An Euler system for GU(2, 1).

Area of Science:

• Number Theory
• Algebraic Geometry
• Representation Theory

Background:

• Euler systems are fundamental objects in modern number theory, providing powerful tools for studying arithmetic objects.
• Automorphic representations, particularly self-dual cuspidal ones, are central to the Langlands program and have deep connections to number fields.
• Shimura varieties offer a rich geometric framework for studying automorphic forms and their associated arithmetic invariants.

Purpose of the Study:

• To construct a new Euler system tailored for a specific class of automorphic representations.
• To associate this Euler system with regular algebraic, essentially conjugate self-dual cuspidal automorphic representations over imaginary quadratic fields.
• To leverage the cohomology of Shimura varieties as the foundational structure for this construction.

Main Methods:

• Utilizing the cohomology of Shimura varieties associated with the general linear group GL(2).
• Employing techniques from algebraic geometry and the theory of automorphic forms.
• Developing a novel construction of an Euler system that captures arithmetic information of the specified automorphic representations.

Main Results:

• Successfully constructed an Euler system associated with regular algebraic, essentially conjugate self-dual cuspidal automorphic representations of GL(2) over imaginary quadratic fields.
• The construction is intrinsically linked to the geometry and cohomology of the relevant Shimura varieties.
• This provides a new arithmetic object for studying these automorphic representations.

Conclusions:

• The constructed Euler system offers a new perspective and tool for investigating the arithmetic of automorphic representations over imaginary quadratic fields.
• This work deepens the understanding of the interplay between number theory, algebraic geometry, and representation theory.
• The methods used may pave the way for constructing Euler systems in more general settings.