A canonical subspace of modular forms of half-integral weight

Abstract

We characterise the space of newforms of weight k + 1/2 on Γ₀(4N), N odd and square-free (studied by the second and third authors with Vasudevan) under the Atkin-Lehner W(4) operator. As an application, we show that the (±1)-eigensubspaces of the W(4) operator on the space of modular forms of weight k + 1/2 on Γ₀(4N) is mapped to modular forms of weight 2k on Γ₀(N), under a class of Shimura maps. The existence of such subspaces having this mapping property was conjectured by Zagier in a private communication. One of the special features of the (±1)-eigensubspaces is that the (2k + 1)-th power of the classical theta series of weight 1/2 belongs to the + eigensubspace and hence this gives interesting congruences for r _2k+1(p ²).