THE RAMANUJAN JOURNAL, Vol. 14, No. 2 (2007), 329-337.

Differential automorphisms for modular forms on $\Gamma_0(4)$

TIMOTHY KILBOURN
Department of Mathematics, University of Illinois, Urbana, IL 61801, USA
Email: tkilbour@math.uiuc.edu

Received February 3, 2004; Accepted July 15, 2004

Abstract. If $k \in \frac{1}{2}\N$, then let $M_k(\Gamma_0(4))$ be the usual space of half integral weight modular forms. Ono constructed differential endomorphisms of $M_k(\Gamma_0(4))$ by using the usual differential operator. Here we construct a similar set of differential endomorphisms using a linear combination of the differential operator and the quasi-modular forms $E_2$, $E_2 \mid V_2$, and $E_2 \mid V_4$. We compute a full set of eigenforms with eigenvalues, and we prove that these endomorphisms are in fact automorphisms.

Keywords: Modular forms, differential operators

2000 Mathematics Classification: Primary 11F11; Secondary 11F25, 11F37