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 Algébriques Cohérents, on coherent cohomology, and Géometrie Algébrique et Géométrie Analytique. 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 couldn't capture as much topology as singular cohomology with integer coefficients. Amongst Serre's 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 étale map – are important. This acted as one important source of inspiration for Grothendieck to develop the étale topology and the corresponding theory ofétale cohomology. These tools, developed in full by Grothendieck and collaborators in Séminaire de géométrie algébrique 4 and SGA 5, provided the tools for the eventual proof of the Weil conjectures by Pierre Deligne.
Other work
From 1959 onward Serre's interests turned towards group theory, number theory, in particular Galois representations and modular forms. Amongst his most original contributions were: his "Conjecture II" on Galois cohomology; his use of group actions on trees ; 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 on mod-p representations that made Fermat's 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.
Honors and awards
Serre, at twenty-seven in 1954, was and still is the youngest person ever to have been 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. 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 and Grand Cross of the Legion of Merit.
