Pub Date : 2023-10-08DOI: 10.2140/ant.2023.17.2181
Mathilde Gerbelli-Gauthier
We give upper bounds on limit multiplicities of certain nontempered representations of unitary groups , conditionally on the endoscopic classification of representations. Our result applies to some cohomological representations, and we give applications to the growth of cohomology of cocompact arithmetic subgroups of unitary groups. The representations considered are transfers of products of characters and discrete series on endoscopic groups, and the bounds are obtained using Arthur’s stabilization of the trace formula and the classification established by Mok, and Kaletha, Minguez, Shin and White.
{"title":"Limit multiplicity for unitary groups and the stable trace formula","authors":"Mathilde Gerbelli-Gauthier","doi":"10.2140/ant.2023.17.2181","DOIUrl":"https://doi.org/10.2140/ant.2023.17.2181","url":null,"abstract":"<p>We give upper bounds on limit multiplicities of certain nontempered representations of unitary groups <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>U</mi><mo stretchy=\"false\">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy=\"false\">)</mo></math>, conditionally on the endoscopic classification of representations. Our result applies to some cohomological representations, and we give applications to the growth of cohomology of cocompact arithmetic subgroups of unitary groups. The representations considered are transfers of products of characters and discrete series on endoscopic groups, and the bounds are obtained using Arthur’s stabilization of the trace formula and the classification established by Mok, and Kaletha, Minguez, Shin and White. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2023-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493545","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-08DOI: 10.2140/ant.2023.17.2097
Arthur-César Le Bras, Alberto Vezzani
We show how to attach to any rigid analytic variety over a perfectoid space a rigid analytic motive over the Fargues–Fontaine curve functorially in and . We combine this construction with the overconvergent relative de Rham cohomology to produce a complex of solid quasicoherent sheaves over , and we show that its cohomology groups are vector bundles if is smooth and proper over or if is quasicompact and is a perfectoid field, thus proving and generalizing a conjecture of Scholze. The main ingredients of the proofs are explicit -homotopies, the motivic proper base change and the formalism of solid quasicoherent sheaves.
{"title":"The de Rham–Fargues–Fontaine cohomology","authors":"Arthur-César Le Bras, Alberto Vezzani","doi":"10.2140/ant.2023.17.2097","DOIUrl":"https://doi.org/10.2140/ant.2023.17.2097","url":null,"abstract":"<p>We show how to attach to any rigid analytic variety <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>V</mi> </math> over a perfectoid space <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>P</mi></math> a rigid analytic motive over the Fargues–Fontaine curve <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi mathvariant=\"bold-script\">𝒳</mi><mo stretchy=\"false\">(</mo><mi>P</mi><mo stretchy=\"false\">)</mo></math> functorially in <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>V</mi> </math> and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>P</mi></math>. We combine this construction with the overconvergent relative de Rham cohomology to produce a complex of solid quasicoherent sheaves over <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi mathvariant=\"bold-script\">𝒳</mi><mo stretchy=\"false\">(</mo><mi>P</mi><mo stretchy=\"false\">)</mo></math>, and we show that its cohomology groups are vector bundles if <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>V</mi> </math> is smooth and proper over <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>P</mi></math> or if <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>V</mi> </math> is quasicompact and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>P</mi></math> is a perfectoid field, thus proving and generalizing a conjecture of Scholze. The main ingredients of the proofs are explicit <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi mathvariant=\"double-struck\">𝔹</mi></mrow><mrow><mn>1</mn></mrow></msup></math>-homotopies, the motivic proper base change and the formalism of solid quasicoherent sheaves. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2023-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71514518","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-08DOI: 10.2140/ant.2023.17.2229
Raf Cluckers, Itay Glazer, Yotam I. Hendel
We provide uniform estimates on the number of -points lying on fibers of flat morphisms between smooth varieties whose fibers have rational singularities, termed (FRS) morphisms. For each individual fiber, the estimates were known by work of Avni and Aizenbud, but we render them uniform over all fibers. The proof technique for individual fibers is based on Hironaka’s resolution of singularities and Denef’s formula, but breaks down in the uniform case. Instead, we use recent results from the theory of motivic integration. Our estimates are moreover equivalent to the (FRS) property, just like in the absolute case by Avni and Aizenbud. In addition, we define new classes of morphisms, called -smooth morphisms (