. For a class of Riemannian manifolds with boundary that includes all negatively curved manifolds with strictly convex boundary, we establish H¨older type stability estimates in the geometric inverse problem of determining the electric potential or the conformal factor from the Dirichlet-to-Neumann map associated with the Schr¨odinger equation and the wave equation. The novelty in this result lies in the fact that we allow some geodesics to be trapped inside the manifold and have infinite length.
{"title":"Stability estimates in inverse problems for the Schrödinger and wave equations with trapping","authors":"V'ictor Arnaiz, C. Guillarmou","doi":"10.4171/rmi/1327","DOIUrl":"https://doi.org/10.4171/rmi/1327","url":null,"abstract":". For a class of Riemannian manifolds with boundary that includes all negatively curved manifolds with strictly convex boundary, we establish H¨older type stability estimates in the geometric inverse problem of determining the electric potential or the conformal factor from the Dirichlet-to-Neumann map associated with the Schr¨odinger equation and the wave equation. The novelty in this result lies in the fact that we allow some geodesics to be trapped inside the manifold and have infinite length.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42785184","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}
For every finitely generated free group F , we construct an irreducible open 3-manifold MF whose end set is homeomorphic to a Cantor set, and with the end homogeneity group of MF isomorphic to F . The end homogeneity group is the group of all selfhomeomorphisms of the end set that extend to homeomorphisms of the entire 3-manifold. This extends an earlier result that constructs, for each finitely generated abelian group G, an irreducible open 3-manifold MG with end homogeneity group G. The method used in the proof of our main result also shows that if G is a group with a Cayley graph in R such that the graph automorphisms have certain nice extension properties, then there is an irreducible open 3-manifold MG with end homogeneity group G.
{"title":"Free groups as end homogeneity groups of 3-manifolds","authors":"D. Garity, Dušan D. Repovš","doi":"10.4171/RMI/1273","DOIUrl":"https://doi.org/10.4171/RMI/1273","url":null,"abstract":"For every finitely generated free group F , we construct an irreducible open 3-manifold MF whose end set is homeomorphic to a Cantor set, and with the end homogeneity group of MF isomorphic to F . The end homogeneity group is the group of all selfhomeomorphisms of the end set that extend to homeomorphisms of the entire 3-manifold. This extends an earlier result that constructs, for each finitely generated abelian group G, an irreducible open 3-manifold MG with end homogeneity group G. The method used in the proof of our main result also shows that if G is a group with a Cayley graph in R such that the graph automorphisms have certain nice extension properties, then there is an irreducible open 3-manifold MG with end homogeneity group G.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46348073","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 the integration theory of two-parameter controlled paths $Y$ allowing us to define integrals of the form begin{equation} int_{[s,t] times [u,v]} Y_{r,r'} ;d(X_{r}, X_{r'}) end{equation} where $X$ is the geometric $p$-rough path that controls $Y$. This extends to arbitrary regularity the definition presented for $2leq p<3$ in the recent paper of Hairer and Gerasimoviv{c}s where it is used in the proof of a version of H"{o}rmander's theorem for a class of SPDEs. We extend the Fubini type theorem of the same paper by showing that this two-parameter integral coincides with the two iterated one-parameter integrals [ int_{[s,t] times [u,v]} Y_{r,r'} ;d(X_{r}, X_{r'}) = int_{s}^{t} int_{u}^{v} Y_{r,r'} ;dX_{r'} ;dX_{r'} = int_{u}^{v} int_{s}^{t} Y_{r,r'} ;dX_{r} ;dX_{r'}. ] A priori these three integrals have distinct definitions, and so this parallels the classical Fubini's theorem for product measures. By extending the two-parameter Young-Towghi inequality in this context, we derive a maximal inequality for the discrete integrals approximating the two-parameter integral. We also extend the analysis to consider integrals of the form begin{equation*} int_{[s,t] times [u,v]} Y_{r,r'} ; d(X_{r}, tilde{X}_{r'}) end{equation*} for possibly different rough paths $X$ and $tilde{X}$, and obtain the corresponding Fubini type theorem. We prove continuity estimates for these integrals in the appropriate rough path topologies. As an application we consider the signature kernel, which has recently emerged as a useful tool in data science, as an example of a two-parameter controlled rough path which also solves a two-parameter rough integral equation.
{"title":"A Fubini type theorem for rough integration","authors":"T. Cass, J. Pei","doi":"10.4171/rmi/1409","DOIUrl":"https://doi.org/10.4171/rmi/1409","url":null,"abstract":"We develop the integration theory of two-parameter controlled paths $Y$ allowing us to define integrals of the form begin{equation} int_{[s,t] times [u,v]} Y_{r,r'} ;d(X_{r}, X_{r'}) end{equation} where $X$ is the geometric $p$-rough path that controls $Y$. This extends to arbitrary regularity the definition presented for $2leq p<3$ in the recent paper of Hairer and Gerasimoviv{c}s where it is used in the proof of a version of H\"{o}rmander's theorem for a class of SPDEs. We extend the Fubini type theorem of the same paper by showing that this two-parameter integral coincides with the two iterated one-parameter integrals [ int_{[s,t] times [u,v]} Y_{r,r'} ;d(X_{r}, X_{r'}) = int_{s}^{t} int_{u}^{v} Y_{r,r'} ;dX_{r'} ;dX_{r'} = int_{u}^{v} int_{s}^{t} Y_{r,r'} ;dX_{r} ;dX_{r'}. ] A priori these three integrals have distinct definitions, and so this parallels the classical Fubini's theorem for product measures. By extending the two-parameter Young-Towghi inequality in this context, we derive a maximal inequality for the discrete integrals approximating the two-parameter integral. We also extend the analysis to consider integrals of the form begin{equation*} int_{[s,t] times [u,v]} Y_{r,r'} ; d(X_{r}, tilde{X}_{r'}) end{equation*} for possibly different rough paths $X$ and $tilde{X}$, and obtain the corresponding Fubini type theorem. We prove continuity estimates for these integrals in the appropriate rough path topologies. As an application we consider the signature kernel, which has recently emerged as a useful tool in data science, as an example of a two-parameter controlled rough path which also solves a two-parameter rough integral equation.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46402375","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}
For any good tilting module T over a ring A, there exists an n-symmetric subcategory E of a module category such that the derived category of the endomorphism ring of T is a recollement of the derived categories of E and A in the sense of Beilinson-Bernstein-Deligne. Thus the kernel of the total left-derived tensor functor induced by the tilting module is triangle equivalent to the derived category of E .
{"title":"Symmetric subcategories, tilting modules, and derived recollements","authors":"Hongxing Chen, Changchang Xi","doi":"10.4171/rmi/1410","DOIUrl":"https://doi.org/10.4171/rmi/1410","url":null,"abstract":"For any good tilting module T over a ring A, there exists an n-symmetric subcategory E of a module category such that the derived category of the endomorphism ring of T is a recollement of the derived categories of E and A in the sense of Beilinson-Bernstein-Deligne. Thus the kernel of the total left-derived tensor functor induced by the tilting module is triangle equivalent to the derived category of E .","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47155929","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 D ⊂ R be a bounded, connected domain with smooth boundary and let −∆u = μ1u be the first nontrivial eigenfunction of the Laplace operator with Neumann boundary conditions. We prove ‖u‖L∞(D) ≤ 60 · ‖u‖L∞(∂D). This shows that the Hot Spots Conjecture cannot fail by an arbitrary factor. An example of Kleefeld shows that the optimal constant is at least 1 + 10−3.
{"title":"An upper bound on the hot spots constant","authors":"S. Steinerberger","doi":"10.4171/rmi/1350","DOIUrl":"https://doi.org/10.4171/rmi/1350","url":null,"abstract":"Let D ⊂ R be a bounded, connected domain with smooth boundary and let −∆u = μ1u be the first nontrivial eigenfunction of the Laplace operator with Neumann boundary conditions. We prove ‖u‖L∞(D) ≤ 60 · ‖u‖L∞(∂D). This shows that the Hot Spots Conjecture cannot fail by an arbitrary factor. An example of Kleefeld shows that the optimal constant is at least 1 + 10−3.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44472546","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 a classical theorem of Sarason that an analytic function of bounded mean oscillation ($BMOA$), is of vanishing mean oscillation if and only if its rotations converge in norm to the original function as the angle of the rotation tends to zero. In a series of two papers Blasco et al. have raised the problem of characterizing all semigroups of holomorphic functions $(varphi_t)$ that can replace the semigroup of rotations in Sarason's Theorem. We give a complete answer to this question, in terms of a logarithmic vanishing oscillation condition on the infinitesimal generator of the semigroup $(varphi_t)$. In addition we confirm the conjecture of Blasco et al. that all such semigroups are elliptic. We also investigate the analogous question for the Bloch and the little Bloch space and surprisingly enough we find that the semigroups for which the Bloch version of Sarason's Theorem holds are exactly the same as in the $BMOA$ case.
{"title":"Holomorphic semigroups and Sarason’s characterization of vanishing mean oscillation","authors":"Nikolaos Chalmoukis, Vassilis Daskalogiannis","doi":"10.4171/RMI/1346","DOIUrl":"https://doi.org/10.4171/RMI/1346","url":null,"abstract":"It is a classical theorem of Sarason that an analytic function of bounded mean oscillation ($BMOA$), is of vanishing mean oscillation if and only if its rotations converge in norm to the original function as the angle of the rotation tends to zero. In a series of two papers Blasco et al. have raised the problem of characterizing all semigroups of holomorphic functions $(varphi_t)$ that can replace the semigroup of rotations in Sarason's Theorem. We give a complete answer to this question, in terms of a logarithmic vanishing oscillation condition on the infinitesimal generator of the semigroup $(varphi_t)$. In addition we confirm the conjecture of Blasco et al. that all such semigroups are elliptic. We also investigate the analogous question for the Bloch and the little Bloch space and surprisingly enough we find that the semigroups for which the Bloch version of Sarason's Theorem holds are exactly the same as in the $BMOA$ case.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44084530","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 that a properly embedded annular end of a surface in H 2 × R with constant mean curvature 0 < H ≤ 12 can not be contained in any horizontal slab. Moreover, we show that a properly embedded surface with constant mean curvature 0 < H ≤ 12 contained in H 2 × [0 , + ∞ ) and with finite topology is necessarily a graph over a simply connected domain of H 2 . For the case H = 12 , the graph is entire. 2020 Mathematics Subject Classification: 53C30, 53A10.
我们证明了在H2×R中，具有常平均曲率0
{"title":"Slab theorem and halfspace theorem for constant mean curvature surfaces in $mathbb H^2timesmathbb R$","authors":"L. Hauswirth, Ana Menezes, Magdalena Rodríguez","doi":"10.4171/rmi/1372","DOIUrl":"https://doi.org/10.4171/rmi/1372","url":null,"abstract":"We prove that a properly embedded annular end of a surface in H 2 × R with constant mean curvature 0 < H ≤ 12 can not be contained in any horizontal slab. Moreover, we show that a properly embedded surface with constant mean curvature 0 < H ≤ 12 contained in H 2 × [0 , + ∞ ) and with finite topology is necessarily a graph over a simply connected domain of H 2 . For the case H = 12 , the graph is entire. 2020 Mathematics Subject Classification: 53C30, 53A10.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42303144","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}
Ruben Henrard, Sondre Kvamme, Adam-Christiaan van Roosmalen, Sven-Ake Wegner
Quasi-abelian categories are abundant in functional analysis and representation theory. It is known that a quasi-abelian category $mathcal{E}$ is a cotilting torsionfree class of an abelian category. In fact, this property characterizes quasi-abelian categories. This ambient abelian category is derived equivalent to the category $mathcal{E}$, and can be constructed as the heart $mathcal{LH}(mathcal{E})$ of a $operatorname{t}$-structure on the bounded derived category $operatorname{D^b}(mathcal{E})$ or as the localization of the category of monomorphisms in $mathcal{E}.$ However, there are natural examples of categories in functional analysis which are not quasi-abelian, but merely one-sided quasi-abelian or even weaker. Examples are the category of $operatorname{LB}$-spaces or the category of complete Hausdorff locally convex spaces. In this paper, we consider additive regular categories as a generalization of quasi-abelian categories that covers the aforementioned examples. These categories can be characterized as pre-torsionfree subcategories of abelian categories. As for quasi-abelian categories, we show that such an ambient abelian category of an additive regular category $mathcal{E}$ can be found as the heart of a $operatorname{t}$-structure on the bounded derived category $operatorname{D^b}(mathcal{E})$, or as the localization of the category of monomorphisms of $mathcal{E}$. In our proof of this last construction, we formulate and prove a version of Auslander's formula for additive regular categories. Whereas a quasi-abelian category is an exact category in a natural way, an additive regular category has a natural one-sided exact structure. Such a one-sided exact category can be 2-universally embedded into its exact hull. We show that the exact hull of an additive regular category is again an additive regular category.
{"title":"The left heart and exact hull of an additive regular category","authors":"Ruben Henrard, Sondre Kvamme, Adam-Christiaan van Roosmalen, Sven-Ake Wegner","doi":"10.4171/RMI/1388","DOIUrl":"https://doi.org/10.4171/RMI/1388","url":null,"abstract":"Quasi-abelian categories are abundant in functional analysis and representation theory. It is known that a quasi-abelian category $mathcal{E}$ is a cotilting torsionfree class of an abelian category. In fact, this property characterizes quasi-abelian categories. This ambient abelian category is derived equivalent to the category $mathcal{E}$, and can be constructed as the heart $mathcal{LH}(mathcal{E})$ of a $operatorname{t}$-structure on the bounded derived category $operatorname{D^b}(mathcal{E})$ or as the localization of the category of monomorphisms in $mathcal{E}.$ However, there are natural examples of categories in functional analysis which are not quasi-abelian, but merely one-sided quasi-abelian or even weaker. Examples are the category of $operatorname{LB}$-spaces or the category of complete Hausdorff locally convex spaces. In this paper, we consider additive regular categories as a generalization of quasi-abelian categories that covers the aforementioned examples. These categories can be characterized as pre-torsionfree subcategories of abelian categories. As for quasi-abelian categories, we show that such an ambient abelian category of an additive regular category $mathcal{E}$ can be found as the heart of a $operatorname{t}$-structure on the bounded derived category $operatorname{D^b}(mathcal{E})$, or as the localization of the category of monomorphisms of $mathcal{E}$. In our proof of this last construction, we formulate and prove a version of Auslander's formula for additive regular categories. Whereas a quasi-abelian category is an exact category in a natural way, an additive regular category has a natural one-sided exact structure. Such a one-sided exact category can be 2-universally embedded into its exact hull. We show that the exact hull of an additive regular category is again an additive regular category.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44532187","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 study removable sets for Newtonian Sobolev functions in metric measure spaces satisfying the usual (local) assumptions of a doubling measure and a Poincar'e inequality. In particular, when restricted to Euclidean spaces, a closed set $Esubset mathbf{R}^n$ with zero Lebesgue measure is shown to be removable for $W^{1,p}(mathbf{R}^n setminus E)$ if and only if $mathbf{R}^n setminus E$ supports a $p$-Poincar'e inequality as a metric space. When $p>1$, this recovers Koskela's result (Ark. Mat. 37 (1999), 291--304), but for $p=1$, as well as for metric spaces, it seems to be new. We also obtain the corresponding characterization for the Dirichlet spaces $L^{1,p}$. To be able to include $p=1$, we first study extensions of Newtonian Sobolev functions in the case $p=1$ from a noncomplete space $X$ to its completion $widehat{X}$. In these results, $p$-path almost open sets play an important role, and we provide a characterization of them by means of $p$-path open, $p$-quasiopen and $p$-finely open sets. We also show that there are nonmeasurable $p$-path almost open subsets of $mathbf{R}^n$, $n geq 2$, provided that the continuum hypothesis is assumed to be true. Furthermore, we extend earlier results about measurability of functions with $L^p$-integrable upper gradients, about $p$-quasiopen, $p$-path and $p$-finely open sets, and about Lebesgue points for $N^{1,1}$-functions, to spaces that only satisfy local assumptions.
{"title":"Removable sets for Newtonian Sobolev spaces and a characterization of $p$-path almost open sets","authors":"Anders Bjorn, Jana Bjorn, P. Lahti","doi":"10.4171/rmi/1419","DOIUrl":"https://doi.org/10.4171/rmi/1419","url":null,"abstract":"We study removable sets for Newtonian Sobolev functions in metric measure spaces satisfying the usual (local) assumptions of a doubling measure and a Poincar'e inequality. In particular, when restricted to Euclidean spaces, a closed set $Esubset mathbf{R}^n$ with zero Lebesgue measure is shown to be removable for $W^{1,p}(mathbf{R}^n setminus E)$ if and only if $mathbf{R}^n setminus E$ supports a $p$-Poincar'e inequality as a metric space. When $p>1$, this recovers Koskela's result (Ark. Mat. 37 (1999), 291--304), but for $p=1$, as well as for metric spaces, it seems to be new. We also obtain the corresponding characterization for the Dirichlet spaces $L^{1,p}$. To be able to include $p=1$, we first study extensions of Newtonian Sobolev functions in the case $p=1$ from a noncomplete space $X$ to its completion $widehat{X}$. In these results, $p$-path almost open sets play an important role, and we provide a characterization of them by means of $p$-path open, $p$-quasiopen and $p$-finely open sets. We also show that there are nonmeasurable $p$-path almost open subsets of $mathbf{R}^n$, $n geq 2$, provided that the continuum hypothesis is assumed to be true. Furthermore, we extend earlier results about measurability of functions with $L^p$-integrable upper gradients, about $p$-quasiopen, $p$-path and $p$-finely open sets, and about Lebesgue points for $N^{1,1}$-functions, to spaces that only satisfy local assumptions.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46391803","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}
Harris B. Daniels, 'Alvaro Lozano-Robledo, J. Morrow
Let $E/mathbb{Q}$ be an elliptic curve, let $overline{mathbb{Q}}$ be a fixed algebraic closure of $mathbb{Q}$, and let $G_{mathbb{Q}}=text{Gal}(overline{mathbb{Q}}/mathbb{Q})$ be the absolute Galois group of $mathbb{Q}$. The action of $G_{mathbb{Q}}$ on the adelic Tate module of $E$ induces the adelic Galois representation $rho_Ecolon G_{mathbb{Q}} to text{GL}(2,widehat{mathbb{Z}}).$ The goal of this paper is to explain how the image of $rho_E$ can be smaller than expected. To this end, we offer a group theoretic categorization of different ways in which an entanglement between division fields can be explained and prove several results on elliptic curves (and more generally, principally polarized abelian varieties) over $mathbb{Q}$ where the entanglement occurs over an abelian extension.
设$E/mathbb{Q}$为一条椭圆曲线，$overline{mathbb{Q}}$为$mathbb{Q}$的一个固定代数闭包，$G_{mathbb{Q}}=text{Gal}(overline{mathbb{Q}}/mathbb{Q})$为$mathbb{Q}$的绝对伽罗瓦群。$G_{mathbb{Q}}$对$E$的adelic Tate模块的作用诱导出adelic Galois表示$rho_Ecolon G_{mathbb{Q}} to text{GL}(2,widehat{mathbb{Z}}).$本文的目的是解释$rho_E$的图像如何比预期的小。为此，我们提供了一种不同的群论分类，其中分域之间的纠缠可以解释，并证明了在$mathbb{Q}$上的椭圆曲线(更一般地说，主要是极化阿贝尔变体)上的几个结果，其中纠缠发生在阿贝尔扩展上。
{"title":"Towards a classification of entanglements of Galois representations attached to elliptic curves","authors":"Harris B. Daniels, 'Alvaro Lozano-Robledo, J. Morrow","doi":"10.4171/rmi/1424","DOIUrl":"https://doi.org/10.4171/rmi/1424","url":null,"abstract":"Let $E/mathbb{Q}$ be an elliptic curve, let $overline{mathbb{Q}}$ be a fixed algebraic closure of $mathbb{Q}$, and let $G_{mathbb{Q}}=text{Gal}(overline{mathbb{Q}}/mathbb{Q})$ be the absolute Galois group of $mathbb{Q}$. The action of $G_{mathbb{Q}}$ on the adelic Tate module of $E$ induces the adelic Galois representation $rho_Ecolon G_{mathbb{Q}} to text{GL}(2,widehat{mathbb{Z}}).$ The goal of this paper is to explain how the image of $rho_E$ can be smaller than expected. To this end, we offer a group theoretic categorization of different ways in which an entanglement between division fields can be explained and prove several results on elliptic curves (and more generally, principally polarized abelian varieties) over $mathbb{Q}$ where the entanglement occurs over an abelian extension.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43287723","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}
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