Compositio Mathematica最新文献

{"title":"Modular forms of half-integral weight on exceptional groups","authors":"Spencer Leslie, Aaron Pollack","doi":"10.1112/s0010437x23007686","DOIUrl":"https://doi.org/10.1112/s0010437x23007686","url":null,"abstract":"<p>We define a notion of modular forms of half-integral weight on the quaternionic exceptional groups. We prove that they have a well-behaved notion of Fourier coefficients, which are complex numbers defined up to multiplication by <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240221201522763-0676:S0010437X23007686:S0010437X23007686_inline1.png\"><span data-mathjax-type=\"texmath\"><span>${pm }1$</span></span></img></span></span>. We analyze the minimal modular form <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240221201522763-0676:S0010437X23007686:S0010437X23007686_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$Theta _{F_4}$</span></span></img></span></span> on the double cover of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240221201522763-0676:S0010437X23007686:S0010437X23007686_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$F_4$</span></span></img></span></span>, following Loke–Savin and Ginzburg. Using <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240221201522763-0676:S0010437X23007686:S0010437X23007686_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$Theta _{F_4}$</span></span></img></span></span>, we define a modular form of weight <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240221201522763-0676:S0010437X23007686:S0010437X23007686_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$tfrac {1}{2}$</span></span></img></span></span> on (the double cover of) <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240221201522763-0676:S0010437X23007686:S0010437X23007686_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$G_2$</span></span></img></span></span>. We prove that the Fourier coefficients of this modular form on <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240221201522763-0676:S0010437X23007686:S0010437X23007686_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$G_2$</span></span></img></span></span> see the <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240221201522763-0676:S0010437X23007686:S0010437X23007686_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$2$</span></span></img></span></span>-torsion in the narrow class groups of totally real cubic fields.</p>","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":"14 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139928034","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":"Stratification of the transverse momentum map","authors":"Maarten Mol","doi":"10.1112/s0010437x23007637","DOIUrl":"https://doi.org/10.1112/s0010437x23007637","url":null,"abstract":"<p>Given a Hamiltonian action of a proper symplectic groupoid (for instance, a Hamiltonian action of a compact Lie group), we show that the transverse momentum map admits a natural constant rank stratification. To this end, we construct a refinement of the canonical stratification associated to the Lie groupoid action (the orbit type stratification, in the case of a Hamiltonian Lie group action) that seems not to have appeared before, even in the literature on Hamiltonian Lie group actions. This refinement turns out to be compatible with the Poisson geometry of the Hamiltonian action: it is a Poisson stratification of the orbit space, each stratum of which is a regular Poisson manifold that admits a natural proper symplectic groupoid integrating it. The main tools in our proofs (which we believe could be of independent interest) are a version of the Marle–Guillemin–Sternberg normal form theorem for Hamiltonian actions of proper symplectic groupoids and a notion of equivalence between Hamiltonian actions of symplectic groupoids, closely related to Morita equivalence between symplectic groupoids.</p>","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":"280 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139772847","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":"Most odd-degree binary forms fail to primitively represent a square","authors":"Ashvin A. Swaminathan","doi":"10.1112/s0010437x23007649","DOIUrl":"https://doi.org/10.1112/s0010437x23007649","url":null,"abstract":"&lt;p&gt;Let &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240206110652704-0043:S0010437X23007649:S0010437X23007649_inline1.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$F$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; be a separable integral binary form of odd degree &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240206110652704-0043:S0010437X23007649:S0010437X23007649_inline2.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$N geq 5$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt;. A result of Darmon and Granville known as ‘Faltings plus epsilon’ implies that the degree-&lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240206110652704-0043:S0010437X23007649:S0010437X23007649_inline3.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$N$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; &lt;span&gt;superelliptic equation&lt;/span&gt; &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240206110652704-0043:S0010437X23007649:S0010437X23007649_inline4.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$y^2 = F(x,z)$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; has finitely many primitive integer solutions. In this paper, we consider the family &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240206110652704-0043:S0010437X23007649:S0010437X23007649_inline5.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$mathscr {F}_N(f_0)$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; of degree-&lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240206110652704-0043:S0010437X23007649:S0010437X23007649_inline6.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$N$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; superelliptic equations with fixed leading coefficient &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240206110652704-0043:S0010437X23007649:S0010437X23007649_inline7.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$f_0 in mathbb {Z} smallsetminus pm mathbb {Z}^2$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt;, ordered by height. For every sufficiently large &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240206110652704-0043:S0010437X23007649:S0010437X23007649_inline8.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$N$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt;, we prove that among equations in the family &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240206110652704-0043:S0010437X23007649:S0010437X23007649_inline9.png\"&gt;&lt;span data-mathjax-type","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":"17 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139772611","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":"On the Bezrukavnikov–Kaledin quantization of symplectic varieties in characteristic p","authors":"Ekaterina Bogdanova, Vadim Vologodsky","doi":"10.1112/s0010437x23007601","DOIUrl":"https://doi.org/10.1112/s0010437x23007601","url":null,"abstract":"<p>We prove that after inverting the Planck constant <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104174227480-0294:S0010437X23007601:S0010437X23007601_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$h$</span></span></img></span></span>, the Bezrukavnikov–Kaledin quantization <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104174227480-0294:S0010437X23007601:S0010437X23007601_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$(X, {mathcal {O}}_h)$</span></span></img></span></span> of symplectic variety <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104174227480-0294:S0010437X23007601:S0010437X23007601_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$X$</span></span></img></span></span> in characteristic <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104174227480-0294:S0010437X23007601:S0010437X23007601_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$p$</span></span></img></span></span> with <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104174227480-0294:S0010437X23007601:S0010437X23007601_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$H^2(X, {mathcal {O}}_X) =0$</span></span></img></span></span> is Morita equivalent to a certain central reduction of the algebra of differential operators on <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104174227480-0294:S0010437X23007601:S0010437X23007601_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$X$</span></span></img></span></span>.</p>","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":"30 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139105273","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":"Boundedness of the p-primary torsion of the Brauer group of an abelian variety","authors":"Marco D'Addezio","doi":"10.1112/s0010437x23007558","DOIUrl":"https://doi.org/10.1112/s0010437x23007558","url":null,"abstract":"<p>We prove that the <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104175731172-0793:S0010437X23007558:S0010437X23007558_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$p^infty$</span></span></img></span></span>-torsion of the transcendental Brauer group of an abelian variety over a finitely generated field of characteristic <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104175731172-0793:S0010437X23007558:S0010437X23007558_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$p&gt;0$</span></span></img></span></span> is bounded. This answers a (variant of a) question asked by Skorobogatov and Zarhin for abelian varieties. To do this, we prove a ‘flat Tate conjecture’ for divisors. We also study other geometric Galois-invariant <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104175731172-0793:S0010437X23007558:S0010437X23007558_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$p^infty$</span></span></img></span></span>-torsion classes of the Brauer group which are not in the transcendental Brauer group. These classes, in contrast with our main theorem, can be infinitely <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104175731172-0793:S0010437X23007558:S0010437X23007558_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$p$</span></span></img></span></span>-divisible. We explain how the existence of these <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104175731172-0793:S0010437X23007558:S0010437X23007558_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$p$</span></span></img></span></span>-divisible towers is naturally related to the failure of surjectivity of specialisation morphisms of Néron–Severi groups in characteristic <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104175731172-0793:S0010437X23007558:S0010437X23007558_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$p$</span></span></img></span></span>.</p>","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":"219 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139105046","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":"Sixfolds of generalized Kummer type and K3 surfaces","authors":"Salvatore Floccari","doi":"10.1112/s0010437x23007625","DOIUrl":"https://doi.org/10.1112/s0010437x23007625","url":null,"abstract":"&lt;p&gt;We prove that any hyper-Kähler sixfold &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104175655345-0074:S0010437X23007625:S0010437X23007625_inline1.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$K$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; of generalized Kummer type has a naturally associated manifold &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104175655345-0074:S0010437X23007625:S0010437X23007625_inline2.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$Y_K$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; of &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104175655345-0074:S0010437X23007625:S0010437X23007625_inline3.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$mathrm {K}3^{[3]}$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; type. It is obtained as crepant resolution of the quotient of &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104175655345-0074:S0010437X23007625:S0010437X23007625_inline4.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$K$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; by a group of symplectic involutions acting trivially on its second cohomology. When &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104175655345-0074:S0010437X23007625:S0010437X23007625_inline5.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$K$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; is projective, the variety &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104175655345-0074:S0010437X23007625:S0010437X23007625_inline6.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$Y_K$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; is birational to a moduli space of stable sheaves on a uniquely determined projective &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104175655345-0074:S0010437X23007625:S0010437X23007625_inline7.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$mathrm {K}3$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; surface &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104175655345-0074:S0010437X23007625:S0010437X23007625_inline8.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$S_K$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt;. As an application of this construction we show that the Kuga–Satake correspondence is algebraic for the K3 surfaces &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104175655345-0074:S0010437X23007625:S0010437X23007625_inline9.png\"&gt;&lt;span ","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":"20 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139105320","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":"Isoparametric hypersurfaces in symmetric spaces of non-compact type and higher rank","authors":"Miguel Domínguez-Vázquez, Víctor Sanmartín-López","doi":"10.1112/s0010437x23007650","DOIUrl":"https://doi.org/10.1112/s0010437x23007650","url":null,"abstract":"<p>We construct inhomogeneous isoparametric families of hypersurfaces with non-austere focal set on each symmetric space of non-compact type and rank <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104173152677-0650:S0010437X23007650:S0010437X23007650_inline1.png\"><span data-mathjax-type=\"texmath\"><span>${geq }3$</span></span></img></span></span>. If the rank is <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104173152677-0650:S0010437X23007650:S0010437X23007650_inline2.png\"><span data-mathjax-type=\"texmath\"><span>${geq }4$</span></span></img></span></span>, there are infinitely many such examples. Our construction yields the first examples of isoparametric families on any Riemannian manifold known to have a non-austere focal set. They can be obtained from a new general extension method of submanifolds from Euclidean spaces to symmetric spaces of non-compact type. This method preserves the mean curvature and isoparametricity, among other geometric properties.</p>","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":"40 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139105001","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":"Dynamics on ℙ1: preperiodic points and pairwise stability","authors":"Laura DeMarco, Niki Myrto Mavraki","doi":"10.1112/s0010437x23007546","DOIUrl":"https://doi.org/10.1112/s0010437x23007546","url":null,"abstract":"&lt;p&gt;DeMarco, Krieger, and Ye conjectured that there is a uniform bound &lt;span&gt;B&lt;/span&gt;, depending only on the degree &lt;span&gt;d&lt;/span&gt;, so that any pair of holomorphic maps &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104180226992-0669:S0010437X23007546:S0010437X23007546_inline4.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$f, g :{mathbb {P}}^1to {mathbb {P}}^1$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; with degree &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104180226992-0669:S0010437X23007546:S0010437X23007546_inline5.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$d$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; will either share all of their preperiodic points or have at most &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104180226992-0669:S0010437X23007546:S0010437X23007546_inline6.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$B$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; in common. Here we show that this uniform bound holds for a Zariski open and dense set in the space of all pairs, &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104180226992-0669:S0010437X23007546:S0010437X23007546_inline7.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$mathrm {Rat}_d times mathrm {Rat}_d$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt;, for each degree &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104180226992-0669:S0010437X23007546:S0010437X23007546_inline8.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$dgeq 2$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt;. The proof involves a combination of arithmetic intersection theory and complex-dynamical results, especially as developed recently by Gauthier and Vigny, Yuan and Zhang, and Mavraki and Schmidt. In addition, we present alternate proofs of the main results of DeMarco, Krieger, and Ye [&lt;span&gt;Uniform Manin-Mumford for a family of genus 2 curves&lt;/span&gt;, Ann. of Math. (2) &lt;span&gt;191&lt;/span&gt; (2020), 949–1001; &lt;span&gt;Common preperiodic points for quadratic polynomials&lt;/span&gt;, J. Mod. Dyn. &lt;span&gt;18&lt;/span&gt; (2022), 363–413] and of Poineau [&lt;span&gt;Dynamique analytique sur&lt;/span&gt; &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104180226992-0669:S0010437X23007546:S0010437X23007546_inline9.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$mathbb {Z}$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; &lt;span&gt;II : Écart uniforme entre Lattès et conjecture de Bogomolov-Fu-Tschinkel&lt;/span&gt;, Preprint (2022), arXiv:2207.01574 [math.NT]]. In fact, we prove a generalization of a conjecture of Bogomolov, Fu, and Tschinkel in a mixed setting of dynamical systems and elliptic ","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":"2 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139105319","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 Grothendieck–Serre conjecture over valuation rings","authors":"Ning Guo","doi":"10.1112/s0010437x23007583","DOIUrl":"https://doi.org/10.1112/s0010437x23007583","url":null,"abstract":"<p>In this article, we establish the Grothendieck–Serre conjecture over valuation rings: for a reductive group scheme <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104180628938-0323:S0010437X23007583:S0010437X23007583_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$G$</span></span></img></span></span> over a valuation ring <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104180628938-0323:S0010437X23007583:S0010437X23007583_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$V$</span></span></img></span></span> with fraction field <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104180628938-0323:S0010437X23007583:S0010437X23007583_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$K$</span></span></img></span></span>, a <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104180628938-0323:S0010437X23007583:S0010437X23007583_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$G$</span></span></img></span></span>-torsor over <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104180628938-0323:S0010437X23007583:S0010437X23007583_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$V$</span></span></img></span></span> is trivial if it is trivial over <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104180628938-0323:S0010437X23007583:S0010437X23007583_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$K$</span></span></img></span></span>. This result is predicted by the original Grothendieck–Serre conjecture and the resolution of singularities. The novelty of our proof lies in overcoming subtleties brought by general nondiscrete valuation rings. By using flasque resolutions and inducting with local cohomology, we prove a non-Noetherian counterpart of Colliot-Thélène–Sansuc's case of tori. Then, taking advantage of techniques in algebraization, we obtain the passage to the Henselian rank-one case. Finally, we induct on Levi subgroups and use the integrality of rational points of anisotropic groups to reduce to the semisimple anisotropic case, in which we appeal to properties of parahoric subgroups in Bruhat–Tits theory to conclude. In the last section, by using extension properties of reflexive sheaves on formal power series over valuation rings and patching of torsors, we prove a variant of Nisnevich's purity conjecture.</p>","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":"34 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139105267","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":"COM volume 160 issue 1 Cover and Back matter","authors":"","doi":"10.1112/s0010437x23007741","DOIUrl":"https://doi.org/10.1112/s0010437x23007741","url":null,"abstract":"","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":" 2","pages":"b1 - b2"},"PeriodicalIF":1.8,"publicationDate":"2023-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138962941","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}