MATHEMATICS:Fermat's Last Theorem's First Cousin

I MATHEMATICS I Fermat's Last Theorem's First Cousin A proof of the Langlands conjecture for function fields answers a question that has puzzled mathematicians for over 3 decades In January 1967, a 30-year-old Princeton Lafforgue, a number theorist at the Univermathematics professor named Robert Lang- sitk de Paris-Sud. lands wrote to Andrk Weil, the dean of the This fall, thanks to Lafforgue, another world's number theorists, asking for his piece of the program finally fell into place. opinion about two new conjectures. "If you In November, Lafforgue gave the first U.S. am willing to read [my letter] as pure spec- presentation of his proof of the "Langlands ulation I would appreciate that," wrote conjecture for function fields" in a series of Langlands; "if not-I'm sure you have a lectures at the Institute for Advanced Study waste basket." in Princeton, New Jersey. A 300-page handWeil never wrote back, but Langlands's written version of Lafforgue's proof has letter turned out to be a Rosetta stone link- been circulating among mathematicians ing two different branches of mathematics. since the summer, but it has not yet been submitted for publication. Nevertheless, the He posited that there was an equivalenexperts seem quite confident that it will hold rather like a French-English dictionarybetween Galois representations and auto- up. "I'm sure it5 a contender for the Fields morphic forms. The former describe the in- medal:' says Peter Sartricate relationships among the solutions to nak, a number theorist equations studied in number theory. The lat- at Princeton University. ter are highly symmetric functions. The Says Langlands, who most familiar examples are the sine and co- attended the lectures, sine functions, which are periodic, or invari- 'There's nothing suspiant under horizontal shifts. Such shifis (for cious about his arguexample, "move left 271 units" or "move ment. He was a man right 4n units") give the same result when who knew exactly what performed in any order. The elementary he was talking about." symmetry of the sine LC\, +, \<I fl?jb -3lb.5rC and cosine functions \ f= is as boring to mathe,"L\L -5-b t L- y*kk -r .k 4 . A maticians as a test .-,L+L.h --,L.. L*~~oL-\ pattern. But Lang\ * -4\ L lands foresaw that the future of number the4. 4 4 -4 4 , . ory lay in under~ t % ,%q) standing functions with more exotic, order-sensitive kinds Number theorist Robert Langbnds made his eponymous conjeaures in a 1967 letter to And&Weil. of pe,.iodicity-hncThe "fields" that Lafforgue refers to tions with the infinite complexity of fractals. For 30 years, Langlands's questionshave nothing to do with fields in physics which are often called the "Langlands pro- (or, for that matter, agriculture). Mathematigram" because of their many ramifica- cians use the word "i5eId" to denote any altions-have been a driving force in number gebraic structure consisting of objects, or theory. The program has led to two Fields elements, that can be added, subtracted, medals (widely considered the equivalent of multiplied, and divided according to the the Nobel Prize for mathematics) for other rules that govern real numbers. In one large mathematicians. Perhaps the greatest mathe- category of fields, number fields, the obmatical achievement of the 20th century, jects are ordinary rational numbers (fracAndrew Wiles's 1994 proof of Fermat's Last tions), or real numbers, or complex numTheorem, can also be viewed as the comple- be*. Many number theorists deal only with tion of a small part of the Langlands pro- these fields, where the classical questions gram. 'The Langlands program ties together of number theory (such as Fermat's Last theories that are a priori very different and Theorem) first arose. very distant h m one another," says Laurent However, there are two other types of ... - -.\ $+-s- h +- -3 9- fields, each distinctive enough in its properties to attract its own coterie of experts. Perhaps the most bizarre are local fields such as the 7-adic numbers, where live such creatures as the number 1 + 7 i 72+ 73i... These numbers are forbidden in the universe of real numbers because they can become infinitely large. Finally, there are the function fields, whose elements are polynomials or quotients of po~ynom~a~s-~or ex2). Although local ample, (2- 3x + IX(x i fieids -and functio; fields are less Familiar than number fields, they often are easier to study. Lafforgue's work proves Langlands's conjectures only in the context of function fields. In 1998, three other mathematicians proved them for local fields as well. That leaves only the central problem of number fields unresolved. "I don't think it will take a miracle," says Langlands. "There's a hump we've got to get over, an insight that's not out there yet." Historically, Langlands's conjectures arose out of an effort to find very general versions of what number theorists call reciprocity laws-patterns governing how whole numbers can be broken down into sums of products of other whole numbers. (The term is a bit of a mis-' nomer, as reciprocity laws have nothing particularly to do with reciprocals.) These laws date back hundreds of years. In the 17th century, Pierre de Fermat enjoyed solving such questions as this: Which prime numbers can be represented as a sum of two squares? For example, 5 is 2* i12;but 7 cannot be written as a sum of squares. He discovered a simple pattern, whose reasons were nevertheless mysterious: An odd prime number can be written as a sum of two squares if it is 1 greater than a multiple of 4, and not if it is 3 greater than a multiple of 4. Thus the pattern is periodic in the same sense that the sine and cosine functions are. As far as representations as sums of squares are concerned, a shift of the whole prime number system by four units to the left would be invisible. Over the centuries, mathematicians discovered a host of other reciprocity laws, The story seemed to reach a glorious conclusion in 1927, when Emil Artin proved a single reciprocity law that encompassed all the others. Although the number theory behind his work was profound, the geometry was rather banal-the "symmetry groups," or sets of periodicities involved, were onedimensional, like the periodicities of the i3 sine and cosine functions. More complicat- * ed patterns eluded mathematicians until 1967, when Langlands's link to automor- 1 phic forms showed mathematicians how 3 they could bring to bear the theory of ndimensional matrices (or n-by-n tables of 8 4 FEBRUARY 2000 VOL 287 SCIENCE www.sciencemag.org s2 numbers). A number, Langlands realized, is just a 1-by-1 matrix in disguise. Just as shifts can be represented by a single number or 1-by-1 matrix-the distance shifted-he hypothesized that the transformations behind more general reciprocity laws could be represented by matrices. The link has remained conjectural, but the confirmation of one special case led to disproportionately large consequences. The seed of Andrew Wiles's monumental work on Fermat's Last Theorem was his proof of Langlands's conjecture for 2-by-2 matrices whose entries are all 0, I, or 2. Lafforgue's tour de force is unlikely to have such dramatic consequences, because function fields lack the eclat of number fields and Fermat's Last Theorem. However, other mathematicians say that its significance will likely become apparent in time. "Lafforgue has proved that two very different-looking things are the same," says Nicholas Katz, an algebraic geometer at Princeton University. "When you do that, it's almost always the case that there are some properties that are very easy to see one way, and incredibly obscure the other way. . . . It's too soon to predict exactly how it's going to work, but I feel strongly that it's very important." -DANA MACKENZIE tween 1978 and 1997, it followed an upward trajectory. The trend-based on an analysis of more than 20 years of observations by amateur bird watchers-was closely associated with the gentle touch of an atmospheric pressure system called the North Atlantic Oscillation (NAO) and the milder winter The more scientists look, the more connections they see between shifts temperatures it brought. in climate and changes in animal behavior and populations The NAO, which embraces much of the Northern Hemisphere, delivers warm, wet For more than 2 decades, climate model- winters to northern Europe during its high Each year for 31 years, biologist Jerram Brown has trekked into the Chiricahua ers have warned that global warming may phase. (When it flips to low gear, bitter cold Mountains of southern Arizona to chronicle transform our environment by pushing corn usually sets in.) For much of the past 3 the rites of spring for a population of Mexi- belts north, expanding deserts, and melting decades, the high phase has dominated and can jays. Brown adheres to his own ritual: ice caps. Now biologists are getting in on the for the dipper, "a warm year is a good year," The biologist from the State University of action, compiling an impressive array of data Srether says. Because the bird dives for its New York, Albany, notes on which date the suggesting that climate changes big and small food on stream bottoms. it has little to eat if females lay their first clutch, then several can have profound effects on species. Clistreams ice over. The researchers found that weeks later he shinnies ur, 15-meter-tall Chi- mate's fingemrints are turning- ur, in obsewa- the bird's ranks swelled after warm vears. huahua pines to band each tions compiled over thanks to increased immigration and a highand every chick. His peryears and decades. er birth rate in the local population. A longThe sheer com- term warming of 2.5 degrees Celsius, severance has paid off with an intriguing obserplexity of ecosys- Srether's team estimates, should boost diptems makes biolovation: The jays are laying per numbers by 58%. gists reluctant to their eggs earlier and earThe team's mathematical model, which start predicting the takes into account random population fluclier each season. By 1998, the first eggs of the season fate of individual tuations and temperature changes, can be arrived 10 days earlier species based on applied to other species as well, says Peter than in 1971. various climate- Kareiva. a biologist with the U.S. National Brown blames global Early bird.Ajarmer nights seem t o be why change scenarios. oceanic and A&ospheric Administration in But with some mod- Seattle. Unlike previous models, Sather's warming for turning the Mexican jays become parents sooner, 2 hands forward on the jays' els forecasting that teases apart the effects of climate change average global temperatures could rise as from those of density dependence, a phe5 reproductive clock. Although Arizona hasn't H much as 4.6 degrees Celsius in the coming nomenon in which mortality rates tend to n necessarily gotten hotter, it has grown less cool. In the months leading up to the breed- century, the new observations "give us a rise, and birth rates fall, as a population's ing season, Brown found, average daily min- handle to think about where things will be size increases. imum temperatures have nudged up 2.7 de- in 2100," says biologist Camille Parmesan Also in tune with the NAO, it would apgrees Celsius in 27 years. A narrower tem- of the University of Texas, Austin. pear, are grazing mammals. Eric Post and Dippers take a dive. The latest obsewaperature range probably encourages earlier Nils Chr. Stenseth of the University of Oslo 3 breeding by allowing birds to conserve ener- tion of a species' response to climate change in Norway analyzed 15 years of data on $ gy on cold nights, when they can burn off involves Norway's nanorthern mammals. They $ about 10% of their weight just to stay warm, tional bird, the dipper. found that 9 of 11 unguBrown says. The warmer air may also roust On page 854 of this late populations-includissue, a team led by insects earlier, which would likewise proing caribou, musk ox, $ vide extra calories for females to funnel into Bernt-Erik Srether and moose, feral goats, and Jarle Tufto of the Northe energetic business of egg production. Soay sheep-declined Strengthening the case against warming, wegian University of following warm NAO g Brown says, is the fact that many other Science and Technolowinters. But the effects 6 species in the Northern Hemisphere, from gy in Trondheim revaried by location. In P birds to frogs, are also breeding earlier than ports that while the maritime areas, the surthey were years ago. "While no one study number of dippers in a vival rates of Soay sheep $ can prove that earlier breeding is caused by population in southern Warm-weather dipper. A gentle NAO and feral goats improved Norway fluctuated be- boosts populations of this bird. global warming," he says, "it all fits in." during mild winters, How Climate Change Alters Rhythms of the Wild A 9 , 5 $ 3 www.sciencernag.org SCIENCE VOL 287 4 FEBRUARY 2000 793