André Weil

How to use this course

André Weil worked across more of mathematics than almost anyone of his century: number theory, algebraic geometry, group theory, differential geometry, harmonic analysis and history. His central idea was an analogy among three kinds of objects: number fields, curves over finite fields, and Riemann surfaces. He used it to prove the Riemann hypothesis for curves over finite fields and to state the conjectures that drove algebraic geometry from Grothendieck to Deligne. He also founded, with friends from the École Normale, the collective author Nicolas Bourbaki.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. Lessons 04 to 07 form a sequence on arithmetic geometry. Lessons 08, 09 and 10 can be read independently.

Timeline

Paris, 1906–1925

André Weil was born in Paris on 6 May 1906. His father, Bernard Weil, was a doctor from an Alsatian Jewish family; his mother, Selma Reinherz, came from a Jewish family of Russian origin that had settled in Austria. His only sibling, Simone, born in 1909, became one of the most original philosophers of the century. The two children were close, intensely competitive and intellectually precocious.

Weil learned to read early and entered the Lycée Montaigne in 1912. During the First World War his father was posted as an army doctor, and the family followed him; Weil attended lycées in Chartres and Laval and was taught privately in between. He found mathematics early and read widely beyond it. At 14 he taught himself Sanskrit, the beginning of a lifelong attachment to Indian literature and to the Greek and Latin classics.

At the Lycée Saint-Louis in Paris he met Jacques Hadamard, who took an interest in him. Weil entered the École Normale Supérieure in 1922, aged 16. He attended Hadamard’s seminar at the Collège de France, where the papers discussed came from the international research literature, and read the great authors directly: Riemann, Fermat, Gauss. He graduated in 1925, first in his class.

Rome, Göttingen and the thesis, 1925–1929

After a summer walking in the Alps, Weil spent the academic year 1925–26 in Rome on a scholarship. He attended lectures by Vito Volterra and Francesco Severi and absorbed the Italian school of algebraic geometry, with its powerful geometric intuition and its uncertain rigour. His first paper (1926) proved the isoperimetric inequality on surfaces of non-positive curvature.

In 1927 a Rockefeller fellowship took him to Göttingen, where he met Richard Courant and Emmy Noether, and began a lasting friendship with Carl Ludwig Siegel. His thesis problem came from Louis Mordell’s 1922 paper on rational points of cubic curves.

Hadamard encouraged him to prove the full conjecture in Mordell’s paper, that a curve of genus at least 2 has only finitely many rational points. Weil instead proved a theorem he could reach: the rational points of the Jacobian of any curve over a number field form a finitely generated group. His thesis, L’arithmétique sur les courbes algébriques, was defended in Paris in 1928 and published in Acta Mathematica in 1929. Mordell’s conjecture waited until Gerd Faltings proved it in 1983. Weil then did his military service, leaving as a lieutenant in 1929.

Aligarh, Strasbourg, Bourbaki, 1930–1939

In 1930 Weil went to India, to Aligarh Muslim University. He had hoped for a chair in French civilization; he was offered mathematics. He stayed for two academic years. After a year at Marseille he moved in 1933 to the University of Strasbourg, where his colleague was Henri Cartan.

The founding of Bourbaki

Cartan complained repeatedly to Weil about the inadequate textbooks available for the calculus course they taught. Weil proposed that the young mathematicians teaching the same course across France meet and write a new treatise. On 10 December 1934 Weil, Cartan, Claude Chevalley, Jean Delsarte, Jean Dieudonné and René de Possel met for lunch at the Café Capoulade in the Latin Quarter. At their first congress, at Besse-en-Chandesse in July 1935, they adopted the pseudonym Nicolas Bourbaki, after a student prank at the École Normale. The project grew from a calculus book into the Éléments de mathématique, a rebuilding of all of mathematics on axiomatic and structural foundations. The first volume appeared in 1939. Weil was the group’s driving force in its first decades; the symbol \(\varnothing\) for the empty set was his choice, borrowed from the Norwegian alphabet.

The Strasbourg years were productive. Weil developed uniform spaces (1937), wrote L’intégration dans les groupes topologiques et ses applications (published 1940), and in 1938 proved a Riemann–Roch theorem for matrix divisors, an early form of the theory of vector bundles on curves. On 30 October 1937 he married Éveline de Possel, formerly the wife of his Bourbaki colleague.

The Mordell–Weil theorem

For the general reader. On a cubic curve, two rational points determine a third, by drawing the line through them. This makes the rational points into a group. Mordell proved in 1922 that a finite list of points generates them all. Weil’s thesis extended this to every curve and every number field, through the Jacobian, a higher-dimensional group attached to the curve.

Let \(E: y^2=x^3+ax+b\) be an elliptic curve over a number field \(K\), with point at infinity \(O\) as identity. Three points sum to \(O\) exactly when they lie on a line. For \(P=(x_1,y_1)\), \(Q=(x_2,y_2)\) with \(x_1\ne x_2\), put \(\lambda=(y_2-y_1)/(x_2-x_1)\); if \(P=Q\) and \(y_1\ne0\), put \(\lambda=(3x_1^2+a)/(2y_1)\). Then

The integer \(r\) is the rank. It is not known to be computable in general, and its behaviour is the subject of the Birch and Swinnerton-Dyer conjecture.

Structure of the proof

Weak Mordell–Weil. \(A(K)/mA(K)\) is finite for some (every) \(m\ge2\). For an elliptic curve with all 2-torsion rational, \(E: y^2=(x-e_1)(x-e_2)(x-e_3)\), the map \(P\mapsto(x-e_1,\,x-e_2)\) into \((K^\times/K^{\times2})^2\) is a homomorphism with kernel \(2E(K)\). Its image consists of classes unramified outside a fixed finite set \(S\) of primes, and there are only finitely many such classes, by finiteness of the class group and the finite generation of \(S\)-units. Weil’s general version uses the same idea with unramified coverings of \(A\).

Descent. A height on \(A(K)\) is a function \(h\ge0\) such that (i) only finitely many points have \(h(P)\le C\) for each \(C\); (ii) for each \(Q\) there is \(c_Q\) with \(h(P+Q)\le2h(P)+c_Q\); (iii) \(h(mP)\ge m^2h(P)-c\). For \(E\) over \(\mathbb Q\) one can take \(h(P)=\log\max(|p|,|q|)\) where \(x(P)=p/q\) in lowest terms. Weil introduced the general theory of heights for this proof.

The Riemann hypothesis for curves

For the general reader. Count the solutions of a polynomial equation in a finite field with \(q\) elements. For a curve, the answer is close to \(q+1\), and Weil proved exactly how close: the error is at most a fixed multiple of \(\sqrt q\). This is equivalent to an analogue of the Riemann hypothesis, the most famous open problem in mathematics, and Weil found the proof while in prison in 1940.

Let \(C\) be a smooth projective curve of genus \(g\) over \(\mathbb F_q\), and \(N_m=\#C(\mathbb F_{q^m})\). Its zeta function is

the product taken over closed points \(x\) of \(C\). With \(T=q^{-s}\) this is an Euler product like Riemann’s \(\zeta(s)=\prod_p(1-p^{-s})^{-1}\). Emil Artin studied such functions for hyperelliptic curves in his thesis, and F. K. Schmidt proved rationality and the functional equation in general using Riemann–Roch.

Helmut Hasse had proved the case \(g=1\) in the 1930s. Weil announced the general case in a note written in April 1940 and sketched a second proof in 1941. The complete proofs required new foundations (Lesson 07) and appeared in two books in 1948.

The proof on \(C\times C\)

Let \(F\) be the \(q\)-power Frobenius map of \(C\), \(\Delta\subset C\times C\) the diagonal and \(\Gamma\) the graph of \(F\). Points of \(\Delta\cap\Gamma\) are fixed points of \(F\), so \(\Delta\cdot\Gamma=N_1\) (the intersections are transversal). Adjunction gives \(\Delta^2=2-2g\), and since \(\Gamma\) is the graph of a map of degree \(q\), \(\Gamma^2=q(2-2g)\). For a divisor \(D\) put \(d_1=D\cdot(\{pt\}\times C)\) and \(d_2=D\cdot(C\times\{pt\})\). The Castelnuovo–Severi inequality, a form of the Hodge index theorem, states \(D^2\le2d_1d_2\). For \(D=r\Delta+s\Gamma\), \(d_1=r+s\) and \(d_2=r+qs\), so

A quadratic form that is non-negative on \(\mathbb Z^2\) is non-negative on \(\mathbb R^2\), so its discriminant is \(\le0\): \((N_1-q-1)^2\le4g^2q\). Applying this over \(\mathbb F_{q^m}\) gives \(\big|\sum_i\alpha_i^m\big|\le2g\,q^{m/2}\) for all \(m\). A standard argument with \(\sum_m\sum_i(\alpha_iT)^m\) then shows \(|\alpha_i|\le\sqrt q\), and the functional equation, which pairs \(\alpha_i\) with \(q/\alpha_i\), forces equality.

The Weil conjectures

For the general reader. After curves, Weil asked about surfaces and higher-dimensional varieties over finite fields. Computing examples with Gauss sums, he noticed that the counts were governed by the shape of the corresponding complex manifold: its number of holes in each dimension. He conjectured that this was always so. Proving it took the invention of new cohomology theories and twenty-five years.

Let \(X\) be a smooth projective variety of dimension \(n\) over \(\mathbb F_q\), \(N_m=\#X(\mathbb F_{q^m})\), and \(Z(X,T)=\exp\big(\sum_m N_mT^m/m\big)\).

Weil’s insight was that these statements would follow from a cohomology theory for varieties over finite fields, with a Lefschetz fixed-point formula \(N_m=\sum_i(-1)^i\operatorname{tr}\big(F^m\mid H^i\big)\) for the Frobenius map \(F\). His evidence came from equations \(a_0x_0^{k}+\cdots+a_rx_r^{k}=0\), whose solution counts he expressed through Gauss and Jacobi sums, using that a Gauss sum has absolute value \(\sqrt q\), a result going back to Gauss himself.

The proofs

• Bernard Dwork (1960): rationality, by \(p\)-adic analysis.
• Alexander Grothendieck, with Michael Artin and Jean-Louis Verdier (1960s): étale cohomology, a Weil cohomology with \(\ell\)-adic coefficients, giving rationality, the functional equation and the Betti number statement.
• Pierre Deligne (1974): the Riemann hypothesis, in La conjecture de Weil I, using monodromy of Lefschetz pencils and a positivity argument with Rankin’s method. He received the Fields Medal in 1978.

Foundations and abelian varieties

For the general reader. The Italian geometers had wonderful intuition but shaky definitions, and their methods worked only over the complex numbers. To make his 1940 proof rigorous, Weil had to rebuild algebraic geometry over any field. His Foundations of Algebraic Geometry (1946) did this, and two books of 1948 constructed the Jacobian of a curve without complex analysis. Grothendieck later replaced Weil’s language with schemes, but Weil’s constructions remained.

In Foundations Weil worked with varieties defined over a field \(k\) inside a large “universal domain” \(\Omega\), algebraically closed of infinite transcendence degree, and with generic points and specializations. He developed intersection multiplicities in this setting. In Variétés abéliennes et courbes algébriques (1948) he defined abstract varieties by gluing affine pieces, introduced abelian varieties over arbitrary fields, and constructed the Jacobian \(J(C)\) of a curve of genus \(g\) as an abelian variety of dimension \(g\) from the symmetric power \(C^{(g)}\).

Divisors and reciprocity

A Weil divisor on a normal variety is a finite formal sum of codimension-one subvarieties. For a curve and a function \(f\), \(\operatorname{div}f=\sum_P\operatorname{ord}_P(f)\,P\), and \(f(D)=\prod_Pf(P)^{n_P}\) for \(D=\sum n_PP\) disjoint from the zeros and poles of \(f\).

The Weil pairing

Construction: for \(Q\in E[n]\) choose \(f\) with \(\operatorname{div}f=n(Q)-n(O)\), and \(g\) with \(g^n=f\circ[n]\). Then \(e_n(P,Q)=g(X+P)/g(X)\) for any \(X\) where both sides are defined. Weil reciprocity is the key to proving the properties. The pairing exists for every abelian variety, pairing \(A[n]\) with the dual abelian variety. It is central in cryptography, in the proof of the Mordell–Weil theorem, and in the theory of Galois representations.

Adeles, Haar measure, Tamagawa numbers

For the general reader. A number can be measured in many ways: by its ordinary size, and by its divisibility by each prime. Weil’s adeles put all these measurements together in one object. Inside it the rational numbers sit like a lattice, and volumes computed there encode deep arithmetic. Weil turned this into a method that unified classical results of Gauss, Minkowski and Siegel.

Integration on groups

In L’intégration dans les groupes topologiques (1940), Weil developed harmonic analysis on locally compact groups: existence and uniqueness of Haar measure, Pontryagin duality, and the Fourier transform. Every locally compact group \(G\) carries a left-invariant Radon measure, unique up to a positive scalar. This made it possible to do analysis on \(p\)-adic and adelic groups as on \(\mathbb R^n\).

Adeles

Let \(K\) be a number field with completions \(K_v\). The adele ring is the restricted product

topologized so that \(\prod_{v\mid\infty}K_v\times\prod_{v\nmid\infty}\mathcal O_v\) is open. Chevalley had introduced the idele group \(\mathbb I_K=\mathbb A_K^\times\) in 1936 for class field theory; Weil’s name “adele” (additive idele) came later, and he made the additive and multiplicative groups the setting for his treatment of the subject.

Tamagawa numbers

For a semisimple algebraic group \(G\) over \(K\), an invariant differential form of top degree gives a canonical measure on \(G(\mathbb A_K)\), and \(\tau(G)=\operatorname{vol}\big(G(\mathbb A_K)/G(K)\big)\) is its Tamagawa number. Weil showed that Siegel’s mass formula for quadratic forms is equivalent to \(\tau(\mathrm{SO}(q))=2\), and computed \(\tau\) for classical groups in his 1959–60 IAS lectures, Adeles and Algebraic Groups. He conjectured \(\tau(G)=1\) for simply connected \(G\). Robert Kottwitz completed the proof over number fields in 1988, building on work of Robert Langlands and Kuo-Fong Lai; Dennis Gaitsgory and Jacob Lurie proved the function field case in 2019.

Chern–Weil theory

For the general reader. The Gauss–Bonnet theorem says that the total curvature of a closed surface is \(2\pi\) times its Euler number, a topological invariant. Weil, with Carl Allendoerfer, extended this to higher-dimensional manifolds in 1943, and later, with Shiing-Shen Chern, built the general machine that turns curvature into topology.

Allendoerfer and Weil proved it by embedding \(M\) locally isometrically in Euclidean space and decomposing it into polyhedra. Chern found an intrinsic proof in 1944, which led directly to Chern classes.

The Chern–Weil homomorphism

Let \(G\) be a Lie group with Lie algebra \(\mathfrak g\), \(P\to M\) a principal \(G\)-bundle with connection form \(\omega\) and curvature \(\Omega=d\omega+\tfrac12[\omega,\omega]\). Let \(I(G)\) be the algebra of \(\mathrm{Ad}\)-invariant polynomials on \(\mathfrak g\).

For a complex vector bundle with \(G=\mathrm{U}(n)\), the Chern classes are defined by \(\det\big(I+\tfrac{i}{2\pi}\Omega\big)=1+c_1+c_2+\cdots\). Weil set out the construction in an unpublished manuscript of 1949, later printed in his collected papers; it was spread by Chern’s lectures and by Henri Cartan’s seminar, which introduced the Weil algebra to describe it.

The Weil representation and modularity

For the general reader. In his later work Weil returned to classical objects, theta functions and modular forms, and reorganized them with representation theory. He also stated, more precisely than anyone before, the conjecture that every elliptic curve over the rationals comes from a modular form. Andrew Wiles’s proof of a large case of that conjecture proved Fermat’s Last Theorem.

The Weil representation (1964)

The symplectic group \(\mathrm{Sp}(2n,\mathbb R)\) acts on the Heisenberg group, and by the Stone–von Neumann theorem this produces a projective representation on \(L^2(\mathbb R^n)\). Weil showed that it becomes a true representation of a double cover, the metaplectic group \(\mathrm{Mp}(2n,\mathbb R)\), over every local field, and adelically. Generators act on \(f\in L^2(\mathbb R^n)\) by

up to constants of absolute value 1. Theta series arise as distributions invariant under the arithmetic subgroup. In 1965 he proved the Siegel–Weil formula, which expresses an Eisenstein series as an average of theta series over a genus of quadratic forms, recovering Siegel’s mass formula in representation-theoretic form.

The converse theorem and modularity (1967)

In the same paper Weil observed that the conjectured functional equations of Hasse–Weil \(L\)-functions of elliptic curves over \(\mathbb Q\) would therefore make them modular, with the level equal to the conductor of the curve. The question had been raised by Yutaka Taniyama in 1955 and made precise by Goro Shimura. The modularity theorem (Wiles and Taylor–Wiles 1995 for semistable curves; Breuil, Conrad, Diamond and Taylor 2001 in general) says that for every elliptic curve \(E/\mathbb Q\) of conductor \(N\) there is a newform \(f\) of weight 2 and level \(N\) with \(a_p(f)=p+1-\#E(\mathbb F_p)\) for all \(p\nmid N\).

War, prison, exile, 1939–1947

Weil spent the spring and summer of 1939 in Scandinavia and was in Finland when war broke out. He had decided not to serve in the army, a decision he later defended at length in his autobiography. When the Soviet Union attacked Finland at the end of November 1939, he was arrested as a suspected spy; letters in Russian and papers mentioning Bourbaki did not help. The Finnish mathematician Rolf Nevanlinna intervened, and Weil was deported. He reached France through Sweden and Britain and was arrested at Le Havre in January 1940 for failing to report for duty.

He spent February to May 1940 in the military prison of Bonne-Nouvelle in Rouen. There he worked well. On 26 March he wrote a long letter to Simone explaining the analogy among number fields, function fields over finite fields and Riemann surfaces, which he compared to a trilingual inscription, a Rosetta stone for mathematics. In April he wrote the note announcing his proof of the Riemann hypothesis for curves over finite fields. At his trial on 3 May 1940 he was sentenced to five years, and asked to serve in a combat unit instead. He was sent to a regiment at Cherbourg. After the fall of France in June he joined his family in Marseille.

In January 1941 Weil and his family sailed from Marseille to New York, with support from the Rockefeller Foundation. Simone reached New York in 1942 but soon left for London to work for the Free French, and died in Ashford, Kent, on 24 August 1943. Weil taught at Haverford College and then for two years at Lehigh University, a period he described as underpaid and overworked. In 1945 he went to the University of São Paulo, where he worked alongside Oscar Zariski. Foundations of Algebraic Geometry appeared in 1946. His daughters Sylvie and Nicolette were born in 1942 and 1946.

Chicago, Princeton, legacy, 1947–1998

In 1947 Weil joined the University of Chicago, where Marshall Stone was building a strong department. His Chicago decade produced the Weil conjectures (1949), Chern–Weil theory, work on Kähler manifolds and theta functions, and a generation of students. He gave plenary addresses at the International Congresses of 1950 in Cambridge, Massachusetts, and 1954 in Amsterdam, and spoke again in Helsinki in 1978. In 1958 he moved to the Institute for Advanced Study in Princeton, where he remained, becoming emeritus in 1976.

His doctoral students included Pierre Cartier, Harley Flanders, Teruhisa Matsusaka and Peter Swinnerton-Dyer. His influence ran much wider, through his books, his letters and his seminars: Serre, Grothendieck, Langlands and Shimura all built on his programme. In later years he turned to history, writing Number Theory: An Approach Through History from Hammurapi to Legendre (1984) and studies of Fermat, Euler and Eisenstein. His autobiography of his early years, Souvenirs d’apprentissage, appeared in 1991. He guarded his sister’s memory and her papers.

He was elected a Foreign Member of the Royal Society in 1966 and was a member of the Paris Academy of Sciences and the US National Academy of Sciences. He received the Wolf Prize with Jean Leray in 1979, the Steele Prize and the Barnard Medal in 1980, and the Kyoto Prize in 1994. He died in Princeton on 6 August 1998, aged 92.

Legacy

• Arithmetic geometry: the Mordell–Weil theorem, heights, and the Riemann hypothesis for curves made number theory and algebraic geometry one subject.
• The Weil conjectures: their proof motivated schemes, étale cohomology, and the theory of motives; Deligne’s 1974 proof is one of the summits of the century.
• Automorphic forms: adeles, Tamagawa numbers, the Weil representation, the converse theorem and the Weil group are basic tools of the Langlands programme.
• Bourbaki: the structural style of the Éléments shaped how mathematics was written and taught for decades.
• Names: Weil divisors, Weil pairing, Weil restriction, Weil cohomology, Weil group, Weil numbers, Weil–Petersson metric, Hasse–Weil zeta functions, Chern–Weil theory.

Connections

People linked to André Weil across this collection. A name without a link has no page here yet. See André Weil in the network of all 337 lives →

Teachers and advisors

• Jacques Hadamard 1928
• Émile Picard 1928

Students

• Harley Flanders
• Peter Swinnerton-Dyer
• Pierre Cartier
• Teruhisa Matsusaka

Influenced by

• Emil Artin
• Emmy Noether 1927
• Erich Hecke
• Helmut Hasse
• Solomon Lefschetz
• Louis Mordell

Influenced

• Alexander Grothendieck
• Jean-Pierre Serre
• Pierre Deligne
• Robert Langlands
• Vladimir Drinfeld
• Goro Shimura

Colleagues and collaborators

• Atle Selberg
• Carl Ludwig Siegel
• Claude Chevalley
• Harish-Chandra 1950-1983
• Oscar Zariski 1945
• Carl Allendoerfer 1943
• Henri Cartan 1933-1991
• Jean Delsarte
• Jean Dieudonné
• René de Possel
• Shiing-Shen Chern

Friends

• Carl Ludwig Siegel 1927-1981
• Rolf Nevanlinna 1939

Family

• Simone Weil

Also mentioned in this life

• Richard Courant
• Peter Gustav Lejeune Dirichlet
• Leonhard Euler
• Pierre de Fermat
• Joseph Fourier
• Georg Frobenius
• Évariste Galois
• Carl Friedrich Gauss
• Werner Heisenberg
• Carl Gustav Jacob Jacobi
• Henri Lebesgue
• Adrien-Marie Legendre
• Sophus Lie
• Hermann Minkowski
• Bernhard Riemann
• Vito Volterra
• John von Neumann
• Andrew Wiles

Fields

• number theory 45 lives
• algebraic geometry 17 lives
• arithmetic geometry 3 lives
• harmonic analysis 3 lives
• differential geometry 15 lives

On CoursesHub

• Differential Geometry

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, André Weil, MacTutor History of Mathematics.
2. André Weil, Wikipedia.
3. A. Weil, Souvenirs d’apprentissage, Birkhäuser, 1991; English translation The Apprenticeship of a Mathematician, 1992.
4. A. Weil, A 1940 letter of André Weil on analogy in mathematics, translated by M. H. Krieger, Notices of the AMS 52 (2005).
5. S. Weil, At Home with André and Simone Weil, Northwestern University Press, 2010.
6. Nicolas Bourbaki, Wikipedia.

Mathematics

1. A. Weil, L’arithmétique sur les courbes algébriques, Acta Math. 52 (1929), 281–315.
2. A. Weil, Sur les courbes algébriques et les variétés qui s’en déduisent and Variétés abéliennes et courbes algébriques, Hermann, 1948.
3. A. Weil, Numbers of solutions of equations in finite fields, Bull. Amer. Math. Soc. 55 (1949), 497–508.
4. A. Weil, Foundations of Algebraic Geometry, AMS, 1946.
5. A. Weil, Basic Number Theory, Springer, 1967; Adeles and Algebraic Groups, Birkhäuser, 1982.
6. A. Weil, Sur certains groupes d’opérateurs unitaires, Acta Math. 111 (1964).
7. A. Weil, Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann. 168 (1967).
8. A. Weil, Œuvres scientifiques / Collected Papers, 3 vols., Springer, 1979.
9. J. S. Milne, The Riemann hypothesis over finite fields: from Weil to the present day, arXiv:1509.00797.
10. P. Deligne, La conjecture de Weil I, Publ. Math. IHÉS 43 (1974).
11. J. H. Silverman, The Arithmetic of Elliptic Curves, 2nd ed., Springer, 2009.
12. Weil conjectures and Mordell–Weil theorem, Wikipedia.