Automorphic forms and the Langlands Conjectures

This chapter came from discussions between Patrick, Mario and myself, all currently visiting the Hausdorff Research Institute for Mathematics in Bonn. The ultimate goal is to formally state some version of the global Langlands reciprocity conjectures for \(\operatorname{GL}_n\) over \(\mathbb{Q}\).

• Definition of an automorphic form for \(\operatorname{GL}_n\) over \(\mathbb{Q}\).
• The finite adeles of the rationals.
• The group \(\operatorname{GL}_n(\mathbb{A}_{\mathbb{Q}})\).
• Slowly-increasing functions

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