{"title":"宫崎式升降机不会消失","authors":"Henry H. Kim, Takuya Yamauchi","doi":"10.1007/s12188-019-00207-6","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper, we study the non-vanishing of the Miyawaki type lift in various situations. In the case of <i>GSpin</i>(2, 10) constructed in Kim and Yamauchi (Math Z 288(1–2):415–437, 2018), we use the fact that the Fourier coefficient at the identity is closely related to the Rankin–Selberg <i>L</i>-function of two elliptic cusp forms. In the case of the original Miyawaki lifts of Siegel cusp forms, we use the fact that certain Fourier coefficients are the Petersson inner product which is non-trivial. This provides infinitely many examples of non-zero Miyawaki lifts. We give explicit examples of degree 24 and weight 24. We also prove a similar result for Miyawaki lifts for unitary groups. Especially, we obtain an unconditional result on non-vanishing of Miyawaki lifts for <span>\\(U(n+1,n+1)\\)</span> for each <span>\\(n\\equiv 3\\)</span> mod 4. In the last section, we prove the non-vanishing of the Miyawaki lifts for infinitely many half-integral weight Siegel cusp forms. We give explicit examples of degree 16 and weight <span>\\(\\frac{29}{2}\\)</span>.\n</p></div>","PeriodicalId":50932,"journal":{"name":"Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg","volume":null,"pages":null},"PeriodicalIF":0.4000,"publicationDate":"2019-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s12188-019-00207-6","citationCount":"3","resultStr":"{\"title\":\"Non-vanishing of Miyawaki type lifts\",\"authors\":\"Henry H. Kim, Takuya Yamauchi\",\"doi\":\"10.1007/s12188-019-00207-6\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>In this paper, we study the non-vanishing of the Miyawaki type lift in various situations. In the case of <i>GSpin</i>(2, 10) constructed in Kim and Yamauchi (Math Z 288(1–2):415–437, 2018), we use the fact that the Fourier coefficient at the identity is closely related to the Rankin–Selberg <i>L</i>-function of two elliptic cusp forms. In the case of the original Miyawaki lifts of Siegel cusp forms, we use the fact that certain Fourier coefficients are the Petersson inner product which is non-trivial. This provides infinitely many examples of non-zero Miyawaki lifts. We give explicit examples of degree 24 and weight 24. We also prove a similar result for Miyawaki lifts for unitary groups. Especially, we obtain an unconditional result on non-vanishing of Miyawaki lifts for <span>\\\\(U(n+1,n+1)\\\\)</span> for each <span>\\\\(n\\\\equiv 3\\\\)</span> mod 4. In the last section, we prove the non-vanishing of the Miyawaki lifts for infinitely many half-integral weight Siegel cusp forms. We give explicit examples of degree 16 and weight <span>\\\\(\\\\frac{29}{2}\\\\)</span>.\\n</p></div>\",\"PeriodicalId\":50932,\"journal\":{\"name\":\"Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2019-09-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1007/s12188-019-00207-6\",\"citationCount\":\"3\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s12188-019-00207-6\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s12188-019-00207-6","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 3
摘要
本文研究了宫崎式升降机在各种情况下的不消失性。在Kim和Yamauchi (Math Z 288(1-2):415 - 437,2018)构建的GSpin(2,10)的情况下,我们使用了恒等处的傅里叶系数与两个椭圆尖形的Rankin-Selberg l函数密切相关的事实。在西格尔尖峰形式的原始Miyawaki提升中,我们使用了某些傅里叶系数是Petersson内积的事实,它是非平凡的。这提供了无限多的非零宫崎骏举的例子。我们给出了24度和24权的明确例子。我们也证明了幺正群的Miyawaki提举的类似结果。特别地,对于每个\(n\equiv 3\) mod 4,我们得到了\(U(n+1,n+1)\)的Miyawaki提升不消失的无条件结果。在最后一节中,我们证明了无穷多个半积分权Siegel尖形的Miyawaki凸的不消失性。我们给出了16度和权重\(\frac{29}{2}\)的明确例子。
In this paper, we study the non-vanishing of the Miyawaki type lift in various situations. In the case of GSpin(2, 10) constructed in Kim and Yamauchi (Math Z 288(1–2):415–437, 2018), we use the fact that the Fourier coefficient at the identity is closely related to the Rankin–Selberg L-function of two elliptic cusp forms. In the case of the original Miyawaki lifts of Siegel cusp forms, we use the fact that certain Fourier coefficients are the Petersson inner product which is non-trivial. This provides infinitely many examples of non-zero Miyawaki lifts. We give explicit examples of degree 24 and weight 24. We also prove a similar result for Miyawaki lifts for unitary groups. Especially, we obtain an unconditional result on non-vanishing of Miyawaki lifts for \(U(n+1,n+1)\) for each \(n\equiv 3\) mod 4. In the last section, we prove the non-vanishing of the Miyawaki lifts for infinitely many half-integral weight Siegel cusp forms. We give explicit examples of degree 16 and weight \(\frac{29}{2}\).
期刊介绍:
The first issue of the "Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg" was published in the year 1921. This international mathematical journal has since then provided a forum for significant research contributions. The journal covers all central areas of pure mathematics, such as algebra, complex analysis and geometry, differential geometry and global analysis, graph theory and discrete mathematics, Lie theory, number theory, and algebraic topology.