MODULAR FORMS OF HALF-INTEGRAL WEIGHTS ON SL(2, Z)

Abstract

In this paper, we prove that, for an integer r with (r, 6) = 1 and 0 < r < 24 and a nonnegative even integer s, the set {eta(24 tau)(r) f(24 tau) is an element of M-s (1)} is isomorphic to S-r+2s-1(new) (6, -(8/r), -(12/r)) circle times (12/.) as Hecke modules under the Shimura correspondence. Here M-s (1) denotes the space of modular forms of weight s on Gamma(0)(1) = SL(2, Z), S-2k(new) (6, is an element of(2), is an element of(3)) is the space of newforms of weight 2k on Gamma(0)(6) that are eigenfunctions with eigenvalues is an element of(2) and is an element of(3) for Atkin-Lehner involutions W-2 and W-3 respectively, and the notation circle times(12/.). There is also an analogous results for the cases (r, 6) = 3.