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A relation between automorphic forms on GL(2) and GL(3).
Summary
This study introduces an Euler product for automorphic cuspidal representations of GL(2,A), proving its entire nature. It demonstrates the existence of a GL(3) lift whose L-function matches the Euler product, confirming Langlands conjectures.
Area of Science:
- Number Theory
- Representation Theory
- Automorphic Forms
Background:
- The study builds upon the theory of automorphic representations and their L-functions.
- It specifically addresses representations of the general linear group GL(n,C).
Purpose of the Study:
- To introduce and analyze a degree 3 Euler product for automorphic cuspidal representations of GL(2,A).
- To establish the existence of a lift to GL(3) whose L-function corresponds to this Euler product.
- To verify conjectures proposed by R. P. Langlands.
Main Methods:
- Construction of an Euler product L(s,pi,rho(2) (2)) for a given automorphic cuspidal representation pi of GL(2,A).
- Proof of the analytic properties (entirety) of this Euler product.
- Demonstration of the existence of an automorphic representation II of GL(3) (the lift of pi).
Main Results:
- An Euler product of degree 3, L(s,pi,rho(2) (2)), is introduced and proven to be entire.
- An automorphic representation II of GL(3) is shown to exist, termed 'the lift of pi'.
- The L-function of II, L(s,II,rho(3)), is proven to be equal to L(s,pi,rho(2) (2)).
Conclusions:
- The results confirm specific conjectures within the broader framework of R. P. Langlands' work.
- This establishes a concrete link between representations of GL(2,A) and GL(3) through their L-functions.
- The study contributes to the understanding of the Langlands program.