P -adic Family of Half-integral Weight Modular Forms via Overconvergent Shintani Lifting

The classical Shintani map (see [Shn]) is the Hecke-equivariant map from the space of cusp forms of integral weight to the space of cusp forms of half-integral weight. In this paper, we will construct a Hecke-equivariant overconvergent Shintani lifting which interpolates the classical Shintani lifting p-adically, following the idea of G. Stevens in [St1]. In consequence, we get a formal q-expansion Θ whose q-coefficients are in an overconvergent distribution ring, which can be thought of p-adic analytic family of overconvergent modular forms of half-integral weight, since the specializations of Θ at the arithmetic weights are the classical cusp forms of half-integral weight.