Background: Alzheimer's Disease (AD) is the most common type of age-related dementia, affecting 6.2 million people aged 65 or older according to CDC data. It is commonly agreed that discovering an effective AD diagnosis biomarker could have enormous public health benefits, potentially preventing or delaying up to 40% of dementia cases. Tau neurofibrillary tangles are the primary driver of downstream neurodegeneration and subsequent cognitive impairment in AD, resulting in structural deformations such as hippocampal atrophy that can be observed in magnetic resonance imaging (MRI) scans. Objective: To build a surface-based model to 1) detect differences between APOE subgroups in patterns of tau deposition and hippocampal atrophy, and 2) use the extracted surface-based features to predict cognitive decline. Methods: Using data obtained from different institutions, we develop a surface-based federated Chow test model to study the synergistic effects of APOE, a previously reported significant risk factor of AD, and tau on hippocampal surface morphometry. Results: We illustrate that the APOE-specific morphometry features correlate with AD progression and better predict future AD conversion than other MRI biomarkers. For example, a strong association between atrophy and abnormal tau was identified in hippocampal subregion cornu ammonis 1 (CA1 subfield) and subiculum in e4 homozygote cohort. Conclusion: Our model allows for identifying MRI biomarkers for AD and cognitive decline prediction and may uncover a corner of the neural mechanism of the influence of APOE and tau deposition on hippocampal morphology.