{"title":"$${textrm{GU}}(2,2)$$ 和 $${textrm{GSp}}_4$$ 的 5 级 L 函数的两变量兰金-塞尔伯格积分","authors":"Antonio Cauchi, Armando Gutierrez Terradillos","doi":"10.1007/s00209-024-03583-9","DOIUrl":null,"url":null,"abstract":"<p>We give a two-variable Rankin–Selberg integral for generic cusp forms on <span>\\(\\textrm{PGL}_4\\)</span> and <span>\\(\\textrm{PGU}_{2,2}\\)</span> which represents a product of exterior square <i>L</i>-functions. As a residue of our integral, we obtain an integral representation on <span>\\(\\textrm{PGU}_{2,2}\\)</span> of the degree 5 <i>L</i>-function of <span>\\({\\textrm{GSp}}_4\\)</span> twisted by the quadratic character of <i>E</i>/<i>F</i> of cuspidal automorphic representations which contribute to the theta correspondence for the pair <span>\\((\\textrm{P}{\\textrm{GSp}}_4,\\textrm{P}{\\textrm{GU}}_{2,2})\\)</span>.</p>","PeriodicalId":18278,"journal":{"name":"Mathematische Zeitschrift","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A two variable Rankin–Selberg integral for $${\\\\textrm{GU}}(2,2)$$ and the degree 5 L-function of $${\\\\textrm{GSp}}_4$$\",\"authors\":\"Antonio Cauchi, Armando Gutierrez Terradillos\",\"doi\":\"10.1007/s00209-024-03583-9\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We give a two-variable Rankin–Selberg integral for generic cusp forms on <span>\\\\(\\\\textrm{PGL}_4\\\\)</span> and <span>\\\\(\\\\textrm{PGU}_{2,2}\\\\)</span> which represents a product of exterior square <i>L</i>-functions. As a residue of our integral, we obtain an integral representation on <span>\\\\(\\\\textrm{PGU}_{2,2}\\\\)</span> of the degree 5 <i>L</i>-function of <span>\\\\({\\\\textrm{GSp}}_4\\\\)</span> twisted by the quadratic character of <i>E</i>/<i>F</i> of cuspidal automorphic representations which contribute to the theta correspondence for the pair <span>\\\\((\\\\textrm{P}{\\\\textrm{GSp}}_4,\\\\textrm{P}{\\\\textrm{GU}}_{2,2})\\\\)</span>.</p>\",\"PeriodicalId\":18278,\"journal\":{\"name\":\"Mathematische Zeitschrift\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-09-03\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematische Zeitschrift\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s00209-024-03583-9\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematische Zeitschrift","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00209-024-03583-9","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
我们给出了在\(\textrm{PGL}_4\)和\(\textrm{PGU}_{2,2}\)上一般尖顶形式的双变量兰金-塞尔伯格积分，它表示外部平方 L 函数的乘积。作为积分的残差，我们在 \(\textrm{PGU}_{2、2}）上的一个积分表示，它是\({textrm{GSp}}_4)的阶 5 L 函数，它被尖顶自形表示的 E/F 的二次方特征扭曲了，这有助于一对 \((textrm{P}{textrm{GSp}}_4,textrm{P}{textrm{GU}}_{2,2})\的θ 对应。）
A two variable Rankin–Selberg integral for $${\textrm{GU}}(2,2)$$ and the degree 5 L-function of $${\textrm{GSp}}_4$$
We give a two-variable Rankin–Selberg integral for generic cusp forms on \(\textrm{PGL}_4\) and \(\textrm{PGU}_{2,2}\) which represents a product of exterior square L-functions. As a residue of our integral, we obtain an integral representation on \(\textrm{PGU}_{2,2}\) of the degree 5 L-function of \({\textrm{GSp}}_4\) twisted by the quadratic character of E/F of cuspidal automorphic representations which contribute to the theta correspondence for the pair \((\textrm{P}{\textrm{GSp}}_4,\textrm{P}{\textrm{GU}}_{2,2})\).
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