Pub Date : 2023-11-29DOI: 10.1353/ajm.2023.a913295
Luiz Gustavo Farah, Justin Holmer, Svetlana Roudenko, Kai Yang
Abstract:
We consider the quadratic Zakharov-Kuznetsov equation $$partial_t u + partial_x Delta u + partial_x u^2=0$$ on $Bbb{R}^3$. A solitary wave solution is given by $Q(x-t,y,z)$, where $Q$ is the ground state solution to $-Q+Delta Q+Q^2=0$. We prove the asymptotic stability of these solitary wave solutions. Specifically, we show that initial data close to $Q$ in the energy space, evolves to a solution that, as $ttoinfty$, converges to a rescaling and shift of $Q(x-t,y,z)$ in $L^2$ in a rightward shifting region $x>delta t-tanthetasqrt{y^2+z^2}$ for $0leqthetaleq{piover 3}-delta$.
摘要:考虑$Bbb{R}^3$上的二次Zakharov-Kuznetsov方程$$partial_t u + partial_x Delta u + partial_x u^2=0$$。孤波解由$Q(x-t,y,z)$给出，其中$Q$是$-Q+Delta Q+Q^2=0$的基态解。我们证明了这些孤立波解的渐近稳定性。具体来说，我们表明，在能量空间中接近$Q$的初始数据演变为一个解决方案，作为$ttoinfty$，收敛于在$0leqthetaleq{piover 3}-delta$的右移区域$x>delta t-tanthetasqrt{y^2+z^2}$中重新缩放和移动$L^2$中的$Q(x-t,y,z)$。
{"title":"Asymptotic stability of solitary waves of the 3D quadratic Zakharov-Kuznetsov equation","authors":"Luiz Gustavo Farah, Justin Holmer, Svetlana Roudenko, Kai Yang","doi":"10.1353/ajm.2023.a913295","DOIUrl":"https://doi.org/10.1353/ajm.2023.a913295","url":null,"abstract":"<p><p>Abstract:</p><p>We consider the quadratic Zakharov-Kuznetsov equation $$partial_t u + partial_x Delta u + partial_x u^2=0$$ on $Bbb{R}^3$. A solitary wave solution is given by $Q(x-t,y,z)$, where $Q$ is the ground state solution to $-Q+Delta Q+Q^2=0$. We prove the asymptotic stability of these solitary wave solutions. Specifically, we show that initial data close to $Q$ in the energy space, evolves to a solution that, as $ttoinfty$, converges to a rescaling and shift of $Q(x-t,y,z)$ in $L^2$ in a rightward shifting region $x>delta t-tanthetasqrt{y^2+z^2}$ for $0leqthetaleq{piover 3}-delta$.</p></p>","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138508592","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 : 2023-11-29DOI: 10.1353/ajm.2023.a913294
Chandrashekhar Khare, Niccoló Ronchetti
Abstract:
We study the derived Hecke action at $p$ on the ordinary $p$-adic cohomology of arithmetic subgroups of reductive groups ${rm G}(Bbb{Q})$. This is the analog at $ell=p$ of derived Hecke actions studied by Venkatesh in the tame case, and is the derived analog of Hida's theory for ordinary Hecke algebras. We show that properties of the derived Hecke action at $p$ are related to deep conjectures in Galois cohomology which are higher analogs of the classical Leopoldt conjecture.
摘要研究了约化群${rm G}(Bbb{Q})$的算术子群$p$-进上同调上$p$处的推导Hecke作用。这是Venkatesh在温顺的情况下研究的推导出的Hecke作用在$ well =p$处的类比，也是Hida的理论对普通Hecke代数的推导出的类比。我们证明了在$p$处推导出的Hecke作用的性质与伽罗瓦上同调中的深度猜想有关，这是经典利奥波德猜想的高级类比。
{"title":"Derived Hecke action at p and the ordinary p-adic cohomology of arithmetic manifolds","authors":"Chandrashekhar Khare, Niccoló Ronchetti","doi":"10.1353/ajm.2023.a913294","DOIUrl":"https://doi.org/10.1353/ajm.2023.a913294","url":null,"abstract":"<p><p>Abstract:</p><p>We study the derived Hecke action at $p$ on the ordinary $p$-adic cohomology of arithmetic subgroups of reductive groups ${rm G}(Bbb{Q})$. This is the analog at $ell=p$ of derived Hecke actions studied by Venkatesh in the tame case, and is the derived analog of Hida's theory for ordinary Hecke algebras. We show that properties of the derived Hecke action at $p$ are related to deep conjectures in Galois cohomology which are higher analogs of the classical Leopoldt conjecture.</p></p>","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138508599","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 : 2023-10-01DOI: 10.1353/ajm.2023.a907704
Scott Ahlgren, Olivia Beckwith, Martin Raum
abstract: The arithmetic properties of the ordinary partition function $p(n)$ have been the topic of intensive study for the past century. Ramanujan proved that there are linear congruences of the form $p(ell n+beta)equiv 0$ $({rm mod};ell)$ for the primes $ell=5,7,11$, and it is known that there are no others of this form. On the other hand, for every prime $ellgeq 5$ there are infinitely many examples of congruences of the form $p(ell Q^m n+beta)equiv 0$ $({rm mod};ell)$ where $Qgeq 5$ is prime and $mgeq 3$. This leaves open the question of the existence of such congruences when $m=1$ or $m=2$ (no examples in these cases are known). We prove in a precise sense that such congruences, if they exist, are exceedingly scarce. Our methods involve a careful study of modular forms of half integral weight on the full modular group which are related to the partition function. Among many other tools, we use work of Radu which describes expansions of such modular forms along square classes at cusps of the modular curve $X(ell Q)$, Galois representations and the arithmetic large sieve.
{"title":"Scarcity of congruences for the partition function","authors":"Scott Ahlgren, Olivia Beckwith, Martin Raum","doi":"10.1353/ajm.2023.a907704","DOIUrl":"https://doi.org/10.1353/ajm.2023.a907704","url":null,"abstract":"abstract: The arithmetic properties of the ordinary partition function $p(n)$ have been the topic of intensive study for the past century. Ramanujan proved that there are linear congruences of the form $p(ell n+beta)equiv 0$ $({rm mod};ell)$ for the primes $ell=5,7,11$, and it is known that there are no others of this form. On the other hand, for every prime $ellgeq 5$ there are infinitely many examples of congruences of the form $p(ell Q^m n+beta)equiv 0$ $({rm mod};ell)$ where $Qgeq 5$ is prime and $mgeq 3$. This leaves open the question of the existence of such congruences when $m=1$ or $m=2$ (no examples in these cases are known). We prove in a precise sense that such congruences, if they exist, are exceedingly scarce. Our methods involve a careful study of modular forms of half integral weight on the full modular group which are related to the partition function. Among many other tools, we use work of Radu which describes expansions of such modular forms along square classes at cusps of the modular curve $X(ell Q)$, Galois representations and the arithmetic large sieve.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135996630","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}
{"title":"The Sn -equivariant top weight Euler characteristic of Mg,n","authors":"Melody Chan, Carel Faber, Søren Galatius, Sam Payne","doi":"10.1353/ajm.2023.a907705","DOIUrl":"https://doi.org/10.1353/ajm.2023.a907705","url":null,"abstract":"abstract: We prove a formula, conjectured by Zagier, for the $S_n$-equivariant Euler characteristic of the top weight cohomology of $scr{M}_{g,n}$.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135324817","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 : 2023-10-01DOI: 10.1353/ajm.2023.a907702
Takeshi Saito
abstract: We prove that the graded quotients of the filtration by ramification groups of any henselian discrete valuation field of residue characteristic $p>0$ are ${bf F}_p$-vector spaces. We define an injection of the character group of each graded quotient to a twisted cotangent space defined using a cotangent complex.
{"title":"Graded quotients of ramification groups of local fields with imperfect residue fields","authors":"Takeshi Saito","doi":"10.1353/ajm.2023.a907702","DOIUrl":"https://doi.org/10.1353/ajm.2023.a907702","url":null,"abstract":"abstract: We prove that the graded quotients of the filtration by ramification groups of any henselian discrete valuation field of residue characteristic $p>0$ are ${bf F}_p$-vector spaces. We define an injection of the character group of each graded quotient to a twisted cotangent space defined using a cotangent complex.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135324818","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 : 2023-10-01DOI: 10.1353/ajm.2023.a907701
Emmanuel Lecouturier
abstract: In 1995, Ehud de Shalit proved an analogue of a conjecture of Mazur-Tate for the modular Jacobian $J_0(p)$. His main result was valid away from the Eisenstein primes. We complete the work of de Shalit by including the Eisenstein primes, and give some applications such as an elementary combinatorial identity involving discrete logarithms of difference of supersingular $j$-invariants. An important tool is our recent work on the so called ``generalized cuspidal $1$-motive''.
1995年，Ehud de Shalit证明了模雅可比矩阵$J_0(p)$的Mazur-Tate猜想的一个类似。他的主要结果在不考虑爱森斯坦素数的情况下是有效的。我们通过引入爱森斯坦素数完成了de Shalit的工作，并给出了一些应用，如涉及超奇异$j$不变量差分离散对数的初等组合恒等式。一个重要的工具是我们最近关于所谓的“广义逆$1$动机”的研究。
{"title":"On the Mazur-Tate conjecture for prime conductor and Mazur's Eisenstein ideal","authors":"Emmanuel Lecouturier","doi":"10.1353/ajm.2023.a907701","DOIUrl":"https://doi.org/10.1353/ajm.2023.a907701","url":null,"abstract":"abstract: In 1995, Ehud de Shalit proved an analogue of a conjecture of Mazur-Tate for the modular Jacobian $J_0(p)$. His main result was valid away from the Eisenstein primes. We complete the work of de Shalit by including the Eisenstein primes, and give some applications such as an elementary combinatorial identity involving discrete logarithms of difference of supersingular $j$-invariants. An important tool is our recent work on the so called ``generalized cuspidal $1$-motive''.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135948525","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 : 2023-10-01DOI: 10.1353/ajm.2023.a907706
Louis Ioos, David Kazhdan, Leonid Polterovich
abstract: We introduce the notion of almost representations of Lie algebras and quantum tori, and establish an Ulam-stability type phenomenon: every irreducible almost representation is close to a genuine irreducible representation. As an application, we prove that geometric quantizations of the two-dimensional sphere and the two-dimensional torus are conjugate in the semi-classical limit up to a small error.
{"title":"Almost representations of algebras and quantization","authors":"Louis Ioos, David Kazhdan, Leonid Polterovich","doi":"10.1353/ajm.2023.a907706","DOIUrl":"https://doi.org/10.1353/ajm.2023.a907706","url":null,"abstract":"abstract: We introduce the notion of almost representations of Lie algebras and quantum tori, and establish an Ulam-stability type phenomenon: every irreducible almost representation is close to a genuine irreducible representation. As an application, we prove that geometric quantizations of the two-dimensional sphere and the two-dimensional torus are conjugate in the semi-classical limit up to a small error.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135996629","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 : 2023-10-01DOI: 10.1353/ajm.2023.a907703
Erwan Faou, Pierre Raphaël
abstract: We introduce specific solutions to the linear harmonic oscillator, named bubbles. They form resonant families of invariant tori of the linear dynamics, with arbitrarily large Sobolev norms. We use these modulated bubbles of energy to construct a class of potentials which are real, smooth, time dependent and uniformly decaying to zero with respect to time, such that the corresponding perturbed quantum harmonic oscillator admits solutions which exhibit a logarithmic growth of Sobolev norms. The resonance mechanism is explicit in space variables and produces highly oscillatory solutions. We then give several recipes to construct similar examples using more specific tools based on the continuous resonant (CR) equation in dimension two.
{"title":"On weakly turbulent solutions to the perturbed linear harmonic oscillator","authors":"Erwan Faou, Pierre Raphaël","doi":"10.1353/ajm.2023.a907703","DOIUrl":"https://doi.org/10.1353/ajm.2023.a907703","url":null,"abstract":"abstract: We introduce specific solutions to the linear harmonic oscillator, named bubbles. They form resonant families of invariant tori of the linear dynamics, with arbitrarily large Sobolev norms. We use these modulated bubbles of energy to construct a class of potentials which are real, smooth, time dependent and uniformly decaying to zero with respect to time, such that the corresponding perturbed quantum harmonic oscillator admits solutions which exhibit a logarithmic growth of Sobolev norms. The resonance mechanism is explicit in space variables and produces highly oscillatory solutions. We then give several recipes to construct similar examples using more specific tools based on the continuous resonant (CR) equation in dimension two.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136119316","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 : 2023-10-01DOI: 10.1353/ajm.2023.a907700
Camille Horbez, Jingyin Huang, Jean Lécureux
abstract: Proper proximality of a countable group is a notion that was introduced by Boutonnet, Ioana and Peterson as a tool to study rigidity properties of certain von Neumann algebras associated to groups or ergodic group actions. In the present paper, we establish the proper proximality of many groups acting on nonpositively curved spaces. First, these include many countable groups $G$ acting properly nonelementarily by isometries on a proper ${rm CAT}(0)$ space $X$. More precisely, proper proximality holds in the presence of rank one isometries or when $X$ is a locally thick affine building with a minimal $G$-action. As a consequence of Rank Rigidity, we derive the proper proximality of all countable nonelementary ${rm CAT}(0)$ cubical groups, and of all countable groups acting properly cocompactly nonelementarily by isometries on either a Hadamard manifold with no Euclidean factor, or on a $2$-dimensional piecewise Euclidean ${rm CAT}(0)$ simplicial complex. Second, we establish the proper proximality of many hierarchically hyperbolic groups. These include the mapping class groups of connected orientable finite-type boundaryless surfaces (apart from a few low-complexity cases), thus answering a question raised by Boutonnet, Ioana, and Peterson. We also prove the proper proximality of all subgroups acting nonelementarily on the curve graph. In view of work of Boutonnet, Ioana and Peterson, our results have applications to structural and rigidity results for von Neumann algebras associated to all the above groups and their ergodic actions.
{"title":"Proper proximality in non-positive curvature","authors":"Camille Horbez, Jingyin Huang, Jean Lécureux","doi":"10.1353/ajm.2023.a907700","DOIUrl":"https://doi.org/10.1353/ajm.2023.a907700","url":null,"abstract":"abstract: Proper proximality of a countable group is a notion that was introduced by Boutonnet, Ioana and Peterson as a tool to study rigidity properties of certain von Neumann algebras associated to groups or ergodic group actions. In the present paper, we establish the proper proximality of many groups acting on nonpositively curved spaces. First, these include many countable groups $G$ acting properly nonelementarily by isometries on a proper ${rm CAT}(0)$ space $X$. More precisely, proper proximality holds in the presence of rank one isometries or when $X$ is a locally thick affine building with a minimal $G$-action. As a consequence of Rank Rigidity, we derive the proper proximality of all countable nonelementary ${rm CAT}(0)$ cubical groups, and of all countable groups acting properly cocompactly nonelementarily by isometries on either a Hadamard manifold with no Euclidean factor, or on a $2$-dimensional piecewise Euclidean ${rm CAT}(0)$ simplicial complex. Second, we establish the proper proximality of many hierarchically hyperbolic groups. These include the mapping class groups of connected orientable finite-type boundaryless surfaces (apart from a few low-complexity cases), thus answering a question raised by Boutonnet, Ioana, and Peterson. We also prove the proper proximality of all subgroups acting nonelementarily on the curve graph. In view of work of Boutonnet, Ioana and Peterson, our results have applications to structural and rigidity results for von Neumann algebras associated to all the above groups and their ergodic actions.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135948778","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 : 2023-05-31DOI: 10.1353/ajm.2023.a897497
Baiying Liu, Bin Xu
abstract:We study certain automorphic descent constructions for symplectic groups, and obtain results related to branching problems of automorphic representations. As a byproduct of the construction, based on the knowledge of the global Vogan packets for ${rm Mp}_2(Bbb{A})$, we give a new approach to prove the result that for an automorphic cuspidal representation of ${rm GL}_2(Bbb{A})$ of symplectic type, if there exists a quadratic twist with positive root number, then there exist quadratic twists with non-zero central $L$-values.
{"title":"Automorphic descent for symplectic groups: The branching problems and L-functions","authors":"Baiying Liu, Bin Xu","doi":"10.1353/ajm.2023.a897497","DOIUrl":"https://doi.org/10.1353/ajm.2023.a897497","url":null,"abstract":"abstract:We study certain automorphic descent constructions for symplectic groups, and obtain results related to branching problems of automorphic representations. As a byproduct of the construction, based on the knowledge of the global Vogan packets for ${rm Mp}_2(Bbb{A})$, we give a new approach to prove the result that for an automorphic cuspidal representation of ${rm GL}_2(Bbb{A})$ of symplectic type, if there exists a quadratic twist with positive root number, then there exist quadratic twists with non-zero central $L$-values.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45855106","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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