Bryan John Birch

Bryan John Birch F.R.S. (born 1931) is a British mathematician. His name has been given to the Birch and Swinnerton-Dyer conjecture.

As a doctoral student at the University of Cambridge, he was officially working under J. W. S. Cassels. More influenced by Harold Davenport, he proved Birch's theorem, one of the definitive results to come out of the Hardy-Littlewood circle method; it shows that odd-degree rational forms in a "large enough" set of variables must have zeroes.

He then worked closely with Peter Swinnerton-Dyer on computations relating to the Hasse-Weil L-functions of elliptic curves. Their subsequently formulated conjecture relating the rank of an elliptic curve to the order of zero of an L-function was a major influence on the development of number theory from the mid-1960s onwards. As of 2006 only partial results have been obtained.

In later work he contributed to algebraic K-theory (Birch-Tate conjecture). He then formulated ideas on the role of Heegner points (he had been one of those reconsidering Kurt Heegner's original work, on the class number one problem, which had not initially regained acceptance). Birch put together the context in which the Gross-Zagier theorem was proved; the correspondence is now published.

He was elected a Fellow of the Royal Society in 1972; was awarded the Senior Whitehead Prize in 1993 and the De Morgan Medal in 2007 both of the London Mathematical Society.

Written Works

*"Computers in Number Theory." (editor). London: Academic Press, 1973.
*"Modular function of one variable IV" (editor) with W. Kuyk. Lecture Notes in Mathematics "476". Berlin: Springer Verlag, 1975.
*"The Collected Works of Harold Davenport." (editor). London: Academic Press, 1977.

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