Jean-Pierre Serre

How to use this course

Jean-Pierre Serre has worked in three fields, and in each he changed the basic tools. His thesis gave topologists the spectral sequence of a fibration and the first general theorems on homotopy groups of spheres. His papers of 1955 and 1956 brought sheaf cohomology into algebraic geometry. From about 1959 he turned to number theory, where his work on Galois representations led, through his 1987 modularity conjecture, to the proof of Fermat’s Last Theorem. He received the first Abel Prize in 2003 and turned 100 in September 2026.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. The lessons follow the order in which he worked: topology (1950–1953), algebraic geometry (1953–1960), and number theory and group theory (from 1959 onward). Serre’s own books are the best companions to almost every lesson, and the reading list at the end says which.

Timeline

Bages, Vauvert and Nîmes, 1926–1945

Jean-Pierre Serre was born on 15 September 1926 in Bages, a village in the Pyrénées-Orientales near Perpignan. His parents, Jean Serre and Adèle Diet, were both pharmacists; his mother had studied pharmacy at the University of Montpellier. He started primary school at Vauvert, in the Gard near Nîmes, in 1932.

In 1937 he entered the Lycée de Nîmes, now the Lycée Alphonse-Daudet. He has recalled that his interest in mathematics began early and that he first learned calculus on his own, from books at home. The war years passed in the provincial south. In 1944 he won first prize in mathematics at the Concours Général, the national competition for lycée pupils, and passed his baccalauréat. He stayed one more year at Nîmes to prepare for the entrance examination of the École Normale Supérieure, and entered it in 1945.

Two features of the later mathematician are already visible here. He learned by reading, alone, and he worked fast. Both would shape a career in which a large part of his influence came from books and letters as much as from papers.

The École Normale and the Cartan seminar, 1945–1951

At the École Normale Supérieure, Serre came under the influence of Henri Cartan, who taught there and ran a weekly seminar that became the centre of French mathematics after the war. The Séminaire Cartan of these years worked through sheaf theory, the new spectral sequences of Jean Leray, and the homological algebra being built by Cartan and Samuel Eilenberg. Serre took the agrégation in 1948 and was given a research post at the CNRS, which he held until 1954.

Around 1948 he joined the Bourbaki group, the collective of French mathematicians writing a unified account of mathematics on axiomatic foundations. He was its youngest member, and he would write and revise Bourbaki drafts for many years. The group’s habits of complete definitions, structural thinking and relentless rewriting suited him.

Leray had invented sheaves and spectral sequences while a prisoner of war, to study the cohomology of a space fibred over another. His versions were technical and tied to particular cohomology theories. Serre’s task, set by Cartan, was to find out what spectral sequences could say about homotopy groups, the most difficult invariants in topology. He saw that Leray’s machinery could be adapted to singular homology and to a very general notion of fibration, which made it applicable to loop spaces and path spaces, where the old theory did not reach.

The thesis and the Fields Medal, 1951–1954

Serre defended his thesis, Homologie singulière des espaces fibrés. Applications, at the Sorbonne in 1951; it appeared that year in the Annals of Mathematics. It introduced what is now called the Serre spectral sequence (Lesson 04) and used it to compute the homology of loop spaces. By applying it to the path-loop fibration he could, for the first time, prove general facts about the homotopy groups of spheres. In 1952 he lectured on these results in Princeton.

Over the next two years he developed the method with Cartan, using Eilenberg–MacLane spaces to “kill” homotopy groups one at a time. In 1953 his paper on classes of abelian groups proved that almost all homotopy groups of spheres are finite and located their first \(p\)-torsion (Lesson 05). At the same time, through the Cartan seminar, he took part in the theory of Stein manifolds, where Cartan’s Theorems A and B, proved with sheaf cohomology, gave a complete account of coherent analytic sheaves.

In 1954 the Fields Medal was awarded at the International Congress of Mathematicians in Amsterdam to Kunihiko Kodaira and to Serre, who was 27. He remains the youngest person ever to receive it. From 1954 to 1956 he taught at the University of Nancy. By then he had already begun to move from topology to algebraic geometry.

The spectral sequence of a fibration

For the general reader. A fibration is a space built by attaching a copy of one space (the fibre) over each point of another (the base), in a way that is twisted globally but trivial locally. A spectral sequence is a bookkeeping device that computes the holes of the total space from the holes of the fibre and base, with corrections. Serre made this device work in exactly the setting topologists needed.

Serre fibrations

A map \(\pi:E\to B\) is a Serre fibration if it has the homotopy lifting property for all cubes \(I^n\): every homotopy \(I^n\times I\to B\) whose start lifts to \(E\) lifts entirely. Fibre bundles are Serre fibrations. The key new example is the path fibration: for a pointed space \(B\), let \(PB\) be the space of paths starting at the base point, and \(\pi(\gamma)=\gamma(1)\). The fibre over the base point is the loop space \(\Omega B\), and \(PB\) is contractible. Serre’s weak hypothesis made this fibration available, and with it the loop space.

Proof idea. Filter the singular chains of \(E\) by how far they reach over skeleta of \(B\), or, as Serre did, by cubical chains whose projection to \(B\) has low dimension in the first coordinates. The associated graded is computed by the homology of the fibres, which are all homotopy equivalent when \(B\) is path-connected. The \(\pi_1\) hypothesis makes the local coefficient system \(H_q(F)\) constant.

The loop space of a sphere

Take \(\Omega S^n\to PS^n\to S^n\) with \(n\ge2\). The base has homology only in degrees \(0\) and \(n\), so \(E^2_{p,q}\) is nonzero only in the columns \(p=0\) and \(p=n\), where it equals \(H_q(\Omega S^n)\). The only possible differential is \(d^n:E^n_{n,q}\to E^n_{0,q+n-1}\). Since \(PS^n\) is contractible, everything except \(E_{0,0}\) must die, so every \(d^n\) is an isomorphism. Starting from \(H_0=\mathbb Z\) this gives

The hero picture at the top of the page shows the case \(n=3\).

Homotopy groups of spheres

For the general reader. The homotopy group \(\pi_i(S^n)\) counts the essentially different ways to wrap an \(i\)-dimensional sphere around an \(n\)-dimensional one. Before Serre only scattered values were known. He proved that, apart from two families of exceptions, all these groups are finite, and he found where their prime factors first appear.

Classes of abelian groups

A Serre class \(\mathcal C\) is a family of abelian groups closed under subgroups, quotients and extensions. Examples: the zero group alone; finite groups; finite groups of order prime to \(p\); finitely generated groups. A homomorphism is a \(\mathcal C\)-isomorphism if its kernel and cokernel lie in \(\mathcal C\). Serre’s idea was to do homotopy theory “modulo \(\mathcal C\)”: to ignore, for example, all finite groups, or all torsion prime to \(p\).

Proof sketch. Work modulo finite groups, that is, rationally. For \(n\) odd, the Serre spectral sequence of the fibrations \(K(\mathbb Z,m-1)\to P\to K(\mathbb Z,m)\) shows by induction that \(H^*(K(\mathbb Z,n);\mathbb Q)\) is an exterior algebra on a class of degree \(n\). Hence the map \(S^n\to K(\mathbb Z,n)\) representing a generator of \(H^n\) is an isomorphism in rational homology. Its homotopy fibre therefore has finite homology in every degree, and the Hurewicz theorem modulo finite groups makes its homotopy groups finite; the long exact sequence then gives \(\pi_i(S^n)\) finite for \(i\neq n\). For \(n\) even, \(H^*(K(\mathbb Z,n);\mathbb Q)\) is a polynomial ring, and one compares \(S^n\) with the total space of a fibration over \(K(\mathbb Z,n)\) that kills the square of the generator; this produces the extra \(\mathbb Z\) in degree \(2n-1\), detected by the Whitehead product \([\iota_n,\iota_n]\).

The proof uses the mod \(p\) cohomology of Eilenberg–MacLane spaces, which Serre computed for \(p=2\) in terms of Steenrod squares; Cartan completed the odd primes. That computation introduced the admissible sequences now used to describe the Steenrod algebra.

Coherent sheaves and duality

For the general reader. In 1955 Serre showed that the sheaf methods built for complex manifolds work for algebraic varieties over any field, using the coarse Zariski topology. The paper, known as FAC, made cohomology the basic language of algebraic geometry and set the stage for Grothendieck’s schemes.

FAC

On a variety \(X\) with structure sheaf \(\mathcal O_X\), a sheaf of \(\mathcal O_X\)-modules is coherent if it is locally the cokernel of a map \(\mathcal O_X^m\to\mathcal O_X^n\). Serre defined cohomology through Čech cochains for the Zariski topology, proved that affine varieties have no higher cohomology for coherent sheaves, and computed projective space. On \(\mathbb P^r\) over a field \(k\), with \(\mathcal O(n)\) the twisting sheaf:

and \(H^i(\mathbb P^r,\mathcal O(n))=0\) for \(0<i<r\).

Serre proved duality first for compact complex manifolds, using Dolbeault cohomology and the theory of distributions, and gave the algebraic version in FAC’s sequel. Grothendieck later extended it to proper morphisms. For a curve of genus \(g\) and a line bundle \(L\) of degree \(d\), duality turns the Riemann–Roch formula into \(h^0(L)-h^0(\omega\otimes L^{-1})=d+1-g\).

Two further results from these years are standard. Serre’s criterion (1957): a Noetherian scheme \(X\) is affine if and only if \(H^1(X,\mathcal I)=0\) for every coherent sheaf of ideals \(\mathcal I\). And at the end of FAC Serre asked whether every finitely generated projective module over \(k[x_1,\dots,x_n]\) is free. Daniel Quillen and Andrei Suslin proved independently in 1976 that it is.

GAGA

For the general reader. A complex projective variety can be studied with polynomials or with holomorphic functions. In 1956 Serre proved that, for compact projective varieties, the two approaches give exactly the same answers. The title of his paper, Géométrie algébrique et géométrie analytique, abbreviated GAGA, became the name of the principle.

Let \(X\) be a complex projective variety and \(X^{\mathrm{an}}\) the associated complex analytic space. Each algebraic coherent sheaf \(\mathcal F\) on \(X\) gives an analytic coherent sheaf \(\mathcal F^{\mathrm{an}}=\mathcal F\otimes_{\mathcal O_X}\mathcal O_{X^{\mathrm{an}}}\).

Proof idea. For \(X=\mathbb P^r\) and \(\mathcal F=\mathcal O(n)\), both sides can be computed directly; the analytic computation uses Cartan’s results on Stein spaces. Every coherent sheaf on \(\mathbb P^r\) has a resolution by sums of \(\mathcal O(n)\), by FAC, and a five-lemma argument extends part 1 to all coherent sheaves. For part 2 one uses Cartan’s Theorem A analogue on \(\mathbb P^r\): every coherent analytic sheaf becomes globally generated after twisting, so it is a cokernel of a map between sums of \(\mathcal O(n)^{\mathrm{an}}\), and such maps are algebraic by part 1.

GAGA is the reason that Hodge theory, a theory about harmonic forms on Kähler manifolds, says something about algebraic varieties. It was generalized to proper schemes by Grothendieck, and its philosophy, that analytic objects over a compact algebraic base are algebraic, recurs in rigid-analytic geometry and in the theory of algebraic stacks.

Local fields and Galois cohomology

For the general reader. Around 1959 Serre turned to number theory. His books Corps locaux (1962) and Cohomologie galoisienne (1964) recast local class field theory and the arithmetic of algebraic groups in the language of group cohomology. They are still the standard references, and the questions he posed in them still guide research.

Definitions

Let \(K\) be a field with separable closure \(K_s\) and absolute Galois group \(G_K=\mathrm{Gal}(K_s/K)\), a profinite group. For a discrete \(G_K\)-module \(A\), the Galois cohomology groups are \(H^i(K,A)=H^i(G_K,A)\), computed with continuous cochains. For a non-abelian group \(G\) defined over \(K\), the pointed set \(H^1(K,G)\) classifies \(K\)-forms of objects with automorphism group \(G\), and torsors under \(G\).

The cohomological dimension \(\mathrm{cd}_p(K)\) is the least \(d\) such that \(H^i(K,A)=0\) for all \(i>d\) and all finite \(p\)-primary \(A\). Finite fields have dimension 1; by Tate’s theorems, \(p\)-adic fields and totally imaginary number fields have dimension 2.

Conjecture II is known for many groups, including the classical groups and the case \(G=\mathrm{SL}_1(D)\) through the Merkurjev–Suslin theorem, and for number fields its analogue is the Hasse principle of Kneser, Harder and Chernousov. It remains open in general, with the groups of type \(E_8\) the hardest case. The book also introduced Serre’s view of cohomological invariants, which he and others developed in later decades.

Trees and amalgams

For the general reader. A tree is a graph with no cycles. In his 1977 book Arbres, amalgames, SL\(_2\), Serre showed that a group acting on a tree can be read off from the pieces of the action: stabilizers of vertices and edges, glued along the quotient graph. The theory, completed with Hyman Bass, is now a basic tool of geometric group theory.

Bass extended part 2 to arbitrary quotient graphs: \(G\) is the fundamental group of a graph of groups. As a corollary, every subgroup of a free group acts freely on the same tree, so it is free (Nielsen–Schreier).

Two examples

\(\mathrm{SL}_2(\mathbb Z)\) acts on a tree embedded in the upper half-plane, with quotient an edge, and so

For a prime \(p\), the vertices of the Bruhat–Tits tree of \(\mathrm{SL}_2(\mathbb Q_p)\) are homothety classes of \(\mathbb Z_p\)-lattices in \(\mathbb Q_p^2\); each vertex has \(p+1\) neighbours. From this action Serre recovered Ihara’s theorem that every torsion-free discrete subgroup of \(\mathrm{SL}_2(\mathbb Q_p)\) is free, and the decomposition \(\mathrm{SL}_2(\mathbb Z[1/p])\cong\mathrm{SL}_2(\mathbb Z)*_{\Gamma_0(p)}\mathrm{SL}_2(\mathbb Z)\).

Serre also defined Euler characteristics for groups with torsion-free subgroups of finite index, by \(\chi(G)=\chi(H)/[G:H]\). For an amalgam of finite groups, \(\chi(A*_CB)=\tfrac1{|A|}+\tfrac1{|B|}-\tfrac1{|C|}\).

Galois representations and modularity

For the general reader. The symmetries of the points of finite order on an elliptic curve form a representation of the Galois group of the rational numbers. Serre proved that, for curves without extra symmetry, this representation is as large as it can be. He then conjectured a precise rule saying which two-dimensional representations come from modular forms. That conjecture, made in 1987, connected Fermat’s Last Theorem to modular forms and was proved in full in 2009.

The open image theorem

Let \(E\) be an elliptic curve over a number field \(K\). For a prime \(\ell\), the Galois group acts on the \(\ell\)-power torsion points, giving \(\rho_{E,\ell^\infty}:G_K\to\mathrm{GL}_2(\mathbb Z_\ell)\), and on the \(\ell\)-torsion, giving \(\bar\rho_{E,\ell}:G_K\to\mathrm{GL}_2(\mathbb F_\ell)\).

The proof combines the Chebotarev density theorem, the classification of subgroups of \(\mathrm{GL}_2(\mathbb F_\ell)\), Raynaud’s results on finite flat group schemes, and the theory of abelian \(\ell\)-adic representations that Serre developed in his 1968 book. Serre asked whether, over \(\mathbb Q\), the bound on exceptional \(\ell\) can be made uniform in \(E\). This uniformity question is still open.

The modularity conjecture

A continuous representation \(\bar\rho:G_{\mathbb Q}\to\mathrm{GL}_2(\bar{\mathbb F}_p)\) is odd if \(\det\bar\rho(c)=-1\) for complex conjugation \(c\). It is modular if it is the reduction of the representation attached to a Hecke eigenform.

The strong form already had a striking consequence in 1987. Gerhard Frey had suggested that a solution \(a^p+b^p=c^p\) gives an elliptic curve \(y^2=x(x-a^p)(x+b^p)\) with strange properties. Serre isolated the precise statement needed, the “epsilon conjecture” that the level can be lowered. Kenneth Ribet proved it in 1986, so that Fermat’s Last Theorem followed from the modularity of semistable elliptic curves. Andrew Wiles, with Richard Taylor, proved that in 1994. The weak form implies the strong form by work of Ribet and others, and Khare and Wintenberger, using Mark Kisin’s modularity lifting theorems, proved the full conjecture.

The Collège de France, 1956–1994

In 1956, at the age of 29, Serre was elected to the chair of Algebra and Geometry at the Collège de France. The post carries one obligation: to lecture each year on new research. For 38 years he used his courses to work out subjects he was learning or creating, and many of them became books: local fields, Galois cohomology, Lie algebras and Lie groups, abelian \(\ell\)-adic representations, trees, the Mordell–Weil theorem, and rational points of curves over finite fields. He retired in 1994 and became honorary professor.

His correspondence shaped the period as much as his papers. The letters with Alexander Grothendieck, mostly from 1955 to 1970 and published in 2001, record the building of modern algebraic geometry: Serre suggests, objects, finds references, and corrects. In 1958 his notion of locally isotrivial fibre spaces led Grothendieck towards the étale topology. In 1960 Serre proved a Kähler analogue of the Weil conjectures, which pointed the way to Pierre Deligne’s proof of the Riemann hypothesis over finite fields in 1974. His letters with John Tate, published in 2015, follow number theory across half a century.

He collaborated with Armand Borel on the Borel–Serre compactification of locally symmetric spaces (1973), with Deligne on modular forms of weight one (1974), with Hyman Bass on trees, and with Tate on good reduction of abelian varieties (1968). He spent many terms at the Institute for Advanced Study and at Harvard. His doctoral students include Michel Broué, Jean-Marc Fontaine and Pierre Gabriel, but his influence on students passed mostly through courses, books and letters.

His later research continued in number theory and group theory: the distribution of eigenvalues of Hecke operators, the number of points of curves over finite fields, finite subgroups of Lie groups, and the arithmetic of algebraic groups.

Honours, books and legacy

Serre’s honours span seven decades. After the Fields Medal (1954) came the Prix Peccot-Vimont of the Collège de France (1955), the Prix Gaston Julia (1970), the Émile Picard Medal of the Académie des Sciences (1971), Foreign Membership of the Royal Society (1974), the Balzan Prize (1985), the CNRS Gold Medal (1987), the Steele Prize for Mathematical Exposition (1995) and the Wolf Prize (2000). In 2003 the Norwegian Academy awarded him the first Abel Prize, citing his key role in giving modern form to topology, algebraic geometry and number theory. He was elected to the Académie des Sciences in 1977 and to academies in the Netherlands, the United States, Sweden, Russia and Norway, among others. He holds the Grand Cross of both the Légion d’honneur and the Ordre national du Mérite.

His collected papers, the Œuvres, run to five volumes, the last published in June 2026 and covering work through 2025. In September 2026 the Collège de France and the Institut Henri Poincaré held a two-day conference to mark his 100th birthday, and he took part remotely with a mathematical lecture.

Legacy

• Topology: the Serre spectral sequence, Serre classes and localization at a prime are the working tools of homotopy theory.
• Algebraic geometry: FAC and GAGA made cohomology of coherent sheaves the language of the subject and prepared the ground for Grothendieck.
• Number theory: Galois cohomology, the open image theorem and the modularity conjecture are central to the Langlands programme and to the proof of Fermat’s Last Theorem.
• Exposition: A Course in Arithmetic, Linear Representations of Finite Groups, Local Fields and Trees remain standard textbooks, noted for brevity and precision.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Jean-Pierre Serre, MacTutor History of Mathematics.
2. Jean-Pierre Serre, Wikipedia.
3. Collège de France, Jean-Pierre Serre, Algebra and Geometry chair (1956–1994).
4. Collège de France, Conference for the 100th birthday of Jean-Pierre Serre, 2026.
5. VnExpress International, Youngest-ever Fields Medal winner Jean-Pierre Serre turns 100, September 2026.
6. P. Colmez and J.-P. Serre (eds.), Grothendieck–Serre Correspondence, AMS and SMF, 2004.
7. P. Colmez and J.-P. Serre (eds.), Correspondance Serre–Tate, SMF, 2015.

Mathematics

1. J.-P. Serre, Homologie singulière des espaces fibrés. Applications, Ann. of Math. 54 (1951).
2. J.-P. Serre, Groupes d’homotopie et classes de groupes abéliens, Ann. of Math. 58 (1953).
3. J.-P. Serre, Faisceaux algébriques cohérents, Ann. of Math. 61 (1955).
4. J.-P. Serre, Un théorème de dualité, Comment. Math. Helv. 29 (1955).
5. J.-P. Serre, Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier 6 (1956).
6. J.-P. Serre, Local Fields, Galois Cohomology, Trees, A Course in Arithmetic, Abelian \(\ell\)-adic Representations and Elliptic Curves, Springer and Benjamin.
7. J.-P. Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972).
8. J.-P. Serre, Sur les représentations modulaires de degré 2 de \(\mathrm{Gal}(\bar{\mathbb Q}/\mathbb Q)\), Duke Math. J. 54 (1987).
9. C. Khare and J.-P. Wintenberger, Serre’s modularity conjecture (I), (II), Invent. Math. 178 (2009).
10. J. McCleary, A User’s Guide to Spectral Sequences, Cambridge, 2001.
11. R. Hartshorne, Algebraic Geometry, Springer, 1977, chapter III and appendix B.