1997 Cole Prize
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The Frank Nelson Cole Prize in Alge-bra is awarded every five years for a notable research memoir in algebra which has ap-peared during the previous five y ears. This prize,as well as the Frank Nelson Cole Prize in Number Theor y
, was founded in honor of Professor Frank
Nelson Cole on the occasion of his retirement as secretary of the American Mathematical So-ciety after twenty-five years and as editor-in-chief of the Bulletin for twenty-one years. The origi-nal fund was donated by Professor Cole from moneys presented to him on his retirement. It has been augmented by contributions from members of the Society, including a gift made in 1929 by Charles A. Cole, Professor Cole’s son,which more than doubled the size of the fund.In recent years, the Cole Prizes have been aug-mented by awards from the Leroy P. Steele Fund and currently amount to ,000.
The twenty-fifth Cole Prize has been awarded to Andrew Wiles . The prize was presented at the Society’s 103rd Annual Meeting in San Diego in January 1997. The Cole Prize was awarded by the Council of the American Mathematical So-ciety, acting through a selection committee con-
sisting of Hyman Bass, Karl Rubin (chair), and Wolfgang Schmidt.
The text below includes the committee’s ci-tation, a brief biographical sketch, and a re-sponse from Andrew Wiles upon receiving the award.
The 1997 Frank Nelson Cole Prize in Number Theory is awarded to Andrew Wiles for his work on the Shimura-Taniyama conjecture and Fer-mat’s Last Theorem, published in “Modular el-liptic curves and Fermat’s Last Theorem”, Ann.of Math. 141(1995), 443–551. Fermat proved his “Last Theorem” for exponent 4 by developing the theory of elliptic curves. But there was no ap-parent connection between elliptic curves and higher exponent Fermat equations, so elliptic curves played no further role in work on Fermat’s Last Theorem for almost 350 years, by which time it had become the most famous unsolved problem in mathematics.
The first person in modern times to connect elliptic curves with Fermat’s Last Theorem was Y. Hellegouarch in the 1970s. Then about ten years ago, G. Frey suggested and K. Ribet proved (building on ideas of B. Mazur and J.-P. Serre) that Fermat’s Last Theorem follows from the Shimura-Taniyama conjecture that every ellip-tic curve defined over the rational numbers is modular. Precisely, if
a n +
b n =
is a counterexample to Fermat’s Last Theorem,then the elliptic curve
y2=x(x−a n)(x+b n)
cannot be modular, thus violating the Shimura-Taniyama conjecture. This result set the stage for Wiles’ work. Using Mazur’s deformation the-ory of Galois representations, recent results on Serre’s conjectures on the modularity of Galois representations, and deep arithmetic properties of Hecke algebras, Wiles (with one key step due jointly to Wiles and R. Taylor) succeeded in prov-ing that all semistable elliptic curves defined over the rational numbers are modular. Although less than the full Shimura-Taniyama conjecture, this result does imply that the elliptic curve given above is modular, thereby proving Fer-mat’s Last Theorem.
Wiles’ work is highly original, a technical tour de force, and a monument to individual perse-verance. In addition, it serves as encouraging ev-idence that the abstract machinery of modern arithmetic algebraic geometry has the power to solve long-standing classical problems.
For further reading see the introduction of Wiles’ cited paper for a very readable account of the history of his attack on Fermat’s Last Theo-rem. Among several other accounts of this work and the excitement surrounding it are four pieces in the Notices of the AMS (July/August 1993, 575–576; March 1994, 185–186; January 1995, 48; July 1995, 743–746) and two in the Bulletin of the AMS (July 1994, 15–38; October 1995, 375–402).
Andrew J. Wiles was born in Cambridge, England, on April 11, 1953. He attended Merton College, Oxford University, starting in 1971, and he re-ceived his B.A. there in 1974. That same year he went to Clare College, Cambridge University, earning his Ph.D. there in 1980. From 1977 until 1980, Wiles was a Junior Research Fellow at Clare College and a Benjamin Peirce Assistant Professor at Harvard University. In 1981 he was a visiting professor at the Sonderforschungs-bereich Theoretische Mathematik in Bonn, and later that year he was a member of the Institute for Advanced Study in Princeton. In 1982 he be-came a professor at Princeton University and in the spring of that year was a visiting professor at the Université de Paris, Orsay.
On a Guggenheim Fellowship he was a visit-ing professor at the Institut des Hautes Études Scientifiques and at the École Normale Supérieure (1985–86). From 1988 to 1990 he was a Royal So-ciety Research Professor at Oxford University. In 1994 he assumed his present position as the Eu-gene Higgins Professor of Mathematics at Prince-ton.
Wiles was elected a Fellow of the Royal Soci-ety, London, in 1989. In 1995 he received the Schock Prize in Mathematics from the Roy al Swedish Academy of Sciences. That same year he was awarded the Prix Fermat, presented by the Université Paul Sabatier and Matra Marconi Space. In 1996 Wiles received the Wolf Prize in Mathematics. Wiles was elected a foreign mem-ber to the U.S. National Academy of Sciences (NAS) in 1996 and also received the 1996 NAS Prize in Mathematics.
Response from Andrew Wiles
It is a pleasure to thank the American Math-ematical Society and the Selection Committee for the award of the Cole Prize in Number Theory. Needless to say on the path to Fermat I benefited enormously from the work of many people, not only of Frey and Ribet, who directly inspired it, but also the many others who knowingly influ-enced my thinking along the way. I thank them all. Finally, I would like to acknowledge my debt to both Pierre and Samuel Fermat.
M ARCH1997N OTICES OF THE AMS347