Coates was born in New South Wales, Australia, and studied at the Australian National University from which he gained a B.Sc degree. He then moved to France, doing further study at the École Normale Superieure in Paris, before moving again to England. There he did postgraduate research at the University of Cambridge, his doctoral dissertation being on p-adic analogues of Baker's method.
In 1969, Coates was appointed assistant professor of mathematics at Harvard University in the United States, before moving again in 1972 to Stanford University where he became an associate professor.
In 1975, he returned to England where he was made a fellow of Emmanuel College, and took up a lectureship. Here he taught Andrew Wiles, the mathematician who went on to find a proof of Fermat's Last Theorem - in which Coates's work on elliptic curves, Iwasawa theory, and p-adic L-functions helped greatly.
In 1977, Coates moved back to Australia, becoming a professor at the Australian National University, where he had been an undergraduate. The following year, he moved back to France, taking up a professorship at the University of Paris XI at Orsay. In 1985, he returned to the École Normale Superieure, this time as professor and director of mathematics.
He was elected a fellow of the Royal Society of London in 1985, and was President of the London Mathematical Society from 1988 to 1990. The latter organisation awarded him the Senior Whitehead Prize in 1997, for "his fundamental research in number theory and for his many contributions to mathematical life both in the UK and internationally".
He now works in the Department of Pure Mathematics and Mathematical Statistics (DPMMS) of the University of Cambridge, where he specialises in number theory, arithmetical algebraic geometry, and Iwasawa theory.
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