A common challenge in testing for pleiotropy, mediation, and replication in genetic association studies is accounting for a composite null hypothesis. For instance, consider testing for pleiotropic SNPs with two outcomes. The null hypothesis for this problem includes the case where a SNP is associated with no phenotypes as well the cases where a SNP is associated with only one phenotype. A similar situation arises in mediation analysis, where we only want to reject the null hypothesis of no mediation effect when the coefficient of interest is non-zero in both the mediator and outcome models, and in testing for replication, where we want to identify SNPs that demonstrate association across multiple GWAS. Popular approaches - such as the Sobel test or maximum p-value test for mediation - often produce highly conservative inference, resulting in lower power. Borrowing ideas from replicability analysis, we extend an empirical Bayes framework to allow for inference in all three settings. Simulation demonstrates that our approach can control false discovery proportion across various scenarios, and we apply our methods to GWAS of lung cancer and heart disease.