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

Abstract. The goal of this paper is to construct the p-adic analytic family of overconvergent half-integral weight modular forms using Hecke-equivariant 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. Glenn Stevens proved in [St1] that there is a Λ-adic lifting of this map to the Hida family of ordinary cusp forms of integral weight via the cohomological description of the Shintani correspondence and consequently constructed a Λ-adic modular eigen form of half-integral weight(For the Λ-adic Shimura lifting see [Hi2]). The natural thing to do is generalizing his result to non-ordinary case, i.e. Coleman’s p-adic analytic family of overconvergent cusp forms of finite slope. For this we will use a slope ≤ h decomposition of compact supported cohomology with values in overconvergent distribution (overconvergent modular symbol), which can be interpreted as a cohomological description of Coleman’s p-adic family, and define the p-adic Hecke algebra for the slope ≤ h part of this cohomology. Then we follow the idea of [St1] in the ordinary case. This construction implies that we found a rigid analytic map at least locally from Coleman-Mazur integral weight Eigen curve to half-integral weight Eigen curve(see [Ram2]), while assumming the global existence of the integral and half-integral weight Eigen curve of higher tame level(see [Buz]). This lifting also can be viewed as the inverse construction of Nick Ramsey’s overconvergent Shimura lifting in [Ram1]. The important feature of overconvergent Shintani lifting is that it’s purely cohomological and algebraic, which depends only on the arithmetic of integral indefinite binary quadratic forms, so that we can describe Hecke operators explicitly on q-expansion of universal overconvergent half-integral weight modular forms using the Hecke action on overconvergent modular symbol.