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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This content will become publicly available on January 1, 2025
COHOMOLOGICAL MACKEY 2-FUNCTORS
We show that the bicategory of finite groupoids and right-free permutation bimodules is a quotient of the bicategory of Mackey 2-motives introduced in [2], obtained by modding out the so-called cohomological relations. This categorifies Yoshida’s theorem for ordinary cohomological Mackey functors and provides a direct connection between Mackey 2-motives and the usual blocks of representation theory.
more » « less- Award ID(s):
- 1901696
- PAR ID:
- 10504171
- Date Published:
- Journal Name:
- Journal of the Institute of Mathematics of Jussieu
- Volume:
- 23
- Issue:
- 1
- ISSN:
- 1474-7480
- Page Range / eLocation ID:
- 279 to 309
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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