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Automorphic L-functions of half-integral weight.
S S Gelbart1, I I Piatetski-Shapiro
1Department of Mathematics, Cornell University, Ithaca, New York 14853.
Summary
We introduce a new theory of L-functions for metaplectic groups, establishing an arithmetic link between genuine and standard representations. This work generalizes Shimura
Area of Science:
- Number Theory
- Representation Theory
- Automorphic Forms
Background:
- The study of L-functions and their connection to the properties of number fields and their associated groups is a central theme in modern mathematics.
- Metaplectic groups, extensions of general linear groups, play a significant role in number theory and harmonic analysis.
- Whittaker models and L-functions are fundamental tools for understanding representations of these groups.
Purpose of the Study:
- To develop a theory of Whittaker models and L-functions for irreducible representations of a metaplectic covering group of GL(2).
- To establish an arithmetical correspondence between genuine cuspidal representations of the metaplectic group and cuspidal representations of GL(2) using these L-functions.
- To generalize and reformulate existing results by G. Shimura concerning such correspondences over the rational numbers (Q).
Main Methods:
- Development of a theory for Whittaker models and L-functions tailored for metaplectic covering groups.
- Utilizing L-functions to construct and analyze an arithmetical correspondence.
- Focusing on "exceptional" cusp forms, defined by their unique determination from a single Fourier coefficient.
Main Results:
- A comprehensive theory of Whittaker models and L-functions for metaplectic GL(2) representations is presented.
- An arithmetical correspondence is established between genuine cuspidal representations of the metaplectic group and cuspidal representations of GL(2).
- The theory generalizes and reformulates G. Shimura's prior work on these correspondences for the ground field Q.
Conclusions:
- The developed theory provides a powerful framework for studying metaplectic groups and their representations.
- The established correspondence deepens the understanding of the relationship between different types of automorphic representations.
- The concept of "exceptional" cusp forms highlights specific cases with significant structural implications.