Adjoint $L$-functions, congruence ideals, and Selmer groups over $\mathrm{GL}_n$

TL;DR

This paper explores the connection between adjoint L-values, congruence ideals, and Selmer groups for automorphic representations of GL_n, providing new insights into their interrelations over number fields.

Contribution

It establishes a relationship between L(1,π,Ad^0) and congruence ideals for automorphic representations, and applies this to Selmer groups over CM fields.

Findings

L(1,π,Ad^0) relates to congruence ideals

Lower bounds on Selmer group sizes derived

Connections between automorphic congruences and L-values

Abstract

In this paper, we relate $L(1,\pi,\mathrm{Ad}^\circ)$ to the congruence ideals for cohomological cuspidal automorphic representations $\pi$ of $\mathrm{GL}_n$ over any number field. We then use this result to deduce relationships between the congruences of automorphic forms and adjoint $L$-functions. For CM fields, we apply the result to obtain a lower bound on the cardinality of certain Selmer groups in terms of $L(1,\pi,\mathrm{Ad}^\circ)$.