Convolutional neural networks and Transformer have made significant progresses in multi-modality medical image super-resolution. However, these methods either have a fixed receptive field for local learning or significant computational burdens for global learning, limiting the super-resolution performance. To solve this problem, State Space Models, notably Mamba, is introduced to efficiently model long-range dependencies in images with linear computational complexity. Relying on the Mamba and the fact that low-resolution images rely on global information to compensate for missing details, while high-resolution reference images need to provide more local details for accurate super-resolution, we propose a global and local Mamba network (GLMamba) for multi-modality medical image super-resolution. To be specific, our GLMamba is a two-branch network equipped with a global Mamba branch and a local Mamba branch. The global Mamba branch captures long-range relationships in low-resolution inputs, and the local Mamba branch focuses more on short-range details in high-resolution reference images. We also use the deform block to adaptively extract features of both branches to enhance the representation ability. A modulator is designed to further enhance deformable features in both global and local Mamba blocks. To fully integrate the reference image for low-resolution image super-resolution, we further develop a multi-modality feature fusion block to adaptively fuse features by considering similarities, differences, and complementary aspects between modalities. In addition, a contrastive edge loss (CELoss) is developed for sufficient enhancement of edge textures and contrast in medical images.