# Adjoint L-functions, congruence ideals, and Selmer groups over GLn

## TL;DR

This paper explores connections between adjoint L-functions, congruence ideals, and Selmer groups in number theory, contributing to the understanding of the Bloch-Kato conjecture.

## Contribution

The paper establishes new relationships between adjoint L-functions and congruence ideals for automorphic representations of GLn over any number field.

## Key findings

- A relationship is established between L(1,π,Ad∘) and congruence ideals for cohomological cuspidal automorphic representations.
- For CM fields, a lower bound on the cardinality of certain Selmer groups is derived using adjoint L-functions.
- The methods developed can be applied to further study congruence modules and their arithmetic connections.

## Abstract

The study of special values of adjoint L-functions and congruence ideals is gradually becoming a classical theme in number theory, driven by the Bloch-Kato conjecture and generalisations of Wiles-Lenstra’s numerical criterion. In this paper, we relate \documentclass[12pt]{minimal}
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\begin{document}$$L(1,\pi ,{{\,\textrm{Ad}\,}}^\circ )$$\end{document}L(1,π,Ad∘) to the congruence ideals for cohomological cuspidal automorphic representations \documentclass[12pt]{minimal}
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\begin{document}$$\pi $$\end{document}π of \documentclass[12pt]{minimal}
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\begin{document}$$\textrm{GL}_n$$\end{document}GLn 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, using the existence of Galois representations, we apply the result to obtain a lower bound on the cardinality of certain Selmer groups in terms of \documentclass[12pt]{minimal}
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\begin{document}$$L(1,\pi ,{{\,\textrm{Ad}\,}}^\circ )$$\end{document}L(1,π,Ad∘). This can be viewed as partial progress on the Bloch-Kato conjecture. The main technical ingredients are a careful study of the cohomology associated with the locally symmetric space of \documentclass[12pt]{minimal}
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\begin{document}$$\textrm{GL}_n$$\end{document}GLn, its relation to automorphic representations, and the establishment of some algebraic properties of the congruence ideals. We anticipate that the methods developed here will find further applications in related problems, particularly in the study of congruence modules and their relation to the arithmetic of automorphic forms.