Newforms of half-integral weight

Abstract

Two very different definitions of a newform of half-integral weight are present and continued to be developed in the literature. The first definition originated with Serre and Stark for forms of weight 1/2 [5], and is analogous to the definition of newform for integral weight forms, which uses forms of lower level and shifts of such forms to characterize the notion of old-forms. The second definition originated with Kohnen for half-integral weight forms of squarefree level [1], who used forms of lower level and their image under the Um2 operator to define the notion of oldforms. The choice of the Um2 operator over the shift operator Bd seems a propitious one, since the U operator commutes with the action of the Shimura lift, while the shift operator B does not. More to the point, Kohnen was able to develop a newform theory on a distinguished subspace of the full space of cusp forms (now referred to as the Kohnen subspace), and obtained a multiplicity-one result (with respect to Hecke eigenvalues) for half-integral weight newforms in this subspace. Even nicer, the multiplicity-one result was established by showing that there is a one-to-one correspondence between newforms of level AN in the subspace and the newforms of integral weight of level N.