## ⓘ Jean-Pierre Serre

Jean-Pierre Serre is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954 and the inaugural Abel Prize in 2003.

### * 1.1. Biography * Personal life

Born in Bages, Pyrenees-Orientales, France, to pharmacist parents, Serre was educated at the Lycee de Nimes and then from 1945 to 1948 at the Ecole Normale Superieure in Paris. He was awarded his doctorate from the Sorbonne in 1951. From 1948 to 1954 he held positions at the Centre National de la Recherche Scientifique in Paris. In 1956 he was elected professor at the College de France, a position he held until his retirement in 1994. His wife, Professor Josiane Heulot-Serre, was a chemist; she also was the director of the Ecole Normale Superieure de Jeunes Filles. Their daughter is the former French diplomat, historian and writer Claudine Monteil. The French mathematician Denis Serre his nephew. He practices skiing, table tennis, and rock climbing in Fontainebleau.

### * 1.2. Biography * Career

From a very young age he was an outstanding figure in the school of Henri Cartan, working on algebraic topology, several complex variables and then commutative algebra and algebraic geometry, where he introduced sheaf theory and homological algebra techniques. Serres thesis concerned the Leray–Serre spectral sequence associated to a fibration. Together with Cartan, Serre established the technique of using Eilenberg–MacLane spaces for computing homotopy groups of spheres, which at that time was one of the major problems in topology.

In his speech at the Fields Medal award ceremony in 1954, Hermann Weyl gave high praise to Serre, and also made the point that the award was for the first time awarded to a non-analyst. Serre subsequently changed his research focus.

### * 1.3. Biography * Algebraic geometry

In the 1950s and 1960s, a fruitful collaboration between Serre and the two-years-younger Alexander Grothendieck led to important foundational work, much of it motivated by the Weil conjectures. Two major foundational papers by Serre were Faisceaux Algebriques Coherents FAC, on coherent cohomology, and Geometrie Algebrique et Geometrie Analytique GAGA.

Even at an early stage in his work Serre had perceived a need to construct more general and refined cohomology theories to tackle the Weil conjectures. The problem was that the cohomology of a coherent sheaf over a finite field couldnt capture as much topology as singular cohomology with integer coefficients. Amongst Serres early candidate theories of 1954–55 was one based on Witt vector coefficients.

Around 1958 Serre suggested that isotrivial principal bundles on algebraic varieties – those that become trivial after pullback by a finite etale map – are important. This acted as one important source of inspiration for Grothendieck to develop the etale topology and the corresponding theory of etale cohomology. These tools, developed in full by Grothendieck and collaborators in Seminaire de geometrie algebrique SGA 4 and SGA 5, provided the tools for the eventual proof of the Weil conjectures by Pierre Deligne.

### * 1.4. Biography * Other work

From 1959 onward Serres interests turned towards group theory, number theory, in particular Galois representations and modular forms.

Amongst his most original contributions were: his "Conjecture II" still open on Galois cohomology; his use of group actions on trees with Hyman Bass; the Borel–Serre compactification; results on the number of points of curves over finite fields; Galois representations in ℓ-adic cohomology and the proof that these representations have often a "large" image; the concept of p -adic modular form; and the Serre conjecture now a theorem on mod- p representations that made Fermats last theorem a connected part of mainstream arithmetic geometry.

In his paper FAC, Serre asked whether a finitely generated projective module over a polynomial ring is free. This question led to a great deal of activity in commutative algebra, and was finally answered in the affirmative by Daniel Quillen and Andrei Suslin independently in 1976. This result is now known as the Quillen–Suslin theorem.

## 2. Honors and awards

Serre, at twenty-seven in 1954, is the youngest ever to be awarded the Fields Medal. He went on to win the Balzan Prize in 1985, the Steele Prize in 1995, the Wolf Prize in Mathematics in 2000, and was the first recipient of the Abel Prize in 2003. He has been awarded other prizes, such as the Gold Medal of the French National Scientific Research Centre National de la Recherche Scientifique, CNRS.

He is a foreign member of several scientific Academies) and has received many honorary degrees. In 2012 he became a fellow of the American Mathematical Society.

Serre has been awarded the highest honors in France as Grand Cross of the Legion of Honour Grand Croix de la Legion dHonneur and Grand Cross of the Legion of Merit Grand Croix de lOrdre National du Merite.

## 3. Bibliography

- "Lectures on N_Xp" 2012, AK Peters, CRC Press
- "Cohomological Invariants in Galois Cohomology 2003 with Skip Garibaldi and Alexander Merkurjev, AMS
- Algebres de Lie Semi-simples Complexes 1966, as Complex Semisimple Lie Algebras 1987, Springer-Verlag
- Representations lineaires des groupes finis 1971, Hermann, as Linear Representations of Finite Groups 1977, Springer-Verlag
- Correspondance Serre-Tate 2015, edited with Pierre Colmez, SMF
- Cohomologie Galoisienne 1964 College de France course 1962–63, as Galois Cohomology 1997, Springer-Verlag
- Lectures on the Mordell-Weil Theorem 1990, Vieweg
- "Lie algebras and Lie groups" 1965 Harvard Lectures, Springer-Verlag.
- "Finite Groups: an Introduction" 2016, Higher Education Press & International Press
- Grothendieck–Serre Correspondence 2003, bilingual edition, edited with Pierre Colmez, SMF-AMS
- Oeuvres/Collected Papers in four volumes 1986 Vol. IV in 2000, Springer-Verlag
- Abelian ℓ-Adic Representations and Elliptic Curves 1968, CRC Press, reissue. Addison-Wesley. 1989.
- Algebre Locale, Multiplicites 1965 College de France course 1957–58, as Local Algebra 2000, Springer-Verlag
- Corps Locaux 1962, Hermann, as Local Fields 1980, Springer-Verlag
- Groupes Algebriques et Corps de Classes 1959, Hermann, translated into English as Algebraic Groups and Class Fields 1988, Springer-Verlag
- Topics in Galois Theory 1992, CRC Press
- Arbres, amalgames, SL 2 1977, SMF, as Trees 1980, Springer-Verlag
- Cours darithmetique 1970, PUF, as A Course in Arithmetic 1973, Springer-Verlag
- "Exposes de seminaires 1950–1999" 2001, SMF

A list of corrections, and updating, of these books can be found on his home page at College de France.