The U_p operator of Atkin on modular functions of level 2 with growth conditions

Abstract

Let p be an odd prime, p ≠ 3, and let g be the polynomial defined by

$$ ( - 1)^{{{(p - 1)} \mathord{\left/ {\vphantom {{(p - 1)} 2}} \right. \kern-\nulldelimiterspace} 2}} g(\lambda ) = \sum\limits_{j = 0}^{{{(p - 1)} \mathord{\left/ {\vphantom {{(p - 1)} 2}} \right. \kern-\nulldelimiterspace} 2}} {({{(\frac{1} {2})_j } \mathord{\left/ {\vphantom {{(\frac{1} {2})_j } {j!}}} \right. \kern-\nulldelimiterspace} {j!}})^2 } \lambda ^j $$

(1)

so that g(λ) is the standard formula for the Hasse invariant of the elliptic curve

$$ Y^2 = X(X - 1)(X - \lambda ). $$

(2)

We shall follow in general the notation of our article [3]. In terms of q-expansions, Atkin [1] has defined the transformation

$$ U_p :\sum a_m q^m \to \sum a_{mp} q^m $$

but without the imposition of growth conditions one may construct eigenvectors with quite arbitrary eigenvalues; indeed formally, for any field element γ,

$$ \theta _j = \sum\limits_{S = 0}^\infty {\gamma ^S q^{p^S } } $$

is trivially eigenvector for eigenvalue γ. Thus to obtain an interesting theory we impose the restriction that U_p be applied to functions satisfying certain growth conditions. To explain these conditions for each pair of positive real numbers b₁,b₂, let L(b₁,b₂) be the space of all functions holomorphic and bounded on the set M$M_{b_1 ,b_2 }$ consisting of all λ such that

$$ \begin{gathered} b_1 > ord g(\lambda ) \hfill \\ b_2 > Max(ord \lambda , ord(1 - \lambda ), ord \lambda ^{ - 1} ). \hfill \\ \end{gathered} $$

(3)