Brief introduction to automorphic forms in this setting

Having made assumptions on \(D\) which makes the theory of automorphic forms over \(D^\times\) far less technical, we will now make it a little more technical by using the modern adelic approach to the theory. Note that many results about the adeles of a number field are proved in the adele miniproject (section Miniproject: Adeles). Our automorphic forms will be certain functions on the units of the ring \(D_{\mathbb{A}^\infty}:=D\otimes_F\mathbb{A}_F^\infty\cong D\otimes_{\mathbb{Q}}\mathbb{A}_{\mathbb{Q}}^\infty\). To prove Fermat's Last Theorem it suffices to work in weight 2 and trivial character, and for simplicity we shall (at least for now) bake these assumptions into our definitions, even though they would be easy to remove (at the expense of having to write “of weight 2 and trivial character” throughout the proof). We remark again that there is no analogue of the holomorphicity condition that one sees in the classical theory, because the symmetric space is a finite set of points rather than the upper half plane. Also there is no analogue of the cuspidality condition because there are no cusps in this setting. Other than the number field \(F\) and the quaternion algebra \(D\), the other variable we will see in our situation will be the level of the forms. The main result in this miniproject will be that the space of weight 2 automorphic forms of level \(U\) is finite-dimensional.

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