Adjoint L-functions, congruence ideals, and Selmer groups over GL n

Area of Science:

• Number Theory
• Algebraic Geometry
• Representation Theory

Background:

• The study of special values of adjoint L-functions and congruence ideals is a developing area in number theory.
• This research is motivated by the Bloch-Kato conjecture and generalizations of the Wiles-Lenstra numerical criterion.

Purpose of the Study:

• To establish a relationship between L(1, π, Ad°) and congruence ideals for automorphic representations of GL(n) over number fields.
• To deduce connections between congruences of automorphic forms and adjoint L-functions.
• To provide a lower bound for Selmer group cardinalities using L(1, π, Ad°) for CM fields, contributing to the Bloch-Kato conjecture.

Main Methods:

• Relating special values of adjoint L-functions to congruence ideals for cohomological cuspidal automorphic representations.
• Analyzing the cohomology of the locally symmetric space of GL(n) and its connection to automorphic representations.
• Establishing algebraic properties of congruence ideals and utilizing Galois representations for CM fields.

Main Results:

• A direct relationship is established between L(1, π, Ad°) and congruence ideals for automorphic representations.
• Congruences of automorphic forms are linked to adjoint L-functions.
• A lower bound for Selmer group sizes is derived in terms of L(1, π, Ad°) for CM fields.

Conclusions:

• The study makes partial progress on the Bloch-Kato conjecture by relating arithmetic and analytic objects.
• The developed methods are expected to have further applications in studying congruence modules and automorphic forms.
• The research highlights the interplay between special values of L-functions, automorphic forms, and algebraic structures.