Fermat's Last Theorem's First Cousin

摘要

In January 1967, a 30-year-old Princeton mathematics professor named Robert Lang-lands wrote to Andre Weil, the dean of the world's number theorists, asking for his opinion about two new conjectures. "If you are willing to read [my letter] as pure speculation I would appreciate that," wrote Langlands; "if not—I'm sure you have a waste basket." Weil never wrote back, but Langlands's letter turned out to be a Rosetta stone linking two different branches of mathematics. He posited that there was an equivalence— rather like a French-English dictionary— between Galois representations and auto-morphic forms. The former describe the intricate relationships among the solutions to equations studied in number theory. The latter are highly symmetric functions. The most familiar examples are the sine and cosine functions, which are periodic, or invariant under horizontal shifts. Such shifts (for example, "move left 2n units" or "move right 4n units") give the same result when performed in any order. The elementary symmetry ot tne sine and cosine functions is as boring to mathematicians as a test pattern. But Lang-lands foresaw that the future of number theory lay in understanding functions with more exotic, order-sensitive kinds of periodicity—func- tions with the infinite complexity of fractals.