Diophantine Equations VI - The Taniyama-Shimura Conjecture

For over 350 years, generations of mathematicians and laymen have tried unsuccessfully to demonstrate "Fermat's Last Theorem (UTF)" until, in 1994, English mathematician Andrew Wiles succeeded and presented a demonstration along with many revolutionary ideas.

By 1900, German industrialist Paul Wolfskehl, who had devised a plan for his suicide, awaited the final moment of his life in a library where he read the latest ideas about UTF when he suddenly saw the possibility of fixing a fault. in the work of the German mathematician Ernest Kummer and perhaps get a demonstration of the theorem. But there was a flaw in his reasoning. By the time he realized the mistake, the time scheduled for his suicide had passed. This is how Paul Wofskehl gave up killing himself and instituted the 100,000 German mark prize, which would be about $ 1 million today, for anyone who could do the trick of discovering a UTF demo. However, more realistic biographies state that Paul Wofskehl instituted the prize because he did not want to leave his wife an inheritance, because he had married by family imposition. The great mathematician David Hilbert of the early twentieth century, when asked why he did not apply for the prize, said that even if he was intensely dedicated to investigating the problem for three years, he probably would not succeed either. It was Andrew Wiles who received this award from the Göttingen Academy of Sciences in 1997, which has now been transformed into current DEM 75,000 due to inflation 90 years and 10 years before the expiration date set by Wofskehl. Wiles was not awarded the Fields Medal, considered the Nobel Prize in Mathematics, but perhaps the Wofskehl Prize has been the greatest honor of all that he has ever received, or will ever receive, for solving the most famous problem in mathematics.

In 1954 two young Japanese mathematicians, Yutaka Taniyama and Goro Shimura, started a fruitful friendship. Shimura had learned that issue 24 of the Mathematische Annalen was not on the library shelf where it should be. Taniyama had removed him. Strikingly, they were both interested in the same article and the same calculations. From this common interest would arise the Taniyama - Shimura conjecture: "Every rational elliptic curve is modular." Conjecture, one of the most significant in mathematics, became known through the work of French mathematician André Weil and inspired the famous and important "Langlands Program", a large mathematical research project that investigates the deep and subtle relationships between the various areas of mathematics. Thirty years after the Taniyama-Shimura Conjecture, no progress had been made towards its solution. In 1986 the German mathematician Gerhard Frey realized that among the numerous results that Taniyama - Shimura's powerful conjecture implied was the UTF. Frey then suggested a new UTF line of attack using a notion called modularity. Frey's idea was refined by French mathematician Jean-Pierre Serre by facilitating the work of mathematician Kenneth Ribet of the University of California at Berkeley. Ribet demonstrated that if the modularity conjecture were true, then the UTF would follow. More precisely, Ribet showed that if every semi-stable elliptic curve is modular, then the UTF is true. That is, assuming the Taniyama - Shimura conjecture for semi - stable rational elliptic curves, follows the UTF.

However who would be able to demonstrate the Taniyama - Shimura conjecture? Quite simply, Andrew Wiles's achievement was precisely to discover a revolutionary demonstration for a small part of this difficult conjecture.

Back to columns