P-Adic family of half-integral weight modular forms via overconvergent Shintani lifting

Abstract

The classical Shintani map (Shintani, Nagoya Math J 58:83–126, 1975) 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 p-adic Hecke-equivariant overconvergent Shintani lifting, for finite slope overconvergent modular forms (Coleman family), which interpolates the classical Shintani lifting p-adically, generalizing the result of G. Stevens in the case of slope 0 modular forms (Hida family) in (Stevens, Contemporary Mathematics, vol 174, 1994) (see the Theorems 3.9 and 3.11). 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 (see the Theorem 4.20). Also the explicit description of Hecke operators on Θ will be given.