On some results of Atkin and Lehner

TLDR: In this article, the authors prove two basic if elementary results related to the theory of automorphic forms for GL2, local and global, and there is no immediate connection between them, but they were both suggested by some results of Atkin and Lehner and Ogg [7] in the classical theory of modular forms.

Abstract: In this Paper I shall prove two basic if elementary results related to the theory of automorphic forms for GL2. One is local and the other global, and there is no immediate connection between them, but they were both suggested by some results of Atkin and Lehner [1] and Ogg [7] in the classical theory of modular forms. They are both formulated and proved in terms of group representations, and in fact the proofs are easy pickings from Jacquet-Langlands [4]. In the last Section I shall outline the relationships between these two theorems and the results of [1] and [7]. Most of this is probably well known, but it may be useful to have it expressed, once, in print. The final few remarks pose what seems to me to be one of the fundamental questions in the subject, which deserves to be better known. Partial generalizations by Miyake of the classical results mentioned were the immediate spur to this paper. Roughly at the same time, and independently, Miyake found essentially the same results ([6]). His proofs, especially for his equivalent form of the global theorem, are fairly close to the ones given here, but follow Weil [11] rather than the methods of representations. The proof of Theorem 2 follows a suggestion of Langlands'. My own original idea (which I did not follow up in detail) was considerably more complicated, although perhaps eventually capable of yielding a refinement, it involved a generalization of a result of Rankin's (see [8] or [9]). I shall use the terminology of [4] generally, but not always, without explicit reference.