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• 作者：Angelika Geroldinger
• 关键词：$$p$$ p ; Adic $$L$$ L ; functions ; Automorphic $$L$$ L ; functions ; Automorphic forms ; Eisenstein cohomology ; Mellin transform ; 11F67 ; 11F70 ; 11F75
• 刊名：The Ramanujan Journal
• 出版年：2015
• 出版时间：December 2015
• 年：2015
• 卷：38
• 期：3
• 页码：641-682
• 全文大小：864 KB
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• 作者单位：Angelika Geroldinger (1)
  
  1. Department of Mathematics, University of Vienna, Würthgasse 1/12, 1190, Wien, Austria
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Number Theory
  Field Theory and Polynomials
  Combinatorics
  Fourier Analysis
  Functions of a Complex Variable
  
• 出版者：Springer U.S.
• ISSN：1572-9303

文摘

We construct \(p\)-adic \(L\)-functions attached to cohomological cuspidal automorphic representations \(\pi \) of \(\text {GL}_3\) over \({\mathbb Q}\). These functions will be defined as \(p\)-adic Mellin transforms of \(h\)-admissible measures. Our strategy is to generalize the construction of \(p\)-adic \(L\)-functions given in Mahnkopf (Compos. Math. 124(3):253-04, 2000) for cuspidal representations of \(\text {GL}_3\) which are cohomological with respect to the trivial coefficient system. This mainly relies on Harder’s method of computing bounds for the denominators of certain Eisenstein cohomology classes, cf. Harder (Kohomologie Arithmetischer Gruppen, 1987). Finally, we establish a functional equation for the \(p\)-adic \(L\)-functions. This, in particular, requires the construction of \(p\)-adic \(L\)-functions attached to the critical integers on the right-hand side of the functional equation. Keywords \(p\)-Adic \(L\)-functions Automorphic \(L\)-functions Automorphic forms Eisenstein cohomology Mellin transform