Traffic congestion significantly impacts urban environments, costing billions annually and contributing notably to greenhouse gas emissions and air pollution. Ridesharing services, facilitated by widespread smartphone adoption, have emerged as a promising mobility-on-demand (MoD) solution to alleviate these issues. With the advent of Autonomous Modular Vehicle Technology (AMVT), characterized by autonomy and modularity, these vehicles (also known as pods) can dynamically connect to form a pod train (platooning) and consolidate passengers en-route, potentially enhancing energy efficiency beyond conventional ridesharing. This study introduces and formulates an AMVT-based autonomous modular ridesharing system (AMRS) that employs a shareability hypergraph approach and integer linear programming (ILP) to optimally match passenger requests and minimize energy consumption. Numerical experiments show that AMRS reduces energy consumption by up to 34% compared to non-ridesharing scenarios, with modular coordination contributing incremental yet scalable savings (up to approximately 1.4% additional reduction over basic ridesharing at high demand levels). Sensitivity analysis indicates AMRS benefits are most pronounced in denser networks and with higher numbers of requests, highlighting conditions under which modular operations effectively complement ridesharing efficiency.