S. A. Altug, A. Shankar, Ila Varma, Kevin H. Wilson
We consider families of quartic number fields whose normal closures over Q have Galois group isomorphic to D4, the symmetries of a square. To any such field L, one can associate the Artin conductor of the corresponding 2-dimensional irreducible Galois representation with image D4. We determine the asymptotic number of such D4-quartic fields ordered by conductor, and compute the leading term explicitly as a mass formula, verifying heuristics of Kedlaya and Wood. Additionally, we are able to impose any local splitting conditions at any finite number of primes (sometimes, at an infinite number of primes), and as a consequence, we also compute the asymptotic number of order 4 elements in class groups and narrow class groups of quadratic fields ordered by discriminant. Traditionally, there have been two approaches to counting quartic fields, using arithmetic invariant theory in combination with geometry-of-number techniques, and applying Kummer theory together with L-function methods. Both of these strategies fall short in the case of D4-quartic fields ordered by conductor since counting quartic fields containing a quadratic subfield with large discriminant is difficult. However, when ordering by conductor, we utilize additional algebraic structure arising from the outer automorphism of D4 combined with both approaches mentioned above to obtain exact asymptotics.
{"title":"The number of $D_4$-fields ordered by conductor","authors":"S. A. Altug, A. Shankar, Ila Varma, Kevin H. Wilson","doi":"10.4171/JEMS/1070","DOIUrl":"https://doi.org/10.4171/JEMS/1070","url":null,"abstract":"We consider families of quartic number fields whose normal closures over Q have Galois group isomorphic to D4, the symmetries of a square. To any such field L, one can associate the Artin conductor of the corresponding 2-dimensional irreducible Galois representation with image D4. We determine the asymptotic number of such D4-quartic fields ordered by conductor, and compute the leading term explicitly as a mass formula, verifying heuristics of Kedlaya and Wood. Additionally, we are able to impose any local splitting conditions at any finite number of primes (sometimes, at an infinite number of primes), and as a consequence, we also compute the asymptotic number of order 4 elements in class groups and narrow class groups of quadratic fields ordered by discriminant. Traditionally, there have been two approaches to counting quartic fields, using arithmetic invariant theory in combination with geometry-of-number techniques, and applying Kummer theory together with L-function methods. Both of these strategies fall short in the case of D4-quartic fields ordered by conductor since counting quartic fields containing a quadratic subfield with large discriminant is difficult. However, when ordering by conductor, we utilize additional algebraic structure arising from the outer automorphism of D4 combined with both approaches mentioned above to obtain exact asymptotics.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":"223 1","pages":"2733-2785"},"PeriodicalIF":2.6,"publicationDate":"2021-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75816555","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}
We formulate a conjecture characterizing smooth projective varieties in positive characteristic whose Frobenius morphism can be lifted modulo $p^2$ - we expect that such varieties, after a finite 'etale cover, admit a toric fibration over an ordinary abelian variety. We prove that this assertion implies a conjecture of Occhetta and Wi'sniewski, which states that in characteristic zero a smooth image of a projective toric variety is a toric variety. To this end we analyse the behaviour of toric varieties in families showing some generization and specialization results. Furthermore, we prove a positive characteristic analogue of Winkelmann's theorem on varieties with trivial logarithmic tangent bundle (generalising a result of Mehta-Srinivas), and thus obtaining an important special case of our conjecture. Finally, using deformations of rational curves we verify our conjecture for homogeneous spaces, solving a problem posed by Buch-Thomsen-Lauritzen-Mehta.
{"title":"Global Frobenius liftability I","authors":"Piotr Achinger, J. Witaszek, Maciej Zdanowicz","doi":"10.4171/JEMS/1063","DOIUrl":"https://doi.org/10.4171/JEMS/1063","url":null,"abstract":"We formulate a conjecture characterizing smooth projective varieties in positive characteristic whose Frobenius morphism can be lifted modulo $p^2$ - we expect that such varieties, after a finite 'etale cover, admit a toric fibration over an ordinary abelian variety. We prove that this assertion implies a conjecture of Occhetta and Wi'sniewski, which states that in characteristic zero a smooth image of a projective toric variety is a toric variety. To this end we analyse the behaviour of toric varieties in families showing some generization and specialization results. Furthermore, we prove a positive characteristic analogue of Winkelmann's theorem on varieties with trivial logarithmic tangent bundle (generalising a result of Mehta-Srinivas), and thus obtaining an important special case of our conjecture. Finally, using deformations of rational curves we verify our conjecture for homogeneous spaces, solving a problem posed by Buch-Thomsen-Lauritzen-Mehta.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":"118 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2021-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81273535","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}
Existing negative results on invalidity of analogues of classical Density and Differentiation Theorems in infinite dimensional spaces are considerably strengthened by a construction of a Gaussian measure γ on a separable Hilbert space H for which the Density Theorem fails uniformly, i.e., there is a set M ⊂ H of positive γ-measure such that lim rց0 sup x∈H γ(B(x, r) ∩M) γB(x, r) = 0.
{"title":"A set of positive Gaussian measure with uniformly zero density everywhere.","authors":"D. Preiss, E. Riss, J. Tiser","doi":"10.4171/JEMS/1058","DOIUrl":"https://doi.org/10.4171/JEMS/1058","url":null,"abstract":"Existing negative results on invalidity of analogues of classical Density and Differentiation Theorems in infinite dimensional spaces are considerably strengthened by a construction of a Gaussian measure γ on a separable Hilbert space H for which the Density Theorem fails uniformly, i.e., there is a set M ⊂ H of positive γ-measure such that lim rց0 sup x∈H γ(B(x, r) ∩M) γB(x, r) = 0.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":"8 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2021-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75437082","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}
We prove a version of the strong Taylor's conjecture for stable graphs: if $G$ is a stable graph whose chromatic number is strictly greater than $beth_2(aleph_0)$ then $G$ contains all finite subgraphs of Sh$_n(omega)$ and thus has elementary extensions of unbounded chromatic number. This completes the picture from our previous work. The main new model theoretic ingredient is a generalization of the classical construction of Ehrenfeucht-Mostowski models to an infinitary setting, giving a new characterization of stability.
{"title":"Infinite stable graphs with large chromatic number II","authors":"Yatir Halevi, Itay Kaplan, S. Shelah","doi":"10.4171/jems/1352","DOIUrl":"https://doi.org/10.4171/jems/1352","url":null,"abstract":"We prove a version of the strong Taylor's conjecture for stable graphs: if $G$ is a stable graph whose chromatic number is strictly greater than $beth_2(aleph_0)$ then $G$ contains all finite subgraphs of Sh$_n(omega)$ and thus has elementary extensions of unbounded chromatic number. This completes the picture from our previous work. The main new model theoretic ingredient is a generalization of the classical construction of Ehrenfeucht-Mostowski models to an infinitary setting, giving a new characterization of stability.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":"104 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2021-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90647866","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}
The values of the so-called {em Dedekind--Rademacher cocycle} at certain real quadratic arguments are shown to be global $p$-units in the narrow Hilbert class field of the associated real quadratic field, as predicted by conjectures of Darmon, Dasgupta, and Vonk. The strategy for proving this result combines an approach of Darmon-Pozzi-Vonk with one crucial extra ingredient: the study of infinitesimal deformations of irregular Hilbert Eisenstein series of weight one in the anti-parallel direction, building on the techniques in earlier work of Betina, Dimitrov, and Pozzi.
{"title":"The values of the Dedekind–Rademacher cocycle at real multiplication points","authors":"H. Darmon, A. Pozzi, Jan Vonk","doi":"10.4171/jems/1344","DOIUrl":"https://doi.org/10.4171/jems/1344","url":null,"abstract":"The values of the so-called {em Dedekind--Rademacher cocycle} at certain real quadratic arguments are shown to be global $p$-units in the narrow Hilbert class field of the associated real quadratic field, as predicted by conjectures of Darmon, Dasgupta, and Vonk. The strategy for proving this result combines an approach of Darmon-Pozzi-Vonk with one crucial extra ingredient: the study of infinitesimal deformations of irregular Hilbert Eisenstein series of weight one in the anti-parallel direction, building on the techniques in earlier work of Betina, Dimitrov, and Pozzi.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":"424 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2021-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75770968","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}
We prove that TR is corepresentable by the reduced topological Hochschild homology of the flat affine line $mathbf{S}[t]$ as a functor defined on the $infty$-category of cyclotomic spectra with values in the $infty$-category of spectra with Frobenius lifts, refining a result of Blumberg-Mandell. We define the notion of an integral topological Cartier module using Barwick's formalism of spectral Mackey functors on orbital $infty$-categories, extending the work of Antieau-Nikolaus in the $p$-typical setting. As an application, we show that TR evaluated on a connective $mathbf{E}_1$-ring admits a description in terms of the spectrum of curves on algebraic K-theory generalizing the work of Hesselholt and Betley-Schlichtkrull.
{"title":"On curves in K-theory and TR","authors":"Jonas McCandless","doi":"10.4171/jems/1347","DOIUrl":"https://doi.org/10.4171/jems/1347","url":null,"abstract":"We prove that TR is corepresentable by the reduced topological Hochschild homology of the flat affine line $mathbf{S}[t]$ as a functor defined on the $infty$-category of cyclotomic spectra with values in the $infty$-category of spectra with Frobenius lifts, refining a result of Blumberg-Mandell. We define the notion of an integral topological Cartier module using Barwick's formalism of spectral Mackey functors on orbital $infty$-categories, extending the work of Antieau-Nikolaus in the $p$-typical setting. As an application, we show that TR evaluated on a connective $mathbf{E}_1$-ring admits a description in terms of the spectrum of curves on algebraic K-theory generalizing the work of Hesselholt and Betley-Schlichtkrull.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":"69 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2021-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82381844","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}
Discovered in the context of card shuffling by Aldous, Diaconis and Shahshahani, the cutoff phenomenon has since then been established in a variety of Markov chains. However, proving cutoff remains a delicate affair, which requires a detailed knowledge of the chain. Identifying the general mechanisms underlying this phase transition -- without having to pinpoint its precise location -- remains one of the most fundamental open problems in the area of mixing times. In the present paper, we make a step in this direction by establishing cutoff for Markov chains with non-negative curvature, under a suitably refined product condition. The result applies, in particular, to random walks on abelian Cayley expanders satisfying a mild degree condition, hence in particular to emph{almost all} abelian Cayley graphs. Our proof relies on a quantitative emph{entropic concentration principle}, which we believe to lie behind all cutoff phenomena.
{"title":"Cutoff for non-negatively curved Markov chains","authors":"J. Salez","doi":"10.4171/jems/1348","DOIUrl":"https://doi.org/10.4171/jems/1348","url":null,"abstract":"Discovered in the context of card shuffling by Aldous, Diaconis and Shahshahani, the cutoff phenomenon has since then been established in a variety of Markov chains. However, proving cutoff remains a delicate affair, which requires a detailed knowledge of the chain. Identifying the general mechanisms underlying this phase transition -- without having to pinpoint its precise location -- remains one of the most fundamental open problems in the area of mixing times. In the present paper, we make a step in this direction by establishing cutoff for Markov chains with non-negative curvature, under a suitably refined product condition. The result applies, in particular, to random walks on abelian Cayley expanders satisfying a mild degree condition, hence in particular to emph{almost all} abelian Cayley graphs. Our proof relies on a quantitative emph{entropic concentration principle}, which we believe to lie behind all cutoff phenomena.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":"22 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2021-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87782281","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}
Daniel Cristofaro-Gardiner, Vincent Humilière, Sobhan Seyfaddini
We resolve three longstanding questions related to the large scale geometry of the group of Hamiltonian diffeomorphisms of the two-sphere, equipped with Hofer's metric. Namely: (1) we resolve the Kapovich-Polterovich question by showing that this group is not quasi-isometric to the real line; (2) more generally, we show that the kernel of Calabi over any proper open subset is unbounded; and (3) we show that the group of area and orientation preserving homeomorphisms of the two-sphere is not a simple group. We also obtain, as a corollary, that the group of area-preserving diffeomorphisms of the open disc, equipped with an area-form of finite area, is not perfect. Central to all of our proofs are new sequences of spectral invariants over the two-sphere, defined via periodic Floer homology.
{"title":"PFH spectral invariants on the two-sphere and the large scale geometry of Hofer’s metric","authors":"Daniel Cristofaro-Gardiner, Vincent Humilière, Sobhan Seyfaddini","doi":"10.4171/jems/1351","DOIUrl":"https://doi.org/10.4171/jems/1351","url":null,"abstract":"We resolve three longstanding questions related to the large scale geometry of the group of Hamiltonian diffeomorphisms of the two-sphere, equipped with Hofer's metric. Namely: (1) we resolve the Kapovich-Polterovich question by showing that this group is not quasi-isometric to the real line; (2) more generally, we show that the kernel of Calabi over any proper open subset is unbounded; and (3) we show that the group of area and orientation preserving homeomorphisms of the two-sphere is not a simple group. We also obtain, as a corollary, that the group of area-preserving diffeomorphisms of the open disc, equipped with an area-form of finite area, is not perfect. Central to all of our proofs are new sequences of spectral invariants over the two-sphere, defined via periodic Floer homology.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":"36 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2021-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74088723","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}
Using quantization techniques, we show that the $delta$-invariant of Fujita-Odaka coincides with the optimal exponent in certain Moser-Trudinger type inequality. Consequently we obtain a uniform Yau-Tian-Donaldson theorem for the existence of twisted K"ahler-Einstein metrics with arbitrary polarizations. Our approach mainly uses pluripotential theory, which does not involve Cheeger-Colding-Tian theory or the non-Archimedean language. A new computable criterion for the existence of constant scalar curvature K"ahler metrics is also given.
{"title":"A quantization proof of the uniform Yau–Tian–Donaldson conjecture","authors":"Kewei Zhang","doi":"10.4171/jems/1373","DOIUrl":"https://doi.org/10.4171/jems/1373","url":null,"abstract":"Using quantization techniques, we show that the $delta$-invariant of Fujita-Odaka coincides with the optimal exponent in certain Moser-Trudinger type inequality. Consequently we obtain a uniform Yau-Tian-Donaldson theorem for the existence of twisted K\"ahler-Einstein metrics with arbitrary polarizations. Our approach mainly uses pluripotential theory, which does not involve Cheeger-Colding-Tian theory or the non-Archimedean language. A new computable criterion for the existence of constant scalar curvature K\"ahler metrics is also given.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":"1 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2021-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77777085","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":"Bounds for twists of GL(3) $L$-functions","authors":"Yongxiao Lin","doi":"10.4171/JEMS/1046","DOIUrl":"https://doi.org/10.4171/JEMS/1046","url":null,"abstract":"","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":"124 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2021-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82957522","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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