Pierre Colmez Explained

Pierre Colmez
Nationality:	French
Fields:	Mathematics
Workplaces:	CNRS Sorbonne University
Alma Mater:	École Normale Supérieure and Grenoble University
Doctoral Advisor:	John H. Coates Jean-Marc Fontaine
Awards:	Fermat Prize (2005)
Children:	Coralie Colmez

Pierre Colmez (born 1962) is a French mathematician and directeur de recherche at the CNRS (IMJ-PRG) known for his work in number theory and p-adic analysis.

Education

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

Research

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.^[1]

• A conjecture and periods of abelian varieties with complex multiplication, a far-reaching generalization of the Chowla-Selberg formula: the Colmez conjecture relating Artin L-functions at

s =0 .^[2]

• A proof of Perrin-Riou's conjectural explicit reciprocity law related to the functional equation of

p -adic L-functions.^[3]

• Several contributions to Fontaine's program of classification of

p -adic representations of the absolute Galois group of a finite extension of Q _{p} , including proofs of conjectures of Fontaine such as "weakly admissible implies admissible"^[4] and the "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

p -adic local Langlands correspondence for GL _{2(Q} _{p)} , via the construction of a functor (known as "Colmez's functor" or "Colmez's Montreal functor"^[10]) from representation of GL _{2(Q} _{p)} to representations of the absolute Galois group of Q _{p} .^[11]

• Comparison theorems for

p -adic algebraic and analytic varieties with applications to a geometrization of the p -adic local Langlands correspondence.^{[12] [13] [14]}

With Jean-Pierre Serre, he co-edited the Correspondance Grothendieck-Serre (2001).^{[15] [16]}

Awards and honors

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.^[17]

Colmez has won the French Go championship four times.^[18]

Personal life

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

External links

• Pierre Colmez' website

Notes and References

1. Résidu en s=1 des fonctions zêta p-adiques, Inventiones mathematicae 91 (1988), 371-389
2. Périodes des variétés abéliennes à multiplication complexe, Annals of Mathematics 138 (1993), 625–683
3. Théorie d'Iwasawa des représentations de de Rham d'un corps local, Annals of Mathematics 148 (1998), 485–571
4. Construction des représentations p-adiques semi-stables (avec J.-M. Fontaine), Inventiones mathematicae 140 (2000), 1–43 (avec Jean-Marc Fontaine)
5. Espaces Vectoriels de dimension finie et représentations de de Rham, Astérisque 319 (2008), 117–186
6. Représentations p-adiques surconvergentes, Inventiones mathematicae 133 (1998), 581–611 (avec Frédéric Cherbonnier)
7. Représentations triangulines de dimension 2, Astérisque 319 (2008), 213–258
8. Espaces de Banach de dimension finie, Journal Institut de Mathématiques de Jussieu 1 (2002), 331–439
9. Arthur-César Le Bras, Espaces de Banach–Colmez et faisceaux cohérents sur la courbe de Fargues–Fontaine, Duke Math. J. 167 (2018), 3455-3532
10. Vytautas Paškūnas, The image of Colmez’s Montreal functor, Publications mathématiques de l'IHÉS 118 (2013), 1–191
11. Représentations de GL _{2(Q} _{p)} et (\varphi,\Gamma) -modules, Astérisque 330 (2010), 281–509
12. p -adic vanishing cycles and syntomic cohomology, Inventiones mathematicae 208 (2017), 1-108 (with Wiesława Nizioł).
13. Cohomologie p -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).
14. Cohomology of p -adic Stein spaces, Inventiones mathematicae 219 (2020), 873–985 (with Wiesława Nizioł and Gabriel Dospinescu).
15. Raynaud, Michel. Michel Raynaud. Book Review: Correspondance Grothendieck-Serre . Notices of the AMS. October 2003. 50. 9. 1085–1086.
16. Book: Serre, J.-P.. Colmez, P.. Grothendieck-Serre Correspondence. 2004. American Mathematical Society; Société Mathématique de France. 978-0-8218-3424-4.

    bilingual edition

    .
17. Book: Colmez, Pierre. Représentations p-adiques d'un corps local. Doc. Math. (Bielefeld) Extra Vol. ICM Berlin, 1998, vol. II. 1998. 153–162. https://www.elibm.org/ft/10011650000.
18. http://ffg.jeudego.org/informations/competitions/chp_histo.php Open - Fédération Française de Go - Jeu de go
19. News: Allow me to explain, Your Honour . . 2 May 2013 . 2 October 2020.
20. Web site: The Mathematician Who Moonlights As a Rock-Band Violinist . Diana . Tsui . . 9 January 2018 . 2 October 2020.