Conventional genome-wide association studies interrogate individual variants against individual traits. We introduce a multivariate adaptive shrinkage (mash) model that allows each variant to be modeled as a mixture of multivariate normal distributions, boosting power when genetic effects across conditions are shared, using summary statistics. We show that controlling local false sign rates accurately and powerfully identifies replicable genetic associations, and that multivariate control furthers the ability to explain complex disease. We apply this framework to the genetic analyses of blood lipid levels, principal predictors and therapeutic targets for coronary artery disease. Our approach yields high concordance between independent datasets, more accurately prioritizes causal genes, and significantly improves polygenic prediction beyond state-of-the-art methods by up to 59% for lipid traits. Importantly, we describe a framework with important implications for genome-wide association studies and polygenic risk score construction.