Global liftings between inner forms of GSp(4)

Pub Date : 2024-05-17 DOI:10.1016/j.jnt.2024.04.010
Mirko Rösner, Rainer Weissauer
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Abstract

For reductive groups G over a number field we discuss automorphic liftings of cohomological cuspidal irreducible automorphic representations π of G(A) to irreducible cohomological automorphic representations of H(A) 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 H=GSp(4) however we address these finer questions. Especially, we prove the recent conjectures of Ibukiyama and Kitayama on paramodular newforms of square-free level.

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GSp(4) 内形式之间的全局升维
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。
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