P-adic family of half-integral weight modular forms via 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 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 Lamda-adic lifting of this map to the Hida family of ordinary cusp forms of integral weight and consequently constructed Lamda-adic modular eigen form of half-integral weight. The natural thing to do is generalizing his result to non-ordinary finite slope case, i.e. Coleman 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, which can be interpreted as a cohomological description of Coleman 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. We also describe Hecke operators explicitly on q-expansion of universal overconvergent half-integral weight modular forms using the Hecke actions on overconvergent modular symbol.