ThisisasurveyofrecentworkonvaluesofRankin-SelbergL-functions of pairs of cohomological automorphic representations that are critical in Deligne’s sense. The base field is assumed to be a CM field. Deligne’s conjecture is stated in the language of motives over Q, and express 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 repre- sentations of unitary groups, which can be identified with the de Rham cohomology of Shimura varieties, and cohomological automorphic representations of GL.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.
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Shimura varieties for unitary groups and the doubling method
ThetheoryofGaloisrepresentationsattachedtoautomorphicrepresenta- tions of GL(n) is largely based on the study of the cohomology of Shimura varieties of PEL type attached to unitary similitude groups. The need to keep track of the similitude factor complicates notation while making no difference to the final result. It is more natural to work with Shimura varieties attached to the unitary groups themselves, which do not introduce these unnecessary complications; however, these are of abelian type, not of PEL type, and the Galois representations on their cohomology differ slightly from those obtained from the more familiar Shimura varieties.
Results on the critical values of the L-functions of these Galois representations have been established by studying the PEL type Shimura varieties. It is not immediately obvious that the automorphic periods for these varieties are the same as for those attached to unitary groups, which appear more naturally in applications of relative trace formulas, such as the refined Gan-Gross-Prasad conjecture (conjecture of Ichino-Ikeda and N. Harris). The present article reconsiders these critical values, using the Shimura varieties attached to unitary groups, and obtains results that can be used more simply in applications.
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- Award ID(s):
- 2001369
- NSF-PAR ID:
- 10320702
- Editor(s):
- Müller, Werner; Shin, Sug Woo; Templier, Nicolas
- Date Published:
- Journal Name:
- Relative Trace Formulas
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1007/978-3-030-68506-5_6
