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

• B. Dwork
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 350)

## 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 . In terms of q-expansions, Atkin  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 Up 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)

## References

1. 1.
Atkin, A. O. Congruence Hecke operators, Proc. Symp. Pure Math. 12, pp.33–40.Google Scholar
2. 2.
Dwork, B. Amer. J. Math. 82(1960), pp.631–648.
3. 3.
Dwork, B. Inv. Math. 12(1971), pp.249–256.