Pierre Colmez

Colmez studied at École Normale Supérieure and obtained his doctorate from the University of Grenoble.

He works on special values of L-functions and ${\displaystyle p}$-adic representations of ${\displaystyle p}$-adic groups at the meeting point of Fontaine's and Langlands' programs. His contributions include:

• A proof of a ${\displaystyle p}$-adic analog of Dirichlet's analytic class number formula.^[1]
• A conjecture: the Colmez conjecture relating Artin L-functions at ${\displaystyle s=0}$ and periods of abelian varieties with complex multiplication, a far-reaching generalization of the Chowla-Selberg formula.^[2]
• A proof of Perrin-Riou's conjectural explicit reciprocity law related to the functional equation of ${\displaystyle p}$-adic L-functions.^[3]
• Several contributions to Fontaine's program of classification of ${\displaystyle p}$-adic representations of the absolute Galois group of a finite extension of ${\displaystyle \mathbb {Q} _{p}}$, including proofs of conjectures of Fontaine such as "weakly admissible implies admissible"^[4] and the "${\displaystyle p}$-adic monodromy conjecture" which describe representations coming from geometry, or the overconvergence of all representations,^[5][6] and the addition of new concepts such as "trianguline representations"^[7] or "Banach-Colmez spaces".^[8][9]
• A construction of the ${\displaystyle p}$-adic local Langlands correspondence for ${\displaystyle \mathrm {GL} _{2}(\mathbb {Q} _{p})}$, via the construction of a functor (known as "Colmez's functor" or "Colmez's Montreal functor"^[10]) from representation of ${\displaystyle \mathrm {GL} _{2}(\mathbb {Q} _{p})}$ to representations of the absolute Galois group of ${\displaystyle \mathbb {Q} _{p}}$.^[11]
• Comparison theorems for ${\displaystyle p}$-adic algebraic and analytic varieties with applications to a geometrization of the ${\displaystyle p}$-adic local Langlands correspondence.^[12][13][14]

With Jean-Pierre Serre, he co-edited the Correspondance Grothendieck-Serre (2001)^[15][16] and the Correspondance Serre-Tate (2015).^[17]

Colmez won the 2005 Fermat Prize for his contributions to the study of L-functions and p-adic Galois representations.

In 1998, he was an invited speaker at the International Congress of Mathematicians in Berlin.^[18]

Colmez has won the French Go championship four times.^[19]

Pierre Colmez and Leila Schneps are the parents of Coralie Colmez.^[20][21] Violinist David Grimal is Colmez's first cousin.