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Rational structures on automorphic representations

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Abstract

This paper proves the existence of global rational structures on spaces of cusp forms of general reductive groups. We identify cases where the constructed rational structures are optimal, which includes the case of \({{\mathrm{\mathrm {GL}}}}(n)\). As an application, we deduce the existence of a natural set of periods attached to cuspidal automorphic representations of \({{\mathrm{\mathrm {GL}}}}(n)\). This has consequences for the arithmetic of special values of L-functions that we discuss in Januszewski (On period relations for automorphic L-functions I, pp. 1–46, arXiv:1504.06973, 2015, On period relations for automorphic L-functions II, pp. 1–65, arXiv:1604.04253, 2015). In the course of proving our results, we lay the foundations for a general theory of Harish-Chandra modules over arbitrary fields of characteristic 0. In this context, a rational character theory, translation functors and an equivariant theory of cohomological induction are developed. We also study descent problems for Harish-Chandra modules in quadratic extensions, where we obtain a complete theory over number fields.

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Acknowledgements

The author thanks Binyong Sun for his hospitality and fruitful discussions, Jacques Tilouine for pointing out that the field of definition of Harish-Chandra modules is related to the L-packet, and Günter Harder for sharing and explaining his work in [22]. The author thanks Jeff Adams for sharing Robert McLean’s and Ran Cui’s work. The author thanks Claus–Günther Schmidt and Anton Deitmar for their comments and remarks on a preliminary version of this paper. Last but not least the author thanks the referee for helpful remarks and the suggestion to include the general reductive case.

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  1. Karlsruher Institut für Technologie, Karlsruhe, Germany

    Fabian Januszewski

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Januszewski, F. Rational structures on automorphic representations. Math. Ann. 370, 1805–1881 (2018). https://doi.org/10.1007/s00208-017-1567-6

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