P-adic L-functions for GL ( 3 ).

Area of Science:

• Number Theory
• Automorphic Forms
• Algebraic Geometry

Background:

• Regular algebraic cuspidal automorphic representations (RACARs) are central objects in number theory.
• Construction of p-adic L-functions is crucial for understanding arithmetic properties of automorphic forms.
• Existing theories often rely on specific conditions like self-duality or restricted ramification.

Purpose of the Study:

• To construct a p-adic L-function for RACARs of GL3(AQ) that are p-nearly-ordinary.
• To interpolate critical L-values in both halves of the critical strip.
• To extend the theory of p-adic L-functions to automorphic representations of 'general type'.

Main Methods:

• Utilizing the theory of spherical varieties to construct a Betti Euler system.
• Developing a norm-compatible system of classes in the Betti cohomology of locally symmetric spaces.
• Working with arbitrary cohomological weights and ramification at p.

Main Results:

• Construction of a bounded measure L_p(Π) on Z_p^* that interpolates critical L-values.
• Proof of conjectures by Coates-Perrin-Riou and Panchishkin for p-nearly-ordinary RACARs.
• First construction of p-adic L-functions for general type RACARs of GL_n(AQ) for n > 2.

Conclusions:

• The study successfully constructs p-adic L-functions for a broad class of automorphic representations.
• This work significantly advances the understanding of p-adic L-functions and their connection to automorphic forms.
• The methods developed open new avenues for studying arithmetic properties of automorphic representations.