# Pierre Colmez

Pierre Colmez
Born1962 (age 60–61)
NationalityFrench
Alma materÉcole Normale Supérieure and Grenoble University
ChildrenCoralie Colmez
AwardsFermat Prize (2005)
Scientific career
FieldsMathematics
InstitutionsParis VI University
Jean-Marc Fontaine

Pierre Colmez (born 1962) is a French mathematician, notable for his work on p-adic analysis.

Colmez studied at École Normale Supérieure and obtained his doctorate from Grenoble University. He 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 of the International Congress of Mathematicians in Berlin.[1] With Jean-Pierre Serre he edited the Correspondance Grothendieck-Serre (2001).[2][3]

Colmez has won the French Go championship four times.[4]

## Mathematical work

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[5] of a ${\displaystyle p}$-adic analog of Dirichlet's analytic class number formula.
• A conjecture[6] "Colmez's 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.
• A proof[7] of Perrin-Riou's conjectural explicit reciprocity law related to the functional equation of ${\displaystyle p}$-adic L-functions.
• 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[8] "weakly admissible implies admissible" and the[9] "${\displaystyle p}$-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 ${\displaystyle p}$-adic local Langlands correspondence for ${\displaystyle \mathrm {GL} _{2}(\mathbb {Q} _{p})}$, via the construction of a functor (known as "Colmez's functor" or[15] "Colmez's Montreal functor") from representation of ${\displaystyle \mathrm {GL} _{2}(\mathbb {Q} _{p})}$ to representations of the absolute Galois group of ${\displaystyle \mathbb {Q} _{p}}$.
• Comparison theorems[16] · [17] · [18] for ${\displaystyle p}$-adic algebraic and analytic varieties with applications to a geometrization of the ${\displaystyle p}$-adic local Langlands correspondence.

## Personal life

Pierre Colmez and Leila Schneps are the parents of Coralie Colmez.[19][20] Violinist David Grimal is Colmez's first cousin.