PIMS-CRM-FIELDS Prize Lecture: Henri Darmon

Modular functions play an important role in many aspects of number theory. The theory of complex multiplication, one of the grand achievements of the subject in the 19th century, asserts that the values of modular functions at quadratic imaginary arguments generate (essentially all) abelian extensions of imaginary quadratic fields. Kronecker's famous ``Jugendtraum", which later came to be known as Hilbert’s twelfth problem concerns the generalization of this theory to other base fields. I will describe an ongoing work in collaboration with Jan Vonk which identifies a class of functions that seem to play the role of modular functions for real quadratic fields. A key difference with the classical setting is that they are meromorphic functions of a p-adic variable (defined in the framework of “rigid analysis” introduced by Tate) rather than of a complex variable. An important role in this theory of ``rigid modular cocycles" is played by the $p$-modular group ${\bf SL}_2({\rm bf Z}[1/p])$ whose cohomology was studied by Serre and Adem.

 

 

Biography

Born in 1965 in Paris, France, Darmon moved to Canada in 1968. He received a bachelor's degree from McGill University in 1987 and a PhD in mathematics from Harvard University in 1991, under the supervision of Benedict Gross. He then held a postdoctoral position at Princeton University, under the mentorship of Andrew Wiles. It was around this time that Wiles gained worldwide fame for his proof of Fermat's Last Theorem.

 

In 1994, Darmon joined the faculty of McGill University, where he is currently a James McGill Professor in the Department of Mathematics and Statistics. His other honors include the André Aisenstadt Prize (1997), the Coxeter-James Prize of the Canadian Mathematical Society (1998), the Ribenboim Prize of the Canadian Number Theory Association (2002), and the John L. Synge Award of the Royal Society of Canada (2008). He was elected a Fellow of the Royal Society of Canada in 2003 and received the 2017 AMS Cole Prize in Number Theory for his contributions to the arithmetic of elliptic curves and modular forms.