Eigenforms of half-integral weight

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Abstract

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.

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