Bryan John Birch

From Wikipedia, the free encyclopedia

Bryan John Birch, F.R.S., 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 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 (somewhat loosely) to the Hasse-Weil L-functions of elliptic curves. The 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.