| Popis: |
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 half-integral 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 k-1 and level dividing N/2 and character X2; Sk/2(N,X,Φ) is the subspace of forms that are Shimura-equivalent to Φ; and S0(N,X) is the subspace generated by single-variable theta-series. 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 L-functions of twists of an elliptic curve in terms of the coefficients of modular forms of half-integral weight. Following Tunnell, this often allows us to give a criterion for the n-th twist of an elliptic curve to have positive rank in terms of the number of representations of certain integers by certain ternary quadratic forms. |
