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On the notion of the parabolic and the cuspidal support of smooth-automorphic forms and smooth-automorphic representations.

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Summary

This study extends critical values of Rankin-Selberg L-functions to general CM-fields and sums of automorphic representations. The findings generalize and complement previous research in number theory.

Keywords:
Critical valueCuspidal automorphicIsobaric sumL-functionPeriod

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Area of Science:

  • Number Theory
  • Automorphic Representations
  • Algebraic Geometry

Background:

  • The study builds upon prior work on critical values of Rankin-Selberg L-functions, specifically referencing results by Grobner and Harris.
  • Existing research by Raghuram and Mahnkopf provides foundational insights into related automorphic representations and L-functions.

Purpose of the Study:

  • To extend the understanding of critical values for Rankin-Selberg L-functions.
  • To generalize these results to arbitrary rank general linear groups over general CM-fields.
  • To incorporate isobaric sums of unitary cuspidal automorphic representations.

Main Methods:

  • Utilizes advanced techniques in the theory of automorphic forms and L-functions.
  • Employs methods for analyzing critical values in the context of algebraic number fields.
  • Leverages the concept of isobaric sums for automorphic representations.

Main Results:

  • A simultaneous extension of critical values for Rankin-Selberg L-functions is proven.
  • The results apply to general CM-fields and cohomological automorphic representations formed by isobaric sums.
  • Theorem 1.9 establishes a significant generalization and complement to existing literature.

Conclusions:

  • The findings significantly advance the theory of automorphic L-functions.
  • This work provides a more comprehensive framework for studying critical values in number theory.
  • The generalization deepens the connection between automorphic forms and algebraic number fields.