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The goal of the crime forecasting problem is to predict different types of crimes for each geographical region (like a neighborhood or censor tract) in the near future. Since nearby regions usually have similar socioeconomic characteristics which indicate similar crime patterns, recent state-of-the-art solutions constructed a distance-based region graph and utilized Graph Neural Network (GNN) techniques for crime forecasting, because the GNN techniques could effectively exploit the latent relationships between neighboring region nodes in the graph if the edges reveal high dependency or correlation. However, this distance-based pre-defined graph can not fully capture crime correlation between regions that are far from each other but share similar crime patterns. Hence, to make a more accurate crime prediction, the main challenge is to learn a better graph that reveals the dependencies between regions in crime occurrences and meanwhile captures the temporal patterns from historical crime records. To address these challenges, we propose an end-to-end graph convolutional recurrent network called HAGEN with several novel designs for crime prediction. Specifically, our framework could jointly capture the crime correlation between regions and the temporal crime dynamics by combining an adaptive region graph learning module with the Diffusion Convolution Gated Recurrent Unit (DCGRU). Based on the homophily assumption of GNN (i.e., graph convolution works better where neighboring nodes share the same label), we propose a homophily-aware constraint to regularize the optimization of the region graph so that neighboring region nodes on the learned graph share similar crime patterns, thus fitting the mechanism of diffusion convolution. Empirical experiments and comprehensive analysis on two real-world datasets showcase the effectiveness of HAGEN.