On a Result of Atkin and Lehner

Abstract

We wish to give a new proof of one of the main results of Atkin and Lehner. Their theory depends, among other things, on a theorem characterizing forms in S_k(Gamma_0(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 the problem as a collection of local problems. Furthermore, the classical phrasing of the theorem makes the resulting local problems trivial; this is in contrast to the method of Casselman, whose local problem relies upon knowledge of the structure of irreducible representations of GL_2(Q_p). 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.