# John Henry Coates

### Quick Info

Born
26 January 1945
Possum Brush, Manning Valley, New South Wales, Australia
Died
9 May 2022
Cambridge, England

Summary
John Coates was an Australian mathematician who was the Sadleirian Professor at Cambridge and worked in algebraic number theory.

### Biography

John Coates' mother, Beryl L Lee, was from an Irish family that had emigrated to Australia following the Irish potato famine of 1845-50. The family settled in Manning Valley as, sometime later, did John's paternal grandfather who was from Ilkley, Yorkshire, England. John's father, J R Coates, taught French at a high school but a series of mental breakdowns led to him retiring from teaching and managing the farm his father owned near the Manning River. John and his older brother were brought up on the farm:-
... it was totally unspoilt bush ... with animals all around so tremendous for a little boy to grow up in.
Sadly his mother died when he was seven years old and after this he spent time living with his aunt at the nearby town of Cundletown also on the Manning River. By this time John was being educated at a very small school, with only one teacher, in a village about five miles away from the farm on which he lived. He remained at the school until he was eleven years old when his elder brother was sent to the High School in Taree, about fifteen miles from his home. John's father managed to arrange for John to go to Taree Primary School for his final primary year. He then joined his brother at Taree High School where he became interested in physics but also received excellent mathematics teaching from Jack Gibson. When John was thirteen years old his father collapsed again with similar health problems that had plagued him earlier in his life.

After graduating from Taree High School, Coates hoped to continue his education at university but fully understood that his father was not in a financial position to pay his fees. He therefore realised that he would have to win a scholarship, but first he took a summer vacation job at BHP Billiton, a big mining company in Newcastle. There was the possibility of obtaining a university scholarship from the company for they funded two or three for those working on internships each year. However, John failed to win a scholarship but he had also applied to the Australian National University for a scholarship and was invited for an interview. He was interviewed by Howard Florey who offered him a scholarship. He began his studies in mathematics, physics and chemistry, still with the intention of taking a physics degree. However, during his first year of study at A.N.U. he found that the mathematics teaching was much better than the physics teaching and he soon decided that he would specialise in mathematics. It was an exciting time at A.N.U. since Bernhard Neumann had been appointed as head of the Department of Mathematics in the Institute of Advanced Studies in 1963, while Hanna Neumann headed the undergraduate teaching. Kurt Mahler had also been appointed at A.N.U. and he taught Coates a course in elementary number theory during the second semester of his first year of study. Coates was fascinated by number theory and this course played a large role in his decision to move from physics to mathematics. In his final year as an undergraduate, Coates did an honours project supervised by Mahler which involved reading one of Mahler's unpublished papers. It was Coates' introduction to research.

During his undergraduate studies, Coates met Julie Turner who was also an undergraduate at A.N.U. in the year below him. She was the daughter of a Member of Parliament and was studying politics. They married in 1966 and had three sons. After the award of a B.Sc., Coates was awarded an A.N.U. scholarship and was advised by Kurt Mahler and Hanna Neumann to go to study for his doctorate at the École Normale Supérieure in Paris. His wife joined him in Paris after completing her undergraduate degree. However, Coates was unhappy with the areas of research in Paris feeling that he did not have the mathematical background to cope with the abstraction. He wrote to Cassels at Cambridge in England asking if he could move to Cambridge. He was accepted so, after spending a year in Paris, he went to Cambridge to complete his doctorate:-
We arrived in Cambridge in mid-summer and I was a student at Trinity College ... they gave us a nice flat above 'The Blue Boar' until October but then we had a delightful flat in Westminster College, above the Master's Lodge.
Coates' thesis advisor at Cambridge was Alan Baker and, as soon as he arrived, he made the decision to start a new research project rather than continue the one he had begun in Paris. His doctoral dissertation was an outstanding piece of work on $p$-adic analogues of Baker's method.

Continuing to move round some of the leading centres for mathematical research, Coates obtained a position as assistant professor of mathematics at Harvard University in the United States in 1969. From Harvard he moved to Stanford University in 1972, where he was an associate professor, and, after a further three years he returned to England to take up a lectureship at the University of Cambridge in 1975. He became a fellow of Emmanuel College during this period at Cambridge and it was during this time that Andrew Wiles was his research student.

At this stage Coates had returned to work in one of the three universities that he had studied in and indeed he continued with this pattern in 1977 when he was appointed as professor at the Australian National University. However, he did not remain for long in his home country, moving back to France in 1978 to a professorship at the University of Paris XI at Orsay. In 1985 he took up the positions of professor and director of mathematics at the École Normale Supérieure. In the same year he was elected to a fellow of the Royal Society of London.

In 1986 Coates returned to Cambridge when he was appointed to the Sadleirian Chair of Mathematics and he was also elected a fellow of Emmanuel College in Cambridge for the second time. In 1991 he became Head of the Department of Pure Mathematics and Mathematical Statistics at Cambridge.

Coates's first major mathematical publications were in 1966-67 when he published four articles on the algebraic approximation of functions. Then, together with A Baker, he extended Mahler's work on fractional parts of powers of rational numbers. In other work he looked at problems relating properties of algebraic number fields to algebraic K-theory.

He then proved certain special cases of Weil's conjecture on elliptic curves. He worked on Iwasawa's theory and wrote a number of articles with Andrew Wiles published around 1977-78 including Kummer's criterion for Hurwitz number, Explicit reciprocity laws and On p-adic L-functions and elliptic units. As stated in [1]:-
His 1977 article on the conjecture of Birch and Swinnerton-Dyer, written jointly with his research student Andrew Wiles, was a landmark contribution to number theory which introduced a panoply of new methods into the field of elliptic curves.
Wiles had studied for his doctorate under Coates at Cambridge from 1974 and this proved an important link in the various strands which led to Wiles's proof of Fermat's Last Theorem. During the 1980s Coates's work was concerned with elliptic curves, Iwasawa theory and p-adic L-functions, all work closely related to the direction that would eventually yield the proof of Fermat's Last Theorem. From [1]:-
Coates's insights into the Iwasawa theory of the symmetric square of an elliptic curve were instrumental in the recent proof by Wiles of the Shimura-Taniyama conjecture for semistable elliptic curves over $\mathbb{Q}$.
Not only was Coates an outstanding researcher but he also had a reputation as a teacher of the highest quality [1]:-
John Coates is an inspired teacher and expositor who has made decisive contributions to the training of research students and junior colleagues, many of whom have gone on to fruitful research careers.
Again his contribution as an editor was singled out for praise in [1]:-
He was for many years an energetic and active editor of Inventiones Mathematicae, one of the premier journals of research mathematics, often making decisive stylistic improvements to articles which he deemed important but insufficiently well crafted.
Coates served as president of the London Mathematical Society during 1988-90 and as vice-president of the International Mathematical Union from 1991 to 1995. During 1992-94 he served as a member of the Council of the Royal Society, then, in 1996, he served on a group set up by the Royal Society to examine the peer review system used for funding research. The London Mathematical Society awarded Coates their 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.

### References (show)

1. Citation for John Henry Coates, Bull. London Math. Soc. 30 (4) (1998), 442.