Émile Picard

How to use this course

Émile Picard’s name is attached to more mathematical objects than almost anyone’s of his generation: the little and great Picard theorems, the Picard iteration and the Picard–Lindelöf theorem, Picard–Vessiot theory, the Picard–Fuchs equation, the Picard group and Picard variety, the Picard–Lefschetz formula, the Picard modular group. Each grew from a different part of his work, yet they share a style. Picard took a classical object, an analytic function, a differential equation, a surface, and asked which structure controlled it.

He was also, for a quarter of a century, the most powerful administrator in French science. This course follows both careers.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proofs or sketches, and exercises with folded solutions. Lessons 04, 05 and 08 share one object, the modular function λ, and are best read in order.

Timeline

Paris, 1856–1877

Charles Émile Picard was born in Paris on 24 July 1856. His father managed a silk factory; his mother was a doctor’s daughter. He had one younger brother. During the Franco-Prussian War his father died in the siege of Paris of 1870.

Picard attended the Lycée Napoléon, later renamed the Lycée Henri IV. He was a strong pupil across subjects. In the entrance examinations he ranked second for the École Polytechnique and first for the École Normale Supérieure. He chose the Normale, then the usual path to research.

Progress was fast. In 1877 he came first in the agrégation, and the same year he defended a doctoral thesis, prepared under Gaston Darboux, on applications of linear complexes to the study of surfaces and twisted curves. He was not yet 21. The ENS kept him for a year as an assistant, and in 1878 he became a lecturer at the University of Paris.

Toulouse and the theorem, 1879–1881

In 1879 Picard was appointed professor at Toulouse, where he taught differential and integral calculus for two years. That year, aged 22, he published in the Comptes rendus a short note proving that an entire function that is not constant takes every complex value with at most one exception. Karl Weierstrass and Felice Casorati had shown that such a function comes arbitrarily close to every value. Picard showed that it actually hits almost all of them.

The proof was striking because it used no estimates at all. It used the elliptic modular function that Hermite had studied, a function with a deep arithmetic origin, to turn a question about growth into Liouville’s theorem (Lesson 04). In 1880 he extended the result to the neighbourhood of an essential singularity (Lesson 05). Mathematicians spent the next fifty years looking for proofs that avoided the modular function, and the search produced a large part of modern geometric function theory.

In 1881 Picard returned to Paris as maître de conférences in mechanics and astronomy at the ENS. The same year he married Marie Hermite, daughter of Charles Hermite, whose lectures had shaped him. He later wrote a biography of his father-in-law. The Académie des Sciences considered him for membership in 1881, but he was judged too young.

The Sorbonne, 1881–1897

The 1880s were Picard’s most productive decade. He built a Galois theory for linear differential equations (Lesson 07), studied period integrals and their differential equations (Lesson 08), began his work on algebraic surfaces (Lesson 09) and, in 1884, investigated the group \(\mathrm{PSL}_2(\mathbb Z[i])\) acting on three-dimensional hyperbolic space (Lesson 10).

When Claude Bouquet died in 1885, the Sorbonne appointed Picard to his chair of differential calculus. Regulations required a professor to be thirty, so until his thirtieth birthday in 1886 he served formally as his own substitute. Prizes followed: the Poncelet Prize in 1886 and the Grand Prix des Sciences Mathématiques in 1888. In 1889 he was elected to the Académie des Sciences.

In 1890 he published a long memoir on partial differential equations in which the method of successive approximations, used before in special cases by Joseph Liouville and others, became a general existence proof (Lesson 06). His three-volume Traité d’analyse (1891–1896) made him the leading teacher of analysis in France. Jacques Hadamard, his doctoral student, called his teaching perhaps the most marvellous he had known. In 1897 Picard exchanged his chair for that of analysis and higher algebra, which allowed him to teach research students. Among those who wrote doctorates under him were Hadamard, Paul Painlevé, Ernest Vessiot, Sergei Bernstein, Gaston Julia, Simion Stoilow and Mihailo Petrović.

The little theorem

For the general reader. A polynomial of degree at least one takes every complex value. The exponential function \(e^z\) never takes the value 0, but it takes every other value. Picard proved in 1879 that this is the worst that can happen: a function defined by a power series converging everywhere, if it is not constant, misses at most one value.

Replacing \(f\) by \((f-a)/(b-a)\), we may assume the omitted values are 0 and 1. Compare: Liouville’s theorem says that if \(f\) omits a whole disc, \(f\) is constant (apply Liouville to \(1/(f-c)\)). Picard needs only two points.

The modular function

Let \(\mathbb H=\{\tau:\operatorname{Im}\tau>0\}\). The elliptic modular function \(\lambda:\mathbb H\to\mathbb C\) can be defined through Jacobi’s theta functions:

Geometrically, \(\lambda(\tau)\) is the value such that the elliptic curve \(y^2=x(x-1)(x-\lambda)\) has period ratio \(\tau\) for a suitable choice of basis. Its key properties:

Picard’s proof

Suppose \(f:\mathbb C\to\mathbb C\setminus\{0,1\}\) is holomorphic. Since \(\mathbb C\) is simply connected and \(\lambda\) is a covering, \(f\) lifts: there is a holomorphic \(g:\mathbb C\to\mathbb H\) with \(\lambda\circ g=f\). Concretely, start with a local inverse of \(\lambda\) near \(f(0)\) and continue \(\lambda^{-1}\circ f\) analytically along paths; the monodromy theorem makes it single-valued. Now the Cayley map \(w\mapsto(w-i)/(w+i)\) sends \(\mathbb H\) into the unit disc, so \(h=(g-i)/(g+i)\) is a bounded entire function. By Liouville \(h\) is constant, so \(g\) and \(f=\lambda\circ g\) are constant. \(\square\)

The argument is short because the hard part, the existence of a holomorphic universal covering of \(\mathbb C\setminus\{0,1\}\) by the disc, is supplied by the modular function. In modern terms, \(\mathbb C\setminus\{0,1\}\) is hyperbolic and \(\mathbb C\) is not, and holomorphic maps contract hyperbolic metrics. Émile Borel gave a proof avoiding modular functions in 1896, and André Bloch’s theorem of 1925 gives one through elementary estimates.

The great theorem

For the general reader. Some functions blow up near a point in a tame way, like \(1/z\). Others, like \(e^{1/z}\), behave wildly: near 0 they swing through almost every value, again and again. Picard proved in 1880 that wild behaviour is total. In every tiny disc around such a point, the function takes every complex value infinitely many times, with at most one exception.

An isolated singularity \(z_0\) of a holomorphic function is removable, a pole, or essential. At an essential singularity the Laurent series has infinitely many negative powers.

The little theorem follows. If \(f\) is entire and not a polynomial, then \(f(1/z)\) has an essential singularity at 0, so \(f\) takes all values but one infinitely often. If \(f\) is a non-constant polynomial, it takes every value.

Proof through normal families

A family of holomorphic functions on a domain is normal if every sequence has a subsequence converging locally uniformly, either to a holomorphic function or to \(\infty\).

Montel proved this using the modular function or, equivalently, Schottky’s 1904 estimate. With it, the great theorem follows quickly. Take \(z_0=0\) and suppose \(f\) omits 0 and 1 on \(0<|z|<r\). Define \(f_n(z)=f(z/2^n)\) on the annulus \(A=\{r/2<|z|<r\}\). The \(f_n\) omit 0 and 1, so a subsequence converges uniformly on the circle \(|z|=3r/4\), either to a holomorphic limit or to \(\infty\). In the first case \(f\) is bounded on circles \(|z|=3r/2^{n_k+2}\) shrinking to 0, so by the maximum principle it is bounded near 0 and the singularity is removable. In the second, \(1/f\) is bounded the same way, and 0 is a pole. Either way it is not essential. If \(f\) took some value only finitely often, the same argument applies on a smaller punctured disc. \(\square\)

The great theorem started value distribution theory. Rolf Nevanlinna’s theory of 1925 measures how often a meromorphic function takes each value, and his second main theorem is a quantitative Picard theorem.

Successive approximations

For the general reader. To solve \(y'=F(x,y)\), guess a solution, plug it into the right-hand side, integrate, and repeat. Under mild conditions the guesses converge to the true solution. Picard made this simple idea, in his 1890 memoir, into a general method for proving that differential equations have solutions, and it is still the standard proof taught today.

The initial value problem \(y'=F(x,y)\), \(y(x_0)=y_0\) is equivalent, for continuous \(F\), to the integral equation

Picard’s iteration sets \(y_0(x)\equiv y_0\) and \(y_{n+1}(x)=y_0+\int_{x_0}^xF(t,y_n(t))\,dt\).

Proof. By induction, each \(y_n\) is defined on \(I=[x_0-h,x_0+h]\) and stays in \(R\), since \(|y_{n+1}(x)-y_0|\le M|x-x_0|\le Mh\le b\). Next, \(|y_1-y_0|\le M|x-x_0|\), and the Lipschitz condition gives

So \(\sum(y_{n+1}-y_n)\) converges uniformly by comparison with the exponential series, and the limit satisfies the integral equation. For uniqueness, if \(y,z\) are two solutions then \(|y-z|\le L|\int|y-z||\), and iterating gives \(|y-z|\le 2b\,(L|x-x_0|)^n/n!\to0\). \(\square\)

The factorial plays the same role as in Volterra’s integral equations. Ernst Lindelöf put the method in this sharp form in 1894. In Stefan Banach’s language of 1922, the integral operator is a contraction on a suitable space, and the theorem becomes an instance of the contraction mapping principle. In the 1890 memoir Picard applied the iteration to ordinary equations and above all to partial differential equations, such as the hyperbolic equation \(\partial^2u/\partial x\partial y=F(x,y,u,u_x,u_y)\) with data on characteristics, and for nonlinear boundary value problems.

Galois theory for differential equations

For the general reader. Évariste Galois explained which polynomial equations can be solved by radicals by attaching a group of symmetries to each equation. In the 1880s Picard did the same for linear differential equations. The symmetries now form a continuous group of matrices, and its structure decides whether the equation can be solved by integrals and exponentials. Picard’s student Ernest Vessiot completed the theory in his 1892 thesis.

Setting

A differential field is a field \(K\) with a derivation \(\partial\); here \(K=\mathbb C(x)\) with \(\partial=d/dx\), whose field of constants is \(\mathbb C\). Consider

Picard called \(G\) the group of transformations of the equation, by analogy with the group of a polynomial. The monodromy group, obtained by continuing solutions around singular points, is a subgroup of \(G\); for Fuchsian equations it is Zariski-dense in \(G\).

Ellis Kolchin gave the rigorous modern form in the 1940s. In 1986 Jerald Kovacic gave an algorithm deciding this for second-order equations.

Periods and the Picard–Fuchs equation

For the general reader. An elliptic curve has two periods, integrals that measure its shape. When the curve varies in a family, the periods vary too, and they obey a linear differential equation. Picard studied these equations for curves and surfaces, building on the work of Lazarus Fuchs. Today the equations bear both names, and they are a basic tool in algebraic geometry, number theory and string theory.

The Legendre family

For \(\lambda\neq0,1\), let \(E_\lambda: y^2=x(x-1)(x-\lambda)\). A period is \(\omega(\lambda)=\oint_\gamma dx/y\) over a closed cycle \(\gamma\) on \(E_\lambda\). One of them, up to a factor 2, is

valid for \(|\lambda|<1\), where \(F\) is Gauss’s hypergeometric function.

Why a differential equation exists. The forms \(dx/y\), \(\partial_\lambda(dx/y)\) and \(\partial_\lambda^2(dx/y)\) are three classes in the two-dimensional de Rham cohomology \(H^1(E_\lambda)\), so some combination with coefficients in \(\mathbb C(\lambda)\) is exact. Integrating over a closed cycle kills exact forms and gives the equation. Finding the coefficients is a computation in the spirit of Picard’s treatment.

Monodromy and λ

Continuing the two periods around \(\lambda=0\) and \(\lambda=1\) acts on them by matrices that generate the group \(\Gamma(2)\), up to sign. The ratio \(\tau=\omega_2/\omega_1\) of two suitably chosen periods lies in \(\mathbb H\), and the map \(\lambda\mapsto\tau\) is the multivalued inverse of the modular function \(\lambda(\tau)\). This is exactly the inverse used in Picard’s proof of Lesson 04.

Algebraic surfaces

For the general reader. An algebraic curve is the set of solutions of a polynomial equation in two variables; an algebraic surface uses three. Bernhard Riemann had understood curves through their integrals. In the 1880s Picard tried the same for surfaces and found new invariants, such as the number of independent curves a surface carries. His treatise with Georges Simart, published in two volumes in 1897 and 1906, was one of the foundations of the Italian school of algebraic geometry.

Integrals of total differentials

On a smooth projective surface \(S\), Picard studied integrals \(\int P\,dx+Q\,dy\) of closed 1-forms, the analogues of abelian integrals on curves. He distinguished integrals of the first kind (everywhere finite) and showed that their number is a birational invariant, now called the irregularity \(q=\dim H^0(S,\Omega^1)\). He showed that a general surface in ordinary space has no such integrals, so that for most surfaces the theory of curves on the surface must be built by other means.

Divisors and the Picard group

A divisor on \(S\) is a formal integer combination of irreducible curves. Two divisors are linearly equivalent if they differ by the divisor of a rational function. In modern language:

Picard introduced \(\rho\) as the maximum number of curves on \(S\) that are independent in the sense that no integer combination is the divisor of a rational function up to algebraic equivalence, and proved that it is finite. Francesco Severi proved the theorem of the base in 1906. Solomon Lefschetz later identified the Néron–Severi group with the integral classes of type \((1,1)\), which gives \(\rho\le h^{1,1}\).

The Picard–Lefschetz formula

Picard studied a surface by slicing it with a pencil of planes, so that it becomes a family of curves \(C_t\) over a line. At finitely many \(t\) the curve acquires a node, where a loop \(\delta\) on the nearby smooth curve, the vanishing cycle, shrinks to a point.

Picard found this for families of curves on surfaces; Lefschetz extended it to all dimensions in 1924, where the sign becomes \((-1)^{n(n+1)/2}\) for fibres of complex dimension \(n\). It is the basis of the Lefschetz pencil method and of the proof of the Weil conjectures by Pierre Deligne.

The Picard modular group

For the general reader. The modular group of integer matrices acts on the hyperbolic plane and creates the tiling in the picture at the top of this page. In 1884 Picard replaced the integers by the Gaussian integers \(a+bi\) and found a group acting on three-dimensional hyperbolic space. It was the first example of what are now called Bianchi groups, and it opened the study of arithmetic groups in higher dimensions.

Hyperbolic 3-space

Let \(\mathbb H^3=\{(z,t):z\in\mathbb C,\ t>0\}\) with metric \((|dz|^2+dt^2)/t^2\). Writing a point as the quaternion \(P=z+tj\), a matrix \(\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\mathrm{SL}_2(\mathbb C)\) acts by \(P\mapsto(aP+b)(cP+d)^{-1}\). In coordinates,

This extends the action of Möbius transformations on the boundary \(\mathbb C\cup\{\infty\}\), and \(\mathrm{PSL}_2(\mathbb C)\) is the full group of orientation-preserving isometries of \(\mathbb H^3\).

Luigi Bianchi treated \(\mathrm{PSL}_2(\mathcal O_d)\) for all imaginary quadratic fields in 1892. Picard also studied related groups of matrices acting on the complex 2-ball, now called Picard modular groups and Picard modular surfaces. Their arithmetic is studied today as part of the Langlands programme.

Secretary of the Academy, 1897–1924

Picard’s later research turned toward analysis applied to physics: elasticity, heat conduction and the propagation of electrical signals along telegraph cables. He wrote widely for a general audience on the state of science, for instance La science moderne et son état actuel (1905), and in 1922 published one of the first French textbooks on relativity and its astronomical applications. From the 1890s until 1937 he also taught mechanics at the École Centrale des Arts et Manufactures, where he trained more than ten thousand engineers. Abroad he was elected an honorary member of the London Mathematical Society in 1898 and a Foreign Member of the Royal Society in 1909.

The First World War took his family. His son Charles, a finance inspector, was killed at Crouy on 15 January 1915. His daughter Madeleine died in 1916, and his son Henry died in 1924 of an illness contracted in the war. In 1917, in the middle of these losses, Picard became permanent secretary of the Académie des Sciences for the mathematical sciences, the most powerful post in French science, which he held until his death.

After the armistice Picard led the reorganization of international science. At the conference held in Paris in November 1918, the Allied academies created the International Research Council with Picard as president and Volterra among its vice-presidents. The Council and its unions excluded scientists from the former Central Powers, a policy Picard supported firmly. The International Congress of Mathematicians of 1920, which he presided over, met in Strasbourg, newly returned to France, and German mathematicians were not invited. The exclusion was eased only in 1926, when Germany was invited to join the Council; it declined.

Loss, honours, legacy, 1924–1941

In 1922 Picard stood for the Académie française and was defeated. On 27 November 1924 he was elected to seat 1, succeeding the statesman Charles de Freycinet, and he was received on 11 February 1926. He wrote many academic eulogies and historical essays, gathered in Discours et mélanges (1922) and Éloges et discours académiques (1931). He received the Grand Cross of the Légion d’honneur in 1932, the Mittag-Leffler Gold Medal of the Swedish Academy in 1937, honorary degrees from five universities and memberships of dozens of learned societies. He was a founding member, in 1930, of the society that created the Ampère Museum near Lyon.

In the Second World War his grandsons were wounded and taken prisoner. Picard died in Paris on 11 December 1941, aged 85, during the German occupation. The Académie des Sciences awards the Émile Picard Medal for mathematics, and his collected works were published in four volumes in 1978–1981.

Legacy

• Complex analysis: the Picard theorems began value distribution theory, Montel’s normal families and Nevanlinna theory, and they remain the model for hyperbolicity in complex geometry.
• Differential equations: Picard iteration is the standard existence proof; Picard–Vessiot theory is the foundation of differential Galois theory and of algorithms in computer algebra.
• Algebraic geometry: the Picard group, Picard variety, Picard number and Picard–Lefschetz theory are core concepts; Alexander Grothendieck’s Picard scheme generalizes them.
• Arithmetic groups: the Picard modular group and Picard modular surfaces.
• Students: Hadamard, Painlevé, Vessiot, Bernstein, Julia, Stoilow, Petrović, Stanisław Zaremba and many others.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Émile Picard, MacTutor History of Mathematics.
2. Émile Picard, Wikipedia; French edition.
3. Académie française, Émile Picard, list of members.
4. E. T. Whittaker, Charles Émile Picard 1856–1941, Obituary Notices of Fellows of the Royal Society 4 (1942).
5. D. E. Rowe, H. Goenner and others, The impact of World War I on relativity: the aftermath, arXiv:1807.02134 (on the International Research Council).
6. O. Lehto, Mathematics Without Borders: A History of the International Mathematical Union, Springer, 1998.

Mathematics

1. É. Picard, Sur une propriété des fonctions entières, Comptes rendus 88 (1879).
2. É. Picard, Mémoire sur les fonctions entières, Annales scientifiques de l’École Normale Supérieure 9 (1880).
3. É. Picard, Mémoire sur la théorie des équations aux dérivées partielles et la méthode des approximations successives, Journal de Mathématiques Pures et Appliquées (1890).
4. É. Picard, Traité d’analyse, 3 vols., Gauthier-Villars, 1891–1896.
5. É. Picard and G. Simart, Théorie des fonctions algébriques de deux variables indépendantes, 2 vols., Gauthier-Villars, 1897 and 1906.
6. E. Vessiot, Sur l’intégration des équations différentielles linéaires, Annales scientifiques de l’ENS 9 (1892).
7. M. van der Put and M. F. Singer, Galois Theory of Linear Differential Equations, Springer, 2003.
8. J. L. Schiff, Normal Families, Springer, 1993 (proofs of the Picard theorems).
9. J. Elstrodt, F. Grunewald and J. Mennicke, Groups Acting on Hyperbolic Space, Springer, 1998.
10. Picard theorem and Picard–Lindelöf theorem, Wikipedia.