Bryan Birch - Selected Publications#

[1] Bryan Birch, Heegner Points: The Beginnings , pp 1-10 of Heegner Points and Rankin L-Series, MSRI Publications Volume 49 (2004).

[2] B. J. Birch and H. P. F. Swinnerton-Dyer, The Hasse problem for rational surfaces, J. Reine Angew. Math. 274/275
(1975), 164–174.

[3] B. J. Birch, Heegner points of elliptic curves, pp441-445 of Symposia Matematica 15, Istituto Nazionale di Alta Matematica 1975

[4] B. J. Birch, Elliptic curves over Q: A progress report, 1969 Number Theory Institute (Proc. Sympos. Pure Math.,
Vol.XX, State Univ. New York, Stony Brook, N.Y., 1969), Amer. Math. Soc., Providence, R.I., 1971, pp. 396–400.

[5] B. J. Birch, Conjectures concerning elliptic curves, Proc. Sympos. Pure Math., Vol. VIII, Amer. Math. Soc., Providence, R.I., 1965, pp. 106–112.

[6] B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic curves. II, J. Reine Angew. Math. 218 (1965), 79–108.

[7] B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic curves. I, J. Reine Angew. Math. 212 (1963), 7–25.

[8] B. J. Birch, Forms in many variables, Proc. Roy. Soc. Ser. A 265 (1961/1962), 245–263.

[9] B. J. Birch and H. Davenport, On a theorem of Davenport and Heilbronn, Acta Math. 100 (1958), 259–279.

[10] B. J. Birch, Homogeneous forms of odd degree in a large number of variables, Mathematika 4 (1957), 102–105.

It is on the occasion of the two articles [6], [7] that Bryan Birch jointly with Sir Peter Swinnerton-Dyer formulated their famous conjecture (known today as the BSD conjecture). This unexpected conjecture predicts the equality of two fundamental yet very different invariants of an elliptic curve, its rank (which is arithmetic in nature) and the order of vanishing of its L-function (which is analytic in nature.) A quite remarkable aspect of the conjecture (beyond its depth) is that it was corroborated by numerical computations (performed on the computer EDSAC-2 of the University of Cambridge) which in 1965 was quite a challenge both from the arithmetic and the computational viewpoint. This conjecture is extremely deep (for instance it would solve the congruent number problem (whether a given rational number is the area of a right triangle with rational lengths) and it is the motivation of much of the developments in arithmetic geometry of the last 50 years.

The paper [4]] has also been very influential as it introduced the notion of modular symbols which are fundamental geometrical objects very useful for computing modular forms (and hence elliptic curves). In particular the Birch-Stephens formula expressing modular symbols in terms of sums of central values of twisted L-functions is still today a major tool for studying these
mysterious quantities (see the recent work of Mazur-Rubin).

The papers [8] and [10] deal with another area where Bryan Birch has excelled, namely the problem of proving the existence of integral solutions to polynomial equations of large degree in several variables via the circle method. Birch's theorems is still nowadays a natural benchmark with which to test the performance of a new approach in the circle method.