In 2000, the Clay Mathematics Institute put a $1-million bounty on each of seven particularly vexing mathematical problems. One of these “Millennium Prize Problems,” the Birch and Swinnerton-Dyer conjecture, suggests that one can simply tell whether there are a finite or infinite number of rational solutions to equations defining elliptic curves over rational numbers. McGill professor Henri Darmon may not have clinched the million bucks yet, but his Birch and Swinnerton-Dyer insights have earned him a prestigious mathematics prize from the American Mathematical Society.
The AMS Frank Nelson Cole Prize in Number Theory is given every three years. Darmon will receive the prize on Thursday, January 5, 2017 at the Joint Mathematics Meetings in Atlanta.
“The Cole Prize is a fantastic recognition,” says Prof. Darmon. “I am particularly gratified that two of my greatest mathematical influences – my PhD supervisor Benedict Gross, and my postdoctoral mentor Andrew Wiles – also received the prize. The first in 1987, for his landmark work on the so-called Gross-Zagier formula, the second in 1997 for his epoch-making proof of Fermat’s Last Theorem and of the Shimura-Taniyama conjecture. My own work comes nowhere near these great classics in terms of influence and importance, but draws on a lot of ideas that they have introduced. Aside from influences, I am also very grateful to my collaborators, who play a tremendous role in my work, and in my personal life.”
The AMS Cole Prize recognizes two of Darmon’s recent papers, which have deepened our understanding of the Birch and Swinnerton-Dyer conjecture and on possible extensions of the theory of complex multiplication: “Generalized Heegner cycles and p-adic Rankin L-series” (co-authored with Massimo Bertolini and Kartik Prasanna, with an appendix by Brian Conrad) and “Diagonal cycles and Euler systems, II: the Birch and Swinnerton-Dyer conjecture for Hasse-Weil-Artin L-functions” (co-authored with Victor Rotger). The papers were published in the Duke Mathematical Journal and the Journal of the AMS, respectively.
Darmon is considered one of the world’s leading number theorists, and the American Mathematical Society notes that the two papers in question are, in fact, “only high points of a long sequence of influential papers.” After completing his bachelor’s degree at McGill in 1987, Darmon earned his PhD from Harvard, then did postdoctoral studies at Princeton. He joined McGill’s Department of Mathematics and Statistics in 1994 and is currently a James McGill Professor.
Darmon says he is “honoured and humbled” to receive the Cole Prize. Still, he says he’s “very far indeed” from claiming the million-dollar purse that awaits whomever solves the Birch and Swinnerton-Dyer conjecture. (Never say never, though: Grigori Perelman won, but turned down, one million dollars for solving one of the other Millennium Prize Problems, the Poincaré conjecture, in 2003.) And that, Darmon says, is perfectly okay.
“The value of thinking about such problems, even if one has very little idea on how to solve them, is that they tend to lead to a lot of valuable spin-offs, and generate important new insights,” he says. “A good example of this is Fermat’s Last Theorem, which was open for over 350 years and became very famous at the turn of the 20th Century when Paul Wolfskehl, a German Jewish industrialist, offered a large prize for its solution. But even before that, entire new theories, like Kummer’s theory of ideals in the 19th century, arose from efforts to solve Fermat’s Last Theorem, and the eventual solution by Andrew Wiles in 1994 ushered in a revolution in the theory of elliptic curves and modular forms. The Birch and Swinnerton-Dyer conjecture has a similar ability to lead to important new mathematics, which is why it fascinates so many of us.”
Darmon notes another mathematical upside to such big prizes going unclaimed: “The interest on the capital of the Millennium Prize helps to fund a lot of mathematical activity, like conferences.”