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Tamagawa received his PhD in 1954 at the University of Tokyo under Shōkichi Iyanaga. Tamagawa was a visiting scholar at the Institute for Advanced Study in 1955/6, 1958, and 1970. He has been on the Yale University faculty since 1963, and became emeritus in 1996.
He introduced the Tamagawa numbers, which are measures for algebraic groups over algebraic number fields. These measures play an essential role in conjectures on arithmetic algebraic geometry, such as those of Spencer Bloch and Kazuya Kato.