Intersection lane detection is an important functionality that affects the safety and efficient operation of autonomous vehicles. Considering the characteristics of intersection scenarios and lane markings, a multi-modal fusion completion method for lane detection based on intersection scene recognition and road rules is proposed. The proposed method utilizes two types of elements, namely zebra crossings and traffic signals, to recognize intersection scenes. By leveraging the fixed topological structure of intersections and traffic regulations, the lane completion problem is decomposed into straight completion and curve completion. For straight completion, the results from previous frames are directly retained. For curve completion, a curve trajectory model is constructed based on the vehicle's kinematic model. By combining the curve model based on road priors and fixed rules with the curve trajectory model based on the CTRV (Constant Turn Rate and Velocity) model, a multi-modal interaction is performed to complete the curves, ensuring compliance with road boundary constraints. This approach obtains a reliable driving trajectory for intersections, providing decision-making support for autonomous vehicles to drive within safety zones.