He works on special values of L-functions and p-adic representations of p-adic groups at the meeting point of Fontaine's and Langlands' programs. His contributions include: • A proof of a p-adic analog of Dirichlet's analytic
class number formula. • A conjecture: the
Colmez conjecture relating Artin L-functions at s=0 and periods of abelian varieties with complex multiplication, a far-reaching generalization of the
Chowla-Selberg formula. • A proof of
Perrin-Riou's conjectural explicit reciprocity law related to the functional equation of p-adic L-functions. • Several contributions to
Fontaine's program of classification of p-adic representations of the
absolute Galois group of a finite extension of \mathbb{Q}_p, including proofs of conjectures of Fontaine such as "weakly admissible implies admissible" and the "p-adic monodromy conjecture" which describe representations coming from geometry, or the overconvergence of all representations, and the addition of new concepts such as "trianguline representations" or "Banach-Colmez spaces". • A construction of the p-adic local
Langlands correspondence for \mathrm{GL}_2(\mathbb{Q}_p), via the construction of a functor (known as "Colmez's functor" or "Colmez's Montreal functor") from representation of \mathrm{GL}_2(\mathbb{Q}_p) to representations of the absolute Galois group of \mathbb{Q}_p. • Comparison theorems for p-adic algebraic and analytic varieties with applications to a geometrization of the p-adic local Langlands correspondence. With
Jean-Pierre Serre, he co-edited the
Correspondance Grothendieck-Serre (2001) and the
Correspondance Serre-Tate (2015). ==Awards and honors==