{"title":"Global liftings between inner forms of GSp(4)","authors":"Mirko Rösner, Rainer Weissauer","doi":"10.1016/j.jnt.2024.04.010","DOIUrl":null,"url":null,"abstract":"<div><p>For reductive groups <em>G</em> over a number field we discuss automorphic liftings of cohomological cuspidal irreducible automorphic representations <em>π</em> of <span><math><mi>G</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span> to irreducible cohomological automorphic representations of <span><math><mi>H</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span> for the quasi-split inner form <em>H</em> of <em>G</em>, and other inner forms as well. We show the existence of nontrivial weak global cohomological liftings in many cases, in particular for the case where <em>G</em> is anisotropic at the archimedean places. A priori, for these weak liftings we do not give a description of the precise nature of the corresponding local liftings at the ramified places, nor do we characterize the image of the lifting. For inner forms of the group <span><math><mi>H</mi><mo>=</mo><mrow><mi>GSp</mi></mrow><mo>(</mo><mn>4</mn><mo>)</mo></math></span> however we address these finer questions. Especially, we prove the recent conjectures of Ibukiyama and Kitayama on paramodular newforms of square-free level.</p></div>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022314X24001173/pdfft?md5=e1a88da9503c55ed7cf8a41d86d6117b&pid=1-s2.0-S0022314X24001173-main.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022314X24001173","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
For reductive groups G over a number field we discuss automorphic liftings of cohomological cuspidal irreducible automorphic representations π of to irreducible cohomological automorphic representations of for the quasi-split inner form H of G, and other inner forms as well. We show the existence of nontrivial weak global cohomological liftings in many cases, in particular for the case where G is anisotropic at the archimedean places. A priori, for these weak liftings we do not give a description of the precise nature of the corresponding local liftings at the ramified places, nor do we characterize the image of the lifting. For inner forms of the group however we address these finer questions. Especially, we prove the recent conjectures of Ibukiyama and Kitayama on paramodular newforms of square-free level.
对于数域上的还原群 G,我们讨论了对于 G 的准分裂内形式 H 以及其他内形式,G(A) 的同调无穷自形表示 π 到 H(A) 的无穷同调自形表示的自形提升。我们证明了在许多情况下,特别是在 G 在拱顶处各向异性的情况下,存在非微不足道的弱全局同调升维。先验地讲,对于这些弱提升,我们并没有给出相应局部提升在斜切处的精确性质,也没有描述提升的图像。然而,对于 H=GSp(4) 群的内形式,我们解决了这些更精细的问题。特别是,我们证明了伊吹山和北山最近关于无平方级的准新形式的猜想。