Modularity of Elliptic Curves and Beyond

MSRI 1999

MODULARITY THEOREM: Every elliptic curve over Q is modular.

Proof. The proof follows a program initiated by Wiles and Taylor-Wiles. See C. Breuil, B. Conrad, F. Diamond, and R. Taylor, On the modularity of elliptic curves over Q.

PSfrag replacements 2 3

243A: y^{2} + y = x^{3} − 1 243B: y^{2} + y = x^{3} + 2

243C: surface 243D: surface

243E: three-fold 243F: three-fold

images of surface 81A

images of 27A: y^{2} + y = x^{3} − 7

The Jacobian of X_{0}(243)

ARTIN’S CONJECTURE: The L-series of any continuous representation Gal(\overline{Q}/Q) → GL_{n}(C) is entire, except possibly at 1.

Results:

— E. Artin, Über eine neue Art von L-Reihen, Abh. Math. Sem. Univ. Hamburg 3 (1924), 89–108.

— J. Buhler, Icosahedral Galois representations, LNM 654, 1978.

— K. Buzzard, M. Dickinson, N. Shepherd-Barron, and R. Taylor, On icosahedral Artin representations.

— G. Frey, On Artin’s conjecture for odd 2-dimensional representations, LNM 1585, 1994.

— E. Hecke, Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen, Math. Z. 6 (1920), 11–51.

— R. P. Langlands, Base change for GL(2), Princeton University Press, Princeton, 1980.

— J. Tunnell, Artin’s conjecture for representations of octahedral type, Bull. AMS 5 (1981), 173–175.