The Weil Conjectures

by Karen Olsson

  Not everyone was deferential. “The life of this remarkable woman still intrigues me while much of what she writes, naturally, is ridiculous to me,” pronounced Flannery O’Connor, in a letter to a friend who’d sent her a collection of Simone’s writings. “Her life is almost a perfect blending of the Comic and the Terrible, which two things may be opposite sides of the same coin.”

  Simone was more often quasi-canonized as a kind of genius or dismissed as a nutjob than she was recognized as something in between, as human. I suppose she herself bears some of the blame for this, determined as she was to detach herself as much as possible from ordinary personhood.

  Outside of mathematical circles André was never known in his adopted country. When an English translation of his memoir appeared in 1992, it was reviewed only in scientific publications. As a public matter, his story ended well before his death, while Simone’s demise got the ball rolling. In this century, though, you might say there’s been, if not a reversal, then a leveling: Simone’s fame and influence have dwindled, while her brother’s mathematical discoveries remain in place, struts supporting later advances made by others.

  Sylvie Weil also published a memoir, Chez les Weil, about her father and her aunt and her own life in the shadow of all that genius. She writes of meeting a man who’d known Simone in London during the war and who remembered her, in specific and convincing detail, as fragile and tired and shy and isolated—which came as a shock to Sylvie, so different was his description from the aunt whom her father and grandparents had remembered, Simone as indestructible, as a force of nature.

  The same man was certain that Simone had known, in 1942, of the deportation of Jews to Nazi camps. What has bewildered and troubled many people, her admirers and critics both, is that she never directly mentioned it, not once in all the reams of writing that spilled out of her before she died.

  12.

  What is it about these bygone thinkers, these dead mathematicians, that captures me? A fond reverence muddled with a strain of muted pity, a distant, tainted love: I can’t exactly name this feeling and so keep resorting to their stories; I’m still trying to describe it or at least circumscribe it, and so (with apologies) I’ll indulge in one more digression.

  In January 1954, a graduate student at the University of Tokyo went to the library to look for Mathematische Annalen, volume 124, only to find that Mathematische Annalen, volume 124, had been checked out weeks earlier by another graduate student, whom he knew in passing. The first student, Goro Shimura, wrote to the second one, Yutaka Taniyama, to ask whether Taniyama might return Mathematische Annalen, volume 124, so he could read an article that described a theory of complex multiplication. Taniyama sent his reply by postcard, addressed from the town of Kisai, where his parents lived. It appeared that the two of them were working on the same problem, Taniyama wrote; maybe they could talk sometime.

  They were drawn to algebraic geometry and so to the work of André Weil, then teaching at the University of Chicago. Not long before writing to Taniyama to ask for Mathematische Annalen, volume 124, Shimura had sent a manuscript to Weil, in Chicago. As an undergraduate Taniyama had read Weil’s Foundations of Algebraic Geometry as well as some of his papers. Both students had found little to admire in the older generation of Japanese mathematicians, whom they deemed full of themselves and all too prone to offering useless remarks in the guise of advice, and so instead they learned from each other and from distant idols.

  At that time you could see, from the center of Tokyo, the crest of Mount Fuji some seventy miles to the west, a view that pollution would eventually obscure. Taniyama lived on the second floor of a run-down wooden structure wishfully named Villa Tranquil Mountains, though in reality it sat on a narrow urban street and bore no resemblance to a villa. A sort of a ramshackle dormitory, it had, on each of its two floors, twelve tiny dwellings—small rooms with a sink attached to one wall—and a shared toilet. Baths were taken at a nearby public bathhouse.

  Taniyama, as Shimura would later remember him, had few interests outside of mathematics. He wore a shimmery suit made from fabric his father had bought cheaply; he ate whale meat and tongue stew at restaurants. He often left his shoes untied. He would work late into the night, sometimes until dawn. I picture the young mathematician at his desk, after midnight, in an undershirt and iridescent pants, hardly aware that he is picking a bit of blubbery gristle from his teeth. He grapples with fleeting modular forms and elliptic functions, while behind the thin walls of the surrounding miniature apartments, his neighbors work troughs into their pillows and dream of childhood friends or long journeys or snow.

  Shimura was an early riser. He worked in the mornings, and often in the afternoons he would make the trip from his university office to Villa Tranquil Mountains. If Taniyama was awake, the two would huddle in his room and riff on the properties of elliptic curves over algebraic fields. If Taniyama was asleep, Shimura would return home and record the fact in a diary he kept with the precision of a captain’s log. For most if not all of any given twenty-four-hour period, at least one of them would be up, and so together they could be a kind of round-the-clock mathematician, able to labor at a problem continuously.

  Age is just a number, they say. Years are also just numbers—if at times they seem unjust, succeeding one another with such brutal linearity. One pleasure in writing arises from the illusion of holding those numbers in your hand, stopping time or running time backward or hopping back and forth from one year to another, making a game of it, huddled in an imaginary tent pretending to be a kid pretending to go camping.

  Math affords a different kind of timelessness. The mathematician manipulates systems of abstract objects that seem to exist (in whatever way they exist) outside of time; she plays in a field of the forever true. And becomes, habitually, lost in thought. She looks up from her desk and notices that the sun has set.

  Think of an ordinary curve, a looping line drawn on a sheet of paper, as a subset of all possible points on the paper. Now generalize that idea to other spaces, other dimensions: conceptual ripples in conceptual dark seas. Think of families of curves and of ways to bundle them together. Think of the path from Shimura’s department office to Villa Tranquil Mountains. Think of the twisting course of the long walks Shimura and Taniyama take through the streets of Tokyo.

  The two young men are at work on a series of tramways, erecting them section by section. One of them rises promptly, reports to the worksite, dons his hard hat, measures the existing track, and steadies the girders underneath it. The other, late at night, sketches unexpected new routes.

  Some afternoons they meander about the botanical garden together, talking about Riemann surfaces, eyeing women as they pass. Or they visit a Shinto shrine where “oracles” are sold, fortunes printed on small pieces of paper. The early riser learns that he will one day have children. The late-nighter unfolds his paper, reads what is written there, folds it back up, and attaches it to a chain-link fence where hundreds of other paper fortunes flap against the metal—this being the place where unwelcome predictions are discarded.

  Age is just a number, but your days are numbered. When Simone Weil died she was thirty-four. When André Weil died he was ninety-two.

  When Yutaka Taniyama died he was thirty-one. As I write this, Goro Shimura is nearing ninety. In 1989, when he was in his early seventies, he published a beautiful essay in the Bulletin of the London Mathematical Society, called “Yutaka Taniyama and His Time: Very Personal Recollections.”

  In 1955, André agrees to attend the International Symposium on Algebraic Number Theory, held in Tokyo and Nikko. Weil himself in Tokyo! Lately a mood of ambition and optimism has spread through the whole country, one embodied by the young mathematicians, Taniyama and Shimura and a handful of others, who await André like they would a prophet. He arrives in the city several weeks before the beginning of the conference, and once they know he’s nearby, they grow impatient, joking anxiously with one another.

  It’s the middle of Au
gust. They have a chance to shake his hand at a university reception, which doesn’t exactly change anything for them, but then again the fact that Weil is in town changes everything, the sounds and sights of their city altered. From the outside everything comes at them more acutely; on the inside everything whirls, stops, whirls in the other direction.

  One day Shimura is summoned to the Prince Hotel Annex, where Weil is staying. The professor wishes to meet with him and discuss his work, the secretary of the mathematics department informs him over the phone. Young Shimura wears a jacket and tie but finds Weil, standing in the lobby, in tan slacks and an open shirt. They speak English, rather formally, as they wander over to a small courtyard and sit down. A shiver vibrates through the younger man’s body, for here he is in a private meeting with the mighty Weil, the name on all those papers and books, now attached to a short bespectacled Frenchman, who directs a young woman to bring them tea and cake.

  No sooner has she turned her back than Weil hits Shimura with questions about his research. Shimura does his best to answer, though he’s sweating in his jacket and his mouth is growing thick with foreign constructions. Is he actually listening to any of this, Shimura wonders, as the older man starts to scribble formulas on hotel stationery. Weil interrupts, then interrupts himself by coughing, then stands and begins to pace, marching toward the other end of the courtyard, where dogwood branches paw over the top of an iron gate.

  He comes and goes, back and forth, and soon enough he’s the only one talking, and talking some more, trying to pack Shimura’s head full of his ideas, though Shimura’s head is already at capacity. After the woman comes back with his order, Weil devours a large piece of cake. Shimura has lost his appetite.

  Once the conference starts, the young mathematicians make prank calls to one another in which they pretend to be Weil. They’ll call someone up and imitate his voice and try to speak English with a French accent. “Hello, zis is Weil,” they’ll say, in their Japanese accents.

  In his research Taniyama took on a thorny open problem, and he would characterize his efforts to untangle it as “hard fighting” against difficulties and a “bitter struggle” of trial and error. Indeed, any significant mathematical undertaking, he said, was a matter of hard fighting and bitter struggle.

  Shimura noticed something else about the way his friend worked: Taniyama was “gifted with the special capability of making many mistakes,” he wrote, “mostly in the right direction.”

  A notion that turns out to be wrong might still point the way forward, provided it’s wrong in the right way. “I envied him for this, and tried in vain to imitate him,” Shimura continued, “but found it quite difficult to make good mistakes.”

  Corazonada is a Spanish word for hunch, and I like how it implicates the heart (corazón) in our intuitions, not just the bent spine that I’m reminded of by the English. At the conference Taniyama discloses his emerging hunch, the heart of his work so far, to Weil, who is eager to discuss it.

  WEIL: Do you think all elliptic functions are uniformized by modular functions?

  TANIYAMA: Modular functions alone will not be enough. I think other special types of automorphic functions are necessary.

  Et cetera.

  Fluttering about the conference is a crew of pretty young hostesses, there to help the foreign visitors. André mentions one of them, Momoko-san, in his letters home, and his daughters are delighted to imagine her as a kind of doll come to life, while Eveline is not so delighted.

  There’s a passage in Proust’s In Search of Lost Time in which the narrator rereads a letter from a lover who has died, and he feels, fleetingly, a joyful surge of expectation, as though he’d been sent back to the time when he first read the letter, while his beloved was still alive. The narrator then reflects upon how, remembering episodes from the past, we may find ourselves suddenly disoriented in time, inhabiting a past moment as though it’s our current reality, so that we briefly, delusionally look forward to a future that’s already gone by. We anticipate prospects that already came to pass (or didn’t) years ago. “The illusion swiftly dies,” Proust writes, “but for a second we felt ourselves driven forward once more: such is the cruelty of memory.”

  I wonder whether Shimura experienced this phenomenon as he wrote his essay, whether in remembering Taniyama he might’ve slipped into looking forward to seeing Taniyama again, to walking through the botanical garden with him, to discussing math problems—only to fall, again and again, back into the present.

  Of course I don’t know, but his essay from the Bulletin of the London Mathematical Society is suffused with a melancholy that gives way, at its close, to grief for his lost friend.

  The idea that Taniyama and Weil bandied about at the conference would crystallize as a conjecture, one that Shimura made precise and conveyed to Weil, who articulated it in a paper that failed to mention either Shimura or Taniyama. This conjecture would be called, at first, the Weil conjecture (distinct from his 1949 Weil conjectures), yet as details of its genesis became more widely known, the name was modified to the Taniyama-Weil conjecture, then the Shimura-Taniyama-Weil conjecture, or the Taniyama-Shimura-Weil conjecture; if you were hostile to Weil you called it the Taniyama-Shimura conjecture or the Shimura-Taniyama conjecture; while if you enjoyed provoking Shimura you might still call it the Taniyama-Weil conjecture. One mathematician, Michael Harris, attempted to sidestep the whole mess by calling it “the conjecture associated with the names of” Taniyama, Shimura, and Weil. Nonetheless, Harris wrote in a blog post, he and a colleague who’d also adopted this construction “walked into an avalanche”—of criticism, presumably. Now it is not uncommon to drop all the names and refer to it as “the modularity conjecture for elliptic curves.”

  It says: every elliptic curve defined over the rational numbers is a modular form. Which is over my head, but the crux of it, I’m told, is another unlikely link between topology and number theory.

  In 1957, Shimura left to spend a year in France. Shortly before his departure, he and Taniyama attended a dinner party given by a young woman named Misako Suzuki, who made fun of Taniyama because he hardly spoke during the meal. Not long after that she and Taniyama became engaged and signed a lease on an apartment. Shimura would call her a “typically pleasant girl from a typically upper middle-class family,” whom he never got to know well.

  It wasn’t Shimura, by then in France, but the superintendent of Villa Tranquil Mountains who visited the apartment on the morning of November 17, 1958, five days after Taniyama’s thirty-first birthday, and found him dead. There is no mention in Shimura’s article of how Taniyama ended his life, only that he did.

  Taniyama left a three-page note on his desk. “Until yesterday, I had no definite intention of killing myself,” it read. “But more than a few must have noticed lately that I have been tired both physically and mentally. As to the cause of my suicide, I don’t quite understand it myself, but it is not the result of a particular incident, nor of a specific matter. Merely may I say, I am in the frame of mind that I lost confidence in my future.”

  His friends and colleagues, Shimura writes, were utterly perplexed, overcome by shock and sadness. A brilliant career, a fiancée with whom he’d recently been furnishing a new apartment. There was so much ahead of him, in that future he decided all of a sudden to abort. Misako Suzuki killed herself two weeks later.

  When he returned from France, Shimura wrote, spring was quickly passing. Though he’d only been away for a year and a half, and the city streets were as flashy and bustling as ever, he knew that an era of his life, “the years of turbulence” as he called them, had ended.

  13.

  At the Tokyo airport, a stooped and wizened André Weil is following Sylvie away from the baggage claim, and she in turn is following a driver, a young man in a cap who met their plane and is guiding them to his car.

  Give me your hand, I cannot see anything, André demands. With my poor eyesight, how am I expected to manage?

  His eyes, sou
py with cataracts, are failing him, everything is failing him, his ears are fitted with hearing aids, his hips are made of plastic. His skin is loose and speckled.

  Rather than take the hand that Sylvie offers him, André seizes her forearm and bears down, and they wobble forward, Sylvie seesawing between her father on one side and her shoulder bag on the other. Lowering him into the car is another delicate operation. He snaps at her and at the driver all the while. At last he lands, and the seat itself breathes a sigh. Neither André nor Sylvie is in any hurry to reach the hotel, where they’ll have to do it all in reverse.

  It’s 1994, and Eveline has been dead for eight years. André has been spending every spring in Paris, rattling around the apartment on rue Auguste-Comte, alone much of the time, though Sylvie, who goes back and forth between New York and Paris, is often in town. Mathematics has passed him by and he’s left with memories of it, of lines of poetry, of his school pranks. A certain wistfulness regarding the Riemann hypothesis, a profound conjecture that he pursued (and proved a special case of) but never managed to slay. Once or twice a year someone thinks of him and publishes a laudatory article or presents him with an award. Then there are those who persist in asking to interview him about his sister, though when it comes to Simone, he’s said all he has to say.

  After he learned that he’d won the Kyoto Prize, Sylvie tried her best to discourage him from traveling to Japan for the ceremony, but in the end she relented and agreed to go with him.

  On the plane to Tokyo he drifted into a reverie about Momoko-san, that lovely girl from decades ago, unfailingly generous . . . My God, he thought as he descended from his dream, she must be at least sixty now. And what would she think if she could see him, in this changed form? Suddenly irritated, he told his daughter to ring the call button, he wanted a coffee.