Titlebook: An Introduction to Automorphic Representations; With a view toward t Jayce R. Getz,Heekyoung Hahn Textbook 2024 Springer Nature Switzerland

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Rankin-Selberg ,-functions,In this chapter we sketch the theory of generic representations and Rankin-Selberg .-functions.

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Known Cases of Global Langlands Functoriality,Over number fields the global Langlands functoriality conjecture is wide open. Despite this, several important cases have been established. We survey some of the hard earned progress in this chapter.

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Orbital Integrals,We define and study orbital integrals. In the next chapter, we explain how the geometric sides of trace formulae are written in terms of orbital integrals.

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Simple Trace Formulae,In this chapter we prove a simple general relative trace formula and overview its various specializations. This recovers (simple versions of) many trace formulae that have appeared in the literature. In particular, one such specialization yields the Petersson-Bruggeman-Kuznetsov trace formula.

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Applications of Trace Formulae,In this chapter we discuss applications of relative trace formulae. Our goal is to provide an introduction to the key motifs of existence and comparison underlying these applications.