Jean-Marc Fontaine

Jean-Marc Fontaine (born March 13, 1944 in Boulogne-Billancourt , † January 29, 2019 ) was a French mathematician who dealt with algebraic geometry and number theory ( p-adic numbers , Galois representations ).

Life

Fontaine studied at the École polytechnique from 1962 , was a scientist at the CNRS from 1965 to 1971 and received his doctorate in 1972. In 1971/72 he was at the University of Paris VI and from 1972 to 1988 at the University of Grenoble (first Maître de conférences , then professor). From 1989 he was a professor at the University of Paris-South XI in Orsay. In 2009 he retired.

plant

In one of his first works, Fontaine classified the -divisible groups on the ring of whole numbers of a local field and constructed the -adic periods, the -adic analog of the field of complex numbers. He introduced a hierarchy of -adic representations of the absolute Galois group of an -adic body (crystalline, semi-stable and De-Rham representations). A multitude of techniques and conjectures in this area originated from him. He is one of the co-founders of the -adic Hodge theory and its application in number theory, for which his seminar at IHES in 1988 (published in Astérisque, Volume 223) is a fundamental reference. He proved that there are no nontrivial Abelian varieties over the rational numbers with good reduction everywhere ( Il n'y a pas de variété abélienne sur Z , Inventiones Mathematicae Vol. 81, 1985, p. 515). From him comes the concept of the geometric Galois representation of the Galois group of a number field (and a conjecture with Barry Mazur as to when global p-adic Galois representations have geometric origins). He also worked on the Bloch - Kato conjectures. ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle p}$

In 1979 he and Jean-Pierre Wintenberger proved a theorem about the isomorphism of the absolute Galois group between an extension of the field of the p-adic numbers (adjunction of the roots for all n) and a corresponding extension ( perfection ) of the field of the Laurent series over the p-adic Numbers, see Fontaine and Wintenberger's theorem . The sentence was expanded by Peter Scholze as part of his theory of the perfectoid spaces and formed an essential starting point for this theory. With Laurent Fargues he introduced a fundamental curve of the p-adic Hodge theory. ${\displaystyle \mathbb {Q} _{p}}$ ${\displaystyle p^{\frac {1}{p^{n}}}}$

His doctoral students include Christophe Breuil (1996), Pierre Colmez and Jean-Pierre Wintenberger .

Honors and memberships

In 1984 he received the Prix Carrières from the French Academy of Sciences. From 2002 he was a full member of the French Academy of Sciences (corresponding member since 1998). In 2002 he received the Gay Lussac Humboldt Prize . In 2014 he was elected a full member of the Academia Europaea . He was invited speaker at the International Congress of Mathematicians in Warsaw 1983 ( Representations p-adique ) and Beijing 2002 ( Analyze p-adique et representations galoisiennes ). From 1994 to 2004 he was a senior member of the Institut Universitaire de France .

Fonts

• Groupes p-divisibles sur les corps locaux . Astérisque, Vol. 47/48, Societe Mathematique de France, 1977.
• Editor: Periodes -adiques ${\displaystyle p}$ . Astérisque, vol. 223, 1994.
• Editor with Pierre Berthelot , Luc Illusie , Kazuya Kato , Michael Rapoport : Cohomologies -adiques et applications arithmétiques ${\displaystyle p}$ . Astérisque Vol. 278/279, 2002.

Web links

• Fontaine in the Mathematics Genealogy Project
• Entry at the Academy of Sciences
• Obituary by Frank Calegari

Individual evidence

1. ↑ Jean-Marc Fontaine died aged 74. Obituary of the Institut des Hautes Études Scientifiques , January 31, 2019. Retrieved March 26, 2019 (English).
2. ^ Mark Kisin , What is a Galois representation? , Notices AMS, June / July 2007, online
3. ^ Jean-Marc Fontaine, Jean-Pierre Wintenberger, Extensions algébrique et corps des normes des extensions APF des corps locaux, CR Acad. Sci. Paris Sér. A-B, Vol. 288, 1089, A441-A444
4. ↑ Fargues-Fontaine curve , ncat lab
5. ^ Directory of members: Jean-Marc Fontaine. Academia Europaea, accessed September 3, 2017 .