Colmez has won the French
Go championship four times.[4]
Mathematical work
He works on special values of L-functions and -adic representations of -adic groups at the meeting point of Fontaine's and Langlands' programs. His contributions include:
A conjecture[6] "Colmez's conjecture" relating Artin L-functions at and periods of abelian varieties with complex multiplication, a far reaching generalization of the
Chowla-Selberg formula.
A proof[7] of
Perrin-Riou's conjectural explicit reciprocity law related to the functional equation of -adic L-functions.
Several contributions to
Fontaine's program of classification of -adic representations of the
absolute Galois group of a finite extension of , including proofs of conjectures of Fontaine such as[8] "weakly admissible implies admissible" and the[9] "-adic monodromy conjecture" which describe representations coming from geometry, or the overconvergence of all representations,[10] and addition of new concepts such as[11] "trianguline representations" or[12] "Banach-Colmez[13] spaces".
A construction[14] of the -adic local
Langlands correspondence for , via the construction of a functor (known as "Colmez's functor" or[15] "Colmez's Montreal functor") from representation of to representations of the absolute Galois group of .
Comparison theorems[16]·[17]·[18] for -adic algebraic and analytic varieties with applications to a geometrization of the -adic local Langlands correspondence.
^Cohomologie -adique de la tour de Drinfeld, le cas de la dimension 1, Journal of the AMS 33 (2020), 311–362 (with
Wiesława Nizioł and Gabriel Dospinescu).
^Cohomology of -adic Stein spaces, Inventiones mathematicae 219 (2020), 873–985 (with
Wiesława Nizioł and Gabriel Dospinescu).