Period Relations and Special Values of Rankin-Selberg L-Functions
Period Relations and Special Values of Rankin-Selberg L-Functions
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DOI：
10.1007/978-3-319-59728-7_9
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发表时间：
2016-08
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影响因子：
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通讯作者：
M. Harris;Jiezhu Lin
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文献类型：
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作者：
M. Harris;Jiezhu Lin
这是对最近关于上同调自同构表示对的 Rankin-SelbergL 函数值的研究，这在德利涅的意义上是批判性的。假设基字段是CM字段。德利涅的猜想是用动机语言表述的，并表达了临界值，直至理性因素，作为同调类上的射影代数簇的代数微分的某些周期的决定因素。自守方法可以证明的结果将某些临界值表示为自守形式的（扭曲）周期积分。使用酉群的上同调自同构表示之间的 Langlands 函子性（可以与 Shimura 簇的 de Rham 上同调和 GL(n) 的上同调自同构表示来识别），自同构周期可以解释为动机周期。我们报告了两位作者（第一作者格罗布纳）和格尔伯洛夫（Guerberoff）的最新结果。
This is a survey of recent work on values of Rankin-SelbergL-functions of pairs of cohomological automorphic representations that arecriticalin Deligne’s sense. The base field is assumed to be a CM field. Deligne’s conjecture is stated in the language of motives over, and expresses the critical values, up to rational factors, as determinants of certain periods of algebraic differentials on a projective algebraic variety over homology classes. The results that can be proved by automorphic methods express certain critical values as (twisted) period integrals of automorphic forms. Using Langlands functoriality between cohomological automorphic representations of unitary groups, which can be identified with the de Rham cohomology of Shimura varieties, and cohomological automorphic representations ofGL(n), the automorphic periods can be interpreted as motivic periods. We report on recent results of the two authors, of the first-named author with Grobner, and of Guerberoff.