Pierre Colmez Explained

Pierre Colmez
Nationality:	French
Fields:	Mathematics
Workplaces:	Paris VI 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, 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

p -adic representations of p -adic groups at the meeting point of Fontaine's and Langlands' programs. His contributions include:

• A proof^[5] of a

p -adic analog of Dirichlet's analytic class number formula.

• A conjecture^[6] "Colmez's 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^[7] 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 Q _{p} , including proofs of conjectures of Fontaine such as^[8] "weakly admissible implies admissible" and the^[9] "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

p -adic local Langlands correspondence for GL _{2(Q} _{p)} , via the construction of a functor (known as "Colmez's functor" or^[15] "Colmez's Montreal functor") from representation of GL _{2(Q} _{p)} to representations of the absolute Galois group of Q _{p} .

• Comparison theorems^{[16] [17] [18]} for

p -adic algebraic and analytic varieties with applications to a geometrization of the 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.

External links

• Pierre Colmez' website

Notes and References

1. 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.
2. Raynaud, Michel. Michel Raynaud. Book Review: Correspondance Grothendieck-Serre . Notices of the AMS. October 2003. 50. 9. 1085–1086.
3. Book: Serre, J.-P.. Colmez, P.. Grothendieck-Serre Correspondence. 2004. American Mathematical Society; Société Mathématique de France. 978-0-8218-3424-4.

   bilingual edition

   .
4. http://ffg.jeudego.org/informations/competitions/chp_histo.php Open - Fédération Française de Go - Jeu de go
5. Résidu en s=1 des fonctions zêta p-adiques, Inventiones mathematicae 91 (1988), 371-389
6. Périodes des variétés abéliennes à multiplication complexe, Annals of Mathematics 138 (1993), 625–683
7. Théorie d'Iwasawa des représentations de de Rham d'un corps local, Annals of Mathematics 148 (1998), 485–571
8. Construction des représentations p-adiques semi-stables (avec J.-M. Fontaine), Inventiones mathematicae 140 (2000), 1–43 (avec Jean-Marc Fontaine)
9. Espaces Vectoriels de dimension finie et représentations de de Rham, Astérisque 319 (2008), 117–186
10. Représentations p-adiques surconvergentes, Inventiones mathematicae 133 (1998), 581–611 (avec Frédéric Cherbonnier)
11. Représentations triangulines de dimension 2, Astérisque 319 (2008), 213–258
12. Espaces de Banach de dimension finie, Journal Institut de Mathématiques de Jussieu 1 (2002), 331–439
13. Arthur-César Le Bras, Espaces de Banach–Colmez et faisceaux cohérents sur la courbe de Fargues–Fontaine, Duke Math. J. 167 (2018), 3455-3532
14. Représentations de GL _{2(Q} _{p)} et (\varphi,\Gamma) -modules, Astérisque 330 (2010), 281–509
15. Vytautas Paškūnas, The image of Colmez’s Montreal functor, Publications mathématiques de l'IHÉS 118 (2013), 1–191
16. p -adic vanishing cycles and syntomic cohomology, Inventiones mathematicae 208 (2017), 1-108 (with Wiesława Nizioł).
17. 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).
18. Cohomology of p -adic Stein spaces, Inventiones mathematicae 219 (2020), 873–985 (with Wiesława Nizioł and Gabriel Dospinescu).
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.