Precipitation is a key part of hydrological circulation and is a sensitive indicator of climate change. The Integrated Multi-satellitE Retrievals for the Global Precipitation Measurement (GPM) mission (IMERG) datasets are widely used for global and regional precipitation investigations. However, their local application is limited by the relatively coarse spatial resolution. Therefore, in this paper, an attention mechanism based convolutional network (AMCN) is proposed to downscale GPM IMERG monthly precipitation data. The proposed method is an end-to-end network, which consists of a global cross-attention module, a multi-factor cross-attention module, and a residual convolutional module, comprehensively considering the potential relationships between precipitation and complicated surface characteristics. In addition, a degradation loss function based on low-resolution precipitation is designed to physically constrain the network training, to improve the robustness of the proposed network under different time and scale variations. The experiments demonstrate that the proposed network significantly outperforms three baseline methods. Finally, a geographic difference analysis method is introduced to further improve the downscaled results by incorporating in-situ measurements for high-quality and fine-scale precipitation estimation.