Title:

Eigenforms of halfintegral weight

Let k be an odd integer and N a positive integer such that 4  N. Let X be a Dirichlet character modulo N. Shimura decomposes the space of halfintegral weight forms Sk/2(N,X) as Sk/2(N,X) = S0(N,X)oOΦSk/2(N,X,Φ) where Φ runs through the newforms of weight k1 and level dividing N/2 and character X2; Sk/2(N,X,Φ) is the subspace of forms that are Shimuraequivalent to Φ; and S0(N,X) is the subspace generated by singlevariable thetaseries. We give an explicit algorithm for computing this decomposition. Once we have the decomposition, we can exploreWaldspurger's theorem expressing the critical values of the Lfunctions of twists of an elliptic curve in terms of the coefficients of modular forms of halfintegral weight. Following Tunnell, this often allows us to give a criterion for the nth twist of an elliptic curve to have positive rank in terms of the number of representations of certain integers by certain ternary quadratic forms.
