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Magnetic resonance fingerprinting (MRF) uses a novel imaging framework to achieve simultaneous imaging of multiple tissue parameters. However, the size of the tissue fingerprint dictionary used in MRF grows exponentially as the number of tissue parameters increases, which may result in prohibitively large dictionaries that require extensive computational resources. Existing methods based on convolutional neural networks (CNN) obtain parameter reconstruction patch-wisely, using only local information of the data and resulting in limited reconstruction speed. In this paper, we propose a novel end-to-end local and global vision transformer (LG-VIT) for MRF parameter reconstruction. The proposed network can fully utilize both local and global correlations in both spatial and temporal dimensions of MRF data. In addition, the proposed method enables fast end-to-end parameter reconstruction, while avoiding the high computational cost of high-dimensional data. Numerical experiments show that the proposed network can achieve significantly faster and more accurate MRF parameter reconstruction over state-of-the-art deep learning-based methods.