On Shimura lifting of modular forms

Abstract

0. Let k,NeN, 4\N. Let M/c+\/2{N,Xo) denote the space of modular forms for Tq(N) of weight k+l/2 with a character xo (modJV). Lifting maps of cusp forms in M^+i^iN^Xo) to modular forms of integral weight was firststudied by Shimura [7] and later by Niwa [4]. The domain of the map is extended to Mk+\ii(N,Xo) by van Asch [1]in case that Xo is real and N = 4p for p prime, and by Pei [5] in case that Xo is real and N/4 is square-free.In the present paper we consider the liftingmap without any condition on N and /0, and extend the domain of the map to Mk+\/2(N,Xo) f°rk >2. To show the assertion, we take some specificmodular forms in Mk+i/2{N,Xo) which together with cusp forms, span M^+1/2(Ar,/0). Further we construct their liftingsexplicitly.It proves our main result.It may be expected to have further application to study of special values of L-series of Hecke eigen cusp forms, as in Zagier [9],Kohnen-Zagier [3] where the liftingof some particular modular forms plavs an important role.