On a Result of Atkin and Lehner

We wish to give a new proof of one of the main results of Atkin-Lehner [1]. That paper depends, among other things, on a slightly strengthened version of Theorem 1 below, which characterizes forms in Sk(Γ0(N)) whose Fourier coefficients satisfy a certain vanishing condition. Our proof involves rephrasing this vanishing condition in terms of representation theory; this, together with an elementary linear algebra argument, allows us to rewrite our problem as a collection of local problems. Furthermore, the classical phrasing of Theorem 1 makes the resulting local problems trivial; this is in contrast to the method of Casselman [3], whose local problem relies upon knowledge of the structure of irreducible representations of GL2(Qp). Our proof is therefore much more accessible to mathematicians who aren’t specialists in the representation theory of p-adic groups; the method is also applicable to other Atkin-Lehner-style problems, such as the level structures that were considered in Carlton [2]. Our proof of Theorem 1 occupies Section 2. In Section 3, we explain the links between this Theorem and the rest of Atkin-Lehner theory; in particular, we show that Theorem 1, together with either the Global Result of Casselman [3] or Theorem 4 of Atkin-Lehner [1], can be used to derive all of the important results of Atkin-Lehner theory.