Pub Date : 2023-12-18DOI: 10.1112/s0010437x20007526
François Charles, Giovanni Mongardi, Gianluca Pacienza
We study families of rational curves on irreducible holomorphic symplectic varieties. We give a necessary and sufficient condition for a sufficiently ample linear system on a holomorphic symplectic variety of $K3^{[n]}$-type to contain a uniruled divisor covered by rational curves of primitive class. In particular, for any fixed $n$, we show that there are only finitely many polarization types of holomorphic symplectic variety of $K3^{[n]}$-type that do not contain such a uniruled divisor. As an application, we provide a generalization of a result due to Beauville–Voisin on the Chow group of $0$-cycles on such varieties.
{"title":"Families of rational curves on holomorphic symplectic varieties and applications to 0-cycles","authors":"François Charles, Giovanni Mongardi, Gianluca Pacienza","doi":"10.1112/s0010437x20007526","DOIUrl":"https://doi.org/10.1112/s0010437x20007526","url":null,"abstract":"<p>We study families of rational curves on irreducible holomorphic symplectic varieties. We give a necessary and sufficient condition for a sufficiently ample linear system on a holomorphic symplectic variety of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231215111814759-0477:S0010437X20007526:S0010437X20007526_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$K3^{[n]}$</span></span></img></span></span>-type to contain a uniruled divisor covered by rational curves of primitive class. In particular, for any fixed <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231215111814759-0477:S0010437X20007526:S0010437X20007526_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$n$</span></span></img></span></span>, we show that there are only finitely many polarization types of holomorphic symplectic variety of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231215111814759-0477:S0010437X20007526:S0010437X20007526_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$K3^{[n]}$</span></span></img></span></span>-type that do not contain such a uniruled divisor. As an application, we provide a generalization of a result due to Beauville–Voisin on the Chow group of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231215111814759-0477:S0010437X20007526:S0010437X20007526_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$0$</span></span></img></span></span>-cycles on such varieties.</p>","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":"54 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138717385","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 : 2023-12-15DOI: 10.1112/s0010437x23007595
Shay Ben-Moshe, Tomer M. Schlank
<p>We define higher semiadditive algebraic K-theory, a variant of algebraic K-theory that takes into account higher semiadditive structure, as enjoyed for example by the <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231214094411104-0539:S0010437X23007595:S0010437X23007595_inline1.png"><span data-mathjax-type="texmath"><span>$mathrm {K}(n)$</span></span></img></span></span>- and <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231214094411104-0539:S0010437X23007595:S0010437X23007595_inline2.png"><span data-mathjax-type="texmath"><span>$mathrm {T}(n)$</span></span></img></span></span>-local categories. We prove that it satisfies a form of the redshift conjecture. Namely, that if <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231214094411104-0539:S0010437X23007595:S0010437X23007595_inline3.png"><span data-mathjax-type="texmath"><span>$R$</span></span></img></span></span> is a ring spectrum of height <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231214094411104-0539:S0010437X23007595:S0010437X23007595_inline4.png"><span data-mathjax-type="texmath"><span>$leq n$</span></span></img></span></span>, then its semiadditive K-theory is of height <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231214094411104-0539:S0010437X23007595:S0010437X23007595_inline5.png"><span data-mathjax-type="texmath"><span>$leq n+1$</span></span></img></span></span>. Under further hypothesis on <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231214094411104-0539:S0010437X23007595:S0010437X23007595_inline6.png"><span data-mathjax-type="texmath"><span>$R$</span></span></img></span></span>, which are satisfied for example by the Lubin–Tate spectrum <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231214094411104-0539:S0010437X23007595:S0010437X23007595_inline7.png"><span data-mathjax-type="texmath"><span>$mathrm {E}_n$</span></span></img></span></span>, we show that its semiadditive algebraic K-theory is of height exactly <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231214094411104-0539:S0010437X23007595:S0010437X23007595_inline8.png"><span data-mathjax-type="texmath"><span>$n+1$</span></span></img></span></span>. Finally, we connect semiadditive K-theory to <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231
我们定义了高半加代数K理论，它是代数K理论的一个变体，考虑到了高半加结构，比如$mathrm {K}(n)$- 和$mathrm {T}(n)$-local categories。我们证明它满足红移猜想的一种形式。也就是说，如果 $R$ 是高度为 $leq n$ 的环谱，那么它的半加 K 理论高度为 $leq n+1$。在进一步假设 $R$ 满足卢宾-塔特谱 $mathrm {E}_n$ 等条件的情况下，我们证明它的半增加代数 K 理论的高度正好是 $n+1$。最后，我们把半加代数 K 理论与 $mathrm {T}(n+1)$ 本地化 K 理论联系起来，证明它们对于任何 $p$ 不可逆环谱和完整的约翰逊-威尔逊谱 $widehat {mathrm {E}(n)}$ 都是重合的。
{"title":"Higher semiadditive algebraic K-theory and redshift","authors":"Shay Ben-Moshe, Tomer M. Schlank","doi":"10.1112/s0010437x23007595","DOIUrl":"https://doi.org/10.1112/s0010437x23007595","url":null,"abstract":"<p>We define higher semiadditive algebraic K-theory, a variant of algebraic K-theory that takes into account higher semiadditive structure, as enjoyed for example by the <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231214094411104-0539:S0010437X23007595:S0010437X23007595_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$mathrm {K}(n)$</span></span></img></span></span>- and <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231214094411104-0539:S0010437X23007595:S0010437X23007595_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$mathrm {T}(n)$</span></span></img></span></span>-local categories. We prove that it satisfies a form of the redshift conjecture. Namely, that if <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231214094411104-0539:S0010437X23007595:S0010437X23007595_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$R$</span></span></img></span></span> is a ring spectrum of height <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231214094411104-0539:S0010437X23007595:S0010437X23007595_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$leq n$</span></span></img></span></span>, then its semiadditive K-theory is of height <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231214094411104-0539:S0010437X23007595:S0010437X23007595_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$leq n+1$</span></span></img></span></span>. Under further hypothesis on <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231214094411104-0539:S0010437X23007595:S0010437X23007595_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$R$</span></span></img></span></span>, which are satisfied for example by the Lubin–Tate spectrum <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231214094411104-0539:S0010437X23007595:S0010437X23007595_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$mathrm {E}_n$</span></span></img></span></span>, we show that its semiadditive algebraic K-theory is of height exactly <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231214094411104-0539:S0010437X23007595:S0010437X23007595_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$n+1$</span></span></img></span></span>. Finally, we connect semiadditive K-theory to <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":"44 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138681159","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 : 2023-12-04DOI: 10.1112/s0010437x2300756x
Sarah Peluse
<p>Fix a prime <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231201185654640-0378:S0010437X2300756X:S0010437X2300756X_inline3.png"><span data-mathjax-type="texmath"><span>$pgeq 11$</span></span></img></span></span>. We show that there exists a positive integer <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231201185654640-0378:S0010437X2300756X:S0010437X2300756X_inline4.png"><span data-mathjax-type="texmath"><span>$m$</span></span></img></span></span> such that any subset of <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231201185654640-0378:S0010437X2300756X:S0010437X2300756X_inline5.png"><span data-mathjax-type="texmath"><span>$mathbb {F}_p^ntimes mathbb {F}_p^n$</span></span></img></span></span> containing no nontrivial configurations of the form <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231201185654640-0378:S0010437X2300756X:S0010437X2300756X_inline7.png"><span data-mathjax-type="texmath"><span>$(x,y)$</span></span></img></span></span>, <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231201185654640-0378:S0010437X2300756X:S0010437X2300756X_inline8.png"><span data-mathjax-type="texmath"><span>$(x,y+z)$</span></span></img></span></span>, <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231201185654640-0378:S0010437X2300756X:S0010437X2300756X_inline9.png"><span data-mathjax-type="texmath"><span>$(x,y+2z)$</span></span></img></span></span>, <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231201185654640-0378:S0010437X2300756X:S0010437X2300756X_inline10.png"><span data-mathjax-type="texmath"><span>$(x+z,y)$</span></span></img></span></span> must have density <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231201185654640-0378:S0010437X2300756X:S0010437X2300756X_inline11.png"><span data-mathjax-type="texmath"><span>$ll 1/log _{m}{n}$</span></span></img></span></span>, where <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231201185654640-0378:S0010437X2300756X:S0010437X2300756X_inline12.png"><span data-mathjax-type="texmath"><span>$log _{m}$</span></span></img></span></span> denotes the <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231201185654640-0378:S0010437X2300756X:S0010437X230075
{"title":"Subsets of without L-shaped configurations","authors":"Sarah Peluse","doi":"10.1112/s0010437x2300756x","DOIUrl":"https://doi.org/10.1112/s0010437x2300756x","url":null,"abstract":"<p>Fix a prime <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231201185654640-0378:S0010437X2300756X:S0010437X2300756X_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$pgeq 11$</span></span></img></span></span>. We show that there exists a positive integer <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231201185654640-0378:S0010437X2300756X:S0010437X2300756X_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$m$</span></span></img></span></span> such that any subset of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231201185654640-0378:S0010437X2300756X:S0010437X2300756X_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$mathbb {F}_p^ntimes mathbb {F}_p^n$</span></span></img></span></span> containing no nontrivial configurations of the form <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231201185654640-0378:S0010437X2300756X:S0010437X2300756X_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$(x,y)$</span></span></img></span></span>, <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231201185654640-0378:S0010437X2300756X:S0010437X2300756X_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$(x,y+z)$</span></span></img></span></span>, <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231201185654640-0378:S0010437X2300756X:S0010437X2300756X_inline9.png\"><span data-mathjax-type=\"texmath\"><span>$(x,y+2z)$</span></span></img></span></span>, <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231201185654640-0378:S0010437X2300756X:S0010437X2300756X_inline10.png\"><span data-mathjax-type=\"texmath\"><span>$(x+z,y)$</span></span></img></span></span> must have density <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231201185654640-0378:S0010437X2300756X:S0010437X2300756X_inline11.png\"><span data-mathjax-type=\"texmath\"><span>$ll 1/log _{m}{n}$</span></span></img></span></span>, where <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231201185654640-0378:S0010437X2300756X:S0010437X2300756X_inline12.png\"><span data-mathjax-type=\"texmath\"><span>$log _{m}$</span></span></img></span></span> denotes the <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231201185654640-0378:S0010437X2300756X:S0010437X230075","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":"20 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138516593","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 : 2023-12-01DOI: 10.1112/s0010437x23007571
Kiran S. Kedlaya, Daxin Xu
We prove a Tannakian form of Drinfeld's lemma for isocrystals on a variety over a finite field, equipped with actions of partial Frobenius operators. This provides an intermediate step towards transferring V. Lafforgue's work on the Langlands correspondence over function fields from $ell$-adic to $p$-adic coefficients. We also discuss a motivic variant and a local variant of Drinfeld's lemma.
我们证明了有限域上具有部分Frobenius算子作用的变异体上等晶体的Drinfeld引理的Tannakian形式。这为将V. Lafforgue关于函数域上的Langlands对应从$ well $-adic系数转移到$p$-adic系数提供了一个中间步骤。我们还讨论了德林菲尔德引理的动机变体和局部变体。
{"title":"Drinfeld's lemma for F-isocrystals, II: Tannakian approach","authors":"Kiran S. Kedlaya, Daxin Xu","doi":"10.1112/s0010437x23007571","DOIUrl":"https://doi.org/10.1112/s0010437x23007571","url":null,"abstract":"<p>We prove a Tannakian form of Drinfeld's lemma for isocrystals on a variety over a finite field, equipped with actions of partial Frobenius operators. This provides an intermediate step towards transferring V. Lafforgue's work on the Langlands correspondence over function fields from <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231130101456566-0653:S0010437X23007571:S0010437X23007571_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$ell$</span></span></img></span></span>-adic to <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231130101456566-0653:S0010437X23007571:S0010437X23007571_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$p$</span></span></img></span></span>-adic coefficients. We also discuss a motivic variant and a local variant of Drinfeld's lemma.</p>","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":"116 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138516591","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 : 2023-11-30DOI: 10.1112/s0010437x23007613
Serge Cantat, Romain Dujardin
We study finite orbits of non-elementary groups of automorphisms of compact projective surfaces. We prove that if the surface and the group are defined over a number field $mathbf {k}$ and the group contains parabolic elements, then the set of finite orbits is not Zariski dense, except in certain very rigid situations, known as Kummer examples. Related results are also established when $mathbf {k} = mathbf {C}$. An application is given to the description of ‘canonical vector heights’ associated to such automorphism groups.
{"title":"Finite orbits for large groups of automorphisms of projective surfaces","authors":"Serge Cantat, Romain Dujardin","doi":"10.1112/s0010437x23007613","DOIUrl":"https://doi.org/10.1112/s0010437x23007613","url":null,"abstract":"<p>We study finite orbits of non-elementary groups of automorphisms of compact projective surfaces. We prove that if the surface and the group are defined over a number field <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231129100658964-0030:S0010437X23007613:S0010437X23007613_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$mathbf {k}$</span></span></img></span></span> and the group contains parabolic elements, then the set of finite orbits is not Zariski dense, except in certain very rigid situations, known as Kummer examples. Related results are also established when <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231129100658964-0030:S0010437X23007613:S0010437X23007613_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$mathbf {k} = mathbf {C}$</span></span></img></span></span>. An application is given to the description of ‘canonical vector heights’ associated to such automorphism groups.</p>","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":"1 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138516590","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 : 2023-11-22DOI: 10.1112/s0010437x23007534
Lynnelle Ye
We generalize bounds of Liu–Wan–Xiao for slopes in eigencurves for definite unitary groups of rank $2$ to slopes in eigenvarieties for definite unitary groups of any rank. We show that for a definite unitary group of rank $n$, the Newton polygon of the characteristic power series of the $U_p$ Hecke operator has exact growth rate $x^{1+2/{n(n-1)}}$, times a constant proportional to the distance of the weight from the boundary of weight space. The proof goes through the classification of forms associated to principal series representations. We also give a consequence for the geometry of these eigenvarieties over the boundary of weight space.
{"title":"Slopes in eigenvarieties for definite unitary groups","authors":"Lynnelle Ye","doi":"10.1112/s0010437x23007534","DOIUrl":"https://doi.org/10.1112/s0010437x23007534","url":null,"abstract":"<p>We generalize bounds of Liu–Wan–Xiao for slopes in eigencurves for definite unitary groups of rank <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231121150929901-0824:S0010437X23007534:S0010437X23007534_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$2$</span></span></img></span></span> to slopes in eigenvarieties for definite unitary groups of any rank. We show that for a definite unitary group of rank <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231121150929901-0824:S0010437X23007534:S0010437X23007534_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$n$</span></span></img></span></span>, the Newton polygon of the characteristic power series of the <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231121150929901-0824:S0010437X23007534:S0010437X23007534_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$U_p$</span></span></img></span></span> Hecke operator has exact growth rate <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231121150929901-0824:S0010437X23007534:S0010437X23007534_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$x^{1+2/{n(n-1)}}$</span></span></img></span></span>, times a constant proportional to the distance of the weight from the boundary of weight space. The proof goes through the classification of forms associated to principal series representations. We also give a consequence for the geometry of these eigenvarieties over the boundary of weight space.</p>","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":"23 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138516589","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 : 2023-11-09DOI: 10.1112/s0010437x23007510
Mark Shusterman
We prove a large finite field version of the Boston–Markin conjecture on counting Galois extensions of the rational function field with a given Galois group and the smallest possible number of ramified primes. Our proof involves a study of structure groups of (direct products of) racks.
{"title":"The tamely ramified geometric quantitative minimal ramification problem","authors":"Mark Shusterman","doi":"10.1112/s0010437x23007510","DOIUrl":"https://doi.org/10.1112/s0010437x23007510","url":null,"abstract":"We prove a large finite field version of the Boston–Markin conjecture on counting Galois extensions of the rational function field with a given Galois group and the smallest possible number of ramified primes. Our proof involves a study of structure groups of (direct products of) racks.","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":" 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135243616","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}
$K3^{[n]}$-type to contain a uniruled divisor covered by rational curves of primitive class. In particular, for any fixed 
$n$, we show that there are only finitely many polarization types of holomorphic symplectic variety of 
$K3^{[n]}$-type that do not contain such a uniruled divisor. As an application, we provide a generalization of a result due to Beauville–Voisin on the Chow group of 
$0$-cycles on such varieties.
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