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

The study of special values of adjoint <i>L</i>-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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>L</mml:mi> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>π</mml:mi> <mml:mo>,</mml:mo> <mml:msup> <mml:mrow><mml:mrow><mml:mspace></mml:mspace> <mml:mtext>Ad</mml:mtext> <mml:mspace></mml:mspace></mml:mrow> </mml:mrow> <mml:mo>∘</mml:mo></mml:msup> <mml:mo>)</mml:mo></mml:mrow> </mml:math> to the congruence ideals for cohomological cuspidal automorphic representations <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>π</mml:mi></mml:math> of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mtext>GL</mml:mtext> <mml:mi>n</mml:mi></mml:msub> </mml:math> over any number field. We then use this result to deduce relationships between the congruences of automorphic forms and adjoint <i>L</i>-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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>L</mml:mi> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>π</mml:mi> <mml:mo>,</mml:mo> <mml:msup> <mml:mrow><mml:mrow><mml:mspace></mml:mspace> <mml:mtext>Ad</mml:mtext> <mml:mspace></mml:mspace></mml:mrow> </mml:mrow> <mml:mo>∘</mml:mo></mml:msup> <mml:mo>)</mml:mo></mml:mrow> </mml:math> . 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mtext>GL</mml:mtext> <mml:mi>n</mml:mi></mml:msub> </mml:math> , 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.