Implicit formulations of the material point method (MPM) are well suited for quasi-static and long-term geomechanical problems. However, their practical adoption has been limited by the need for accurate Jacobian matrices in Newton-type solvers, particularly for nonlinear and history-dependent constitutive models. In this presentation, we introduce GeoWarp, an open-source, GPU-accelerated implicit MPM framework that integrates reverse-mode automatic differentiation (AD) to enable exact and fully automated Jacobian computation. Built on NVIDIA Warp, GeoWarp combines implicit time integration, kernel-level differentiability, and GPU parallelism, eliminating the need for manual derivation of consistent tangent operators. To address the computational cost typically associated with AD, a sparse Jacobian construction strategy is developed that exploits the localized particle--grid interactions intrinsic to MPM. The resulting algorithm requires only a small, fixed number of backward passes determined by interpolation support, independent of the global problem size. The framework is demonstrated through forward and inverse simulations involving large-deformation elastoplasticity and coupled poromechanics. Numerical results show robust quadratic convergence of Newton iterations, accurate agreement with analytical and experimental benchmarks, and scalable GPU performance. GeoWarp provides a unified and extensible platform for implicit and differentiable MPM, enabling advanced geomechanical simulation as well as gradient-based inverse analysis and optimization.