $| b(X_n) |^2/n$ tends to a constant in probability, where $X_n$ is the trajectory of the random walk $(G,mu)$. As a corollary, we obtain sufficient conditions for $G$ to satisfy Shalom's property $H_{mathrm{FD}}$. Another application is a convergence to a constant in probability of $mu^{*n}(e) -mu^{*n}(g)$, $ngg m$, normalized by its average with respect to $mu^{*m}$, for any finitely generated amenable group without infinite virtually Abelian quotients. Finally, we show that the harmonic equivariant mapping of $G$ to a Hilbert space obtained as an $U$-ultralimit of normalized $mu^{*n}- g mu^{*n}$ can depend on the ultrafilter $U$ for some groups.
给定一个系数为正交表示的$1$-环$b$，我们证明了$b$的任何有限维和当且仅当$| b(X_n) |^2/n$在概率上趋于常数，其中$X_n$是随机漫步$(G，mu)$的轨迹。作为推论，我们得到了$G$满足Shalom的性质$H_{ mathm {FD}}$的充分条件。另一个应用是收敛于一个概率常数$mu^{*n}(e) -mu^{*n}(g)$， $ngg m$，由其对$mu^{*m}$的平均值归一化，适用于任何有限生成的没有无限虚阿贝尔商的可服从群。最后，我们证明了$G$到Hilbert空间的调和等变映射作为归一化$mu^{*n}- G mu^{*n}$的$U$-超极限可以依赖于某些群的超滤波器$U$。
{"title":"Finite-dimensional representations constructed from random walks","authors":"A. Erschler, N. Ozawa","doi":"10.4171/CMH/444","DOIUrl":"https://doi.org/10.4171/CMH/444","url":null,"abstract":"Given a $1$-cocycle $b$ with coefficients in an orthogonal representation, we show that any finite dimensional summand of $b$ is cohomologically trivial if and only if $| b(X_n) |^2/n$ tends to a constant in probability, where $X_n$ is the trajectory of the random walk $(G,mu)$. As a corollary, we obtain sufficient conditions for $G$ to satisfy Shalom's property $H_{mathrm{FD}}$. Another application is a convergence to a constant in probability of $mu^{*n}(e) -mu^{*n}(g)$, $ngg m$, normalized by its average with respect to $mu^{*m}$, for any finitely generated amenable group without infinite virtually Abelian quotients. Finally, we show that the harmonic equivariant mapping of $G$ to a Hilbert space obtained as an $U$-ultralimit of normalized $mu^{*n}- g mu^{*n}$ can depend on the ultrafilter $U$ for some groups.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2016-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/CMH/444","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70841904","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 consider the problem of defining and computing real analogs of polynomial Hurwitz numbers, in other words, the problem of counting properly normalized real polynomials with fixed ramification profiles over real branch points. We show that, provided the polynomials are counted with an appropriate sign, their number does not depend on the order of the branch points on the real line. We study generating series for the invariants thus obtained, determine necessary and sufficient conditions for the vanishing and nonvanishing of these generating series, and obtain a logarithmic asymptotic for the invariants as the degree of the polynomials tends to infinity.
{"title":"Hurwitz numbers for real polynomials","authors":"I. Itenberg, D. Zvonkine","doi":"10.4171/CMH/440","DOIUrl":"https://doi.org/10.4171/CMH/440","url":null,"abstract":"We consider the problem of defining and computing real analogs of polynomial Hurwitz numbers, in other words, the problem of counting properly normalized real polynomials with fixed ramification profiles over real branch points. We show that, provided the polynomials are counted with an appropriate sign, their number does not depend on the order of the branch points on the real line. We study generating series for the invariants thus obtained, determine necessary and sufficient conditions for the vanishing and nonvanishing of these generating series, and obtain a logarithmic asymptotic for the invariants as the degree of the polynomials tends to infinity.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2016-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/CMH/440","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70841386","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}
Assuming two deep but standard conjectures from the Langlands Programme, we prove that the asymptotic Fermat's Last Theorem holds for imaginary quadratic fields Q(sqrt{-d}) with -d=2, 3 mod 4. For a general number field K, again assuming standard conjectures, we give a criterion based on the solutions to a certain S-unit equation, which if satisfied implies the asymptotic Fermat's Last Theorem.
假设朗兰兹纲领中的两个深刻而标准的猜想，我们证明了渐近费马大定理对于虚二次域Q(sqrt{d}) (d= 2,3 mod 4)成立。对于一般的数域K，同样在假设标准猜想的情况下，我们给出了一个基于s -单位方程解的判据，如果该判据满足，则暗示了渐近费马大定理。
{"title":"On the asymptotic Fermat’s last theorem over number fields","authors":"Mehmet Haluk Sengun, S. Siksek","doi":"10.4171/CMH/437","DOIUrl":"https://doi.org/10.4171/CMH/437","url":null,"abstract":"Assuming two deep but standard conjectures from the Langlands Programme, we prove that the asymptotic Fermat's Last Theorem holds for imaginary quadratic fields Q(sqrt{-d}) with -d=2, 3 mod 4. For a general number field K, again assuming standard conjectures, we give a criterion based on the solutions to a certain S-unit equation, which if satisfied implies the asymptotic Fermat's Last Theorem.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2016-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/CMH/437","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70841408","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}
Emmanuel Kowalski and William Sawin proved, using a deep independence result of Kloosterman sheaves, that the polygonal paths joining the partial sums of the normalized classical Kloosterman sums S(a,b0;p)/p^{1/2} converge in the sense of finite distributions to a specific random Fourier series, as a varies over (Z/pZ)^*, b0 is fixed in (Z/pz)* and p tends to infinity among the odd prime numbers. This article considers the case of S(a,b0;p^n)/p^{n/2}, as a varies over (Z/p^nZ)^*, b0 is fixed in (Z/p^nZ)^*, p tends to infinity among the odd prime numbers and n>=2 is a fixed integer. A convergence in law in the Banach space of complex-valued continuous function on [0,1] is also established, as (a,b) varies over (Z/p^nZ)*.(Z/p^nZ)*, p tends to infinity among the odd prime numbers and n>=2 is a fixed integer. This is the analogue of the result obtained by Emmanuel Kowalski and William Sawin in the prime moduli case.
{"title":"Kloosterman paths of prime powers moduli","authors":"G. Ricotta, E. Royer","doi":"10.4171/cmh/442","DOIUrl":"https://doi.org/10.4171/cmh/442","url":null,"abstract":"Emmanuel Kowalski and William Sawin proved, using a deep independence result of Kloosterman sheaves, that the polygonal paths joining the partial sums of the normalized classical Kloosterman sums S(a,b0;p)/p^{1/2} converge in the sense of finite distributions to a specific random Fourier series, as a varies over (Z/pZ)^*, b0 is fixed in (Z/pz)* and p tends to infinity among the odd prime numbers. This article considers the case of S(a,b0;p^n)/p^{n/2}, as a varies over (Z/p^nZ)^*, b0 is fixed in (Z/p^nZ)^*, p tends to infinity among the odd prime numbers and n>=2 is a fixed integer. A convergence in law in the Banach space of complex-valued continuous function on [0,1] is also established, as (a,b) varies over (Z/p^nZ)*.(Z/p^nZ)*, p tends to infinity among the odd prime numbers and n>=2 is a fixed integer. This is the analogue of the result obtained by Emmanuel Kowalski and William Sawin in the prime moduli case.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2016-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/cmh/442","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70841943","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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