About the Abel Prize
On 20 March 2018, the Norwegian Academy of Science and Letters announced the award of the 2018 Abel Prize to mathematician Robert P. Langlands, of the Institute for Advanced Study at Princeton, “for his visionary programme connecting the theory of representations with number theory”.
The Abel Prize, which was established in 2002, can be considered the Nobel Prize of Mathematics (see Abel, Nobel, Fields on this blog). Each year, the prize is awarded to one or more mathematicians in order to recognise a pioneering scientific achievement in the field of mathematics. What is this visionary programme developed by Langlands that has earned him such a distinguished award?
In very general terms, Langlands’ programme establishes a connection between two very different parts of mathematics: number theory and the theory of automorphic representations. Traditionally, mathematics is divided into different areas, such as geometry, analysis, or algebra. Over the centuries, as the body of mathematical knowledge has grown, so has the specialisation, so that in each area different fields are delimited and developed independently, each with its own language and techniques. It is not uncommon for years of study to be required to learn the tools and reach the frontier of knowledge in a particular field. Because of this, when a connection is made between two of these fields, it is often a very exciting event for the mathematical community. The immediate consequence is a very fruitful development in both areas, as the knowledge and tools developed in one field can be applied to study the other. Many of the most spectacular breakthroughs occur when one of these connections takes place. Perhaps the best known example of this phenomenon is the introduction of analytic geometry, by René Descartes and Pierre de Fermat, where algebraic equations are used to study geometric objects, such as curves.
Langlands’ programme can be formulated in a very precise way, as a dictionary between two categories of objects: Galois representations (in the field of classical number theory) and automorphic representations of reductive groups (in the field of harmonic analysis in reductive groups). This dictionary makes it possible to translate classical number theory questions in terms of automorphic representations; for example, this connection has allowed the proof, after more than 300 years, of Fermat’s famous last theorem, by Andrew Wiles (who received the Abel prize in 2016 for this proof). The concrete statement of Langlands’ conjectures requires a very specific language. Here we try to give some rough ideas about them.
Fermat’s first observations
Let us begin illustrating the relationship between number theory and representation theory with a classical result: characterise those prime numbers \(p\) that can be written as a sum of two squares, that is, in the form \(p=a^2 + b^2\), where \(a\) and \(b\) are two natural numbers. The following result is due to Fermat (although the first known proof is due to Euler):
An odd prime \(p\) is written as a sum of two squares if and only if \(p\) is congruent to \(1\) modulo \(4\).
That is, we can write an odd prime \(p\) as a sum of two squares if when we divide \(p\) by \(4\), we obtain as remainder the number \(1\). For example, the primes \(5\), \(13\), \(17\), \(29\) can be written as a sum of two squares: \(5= 1^2 + 2^2\), \(13= 2^2 + 3^2\), \(17= 1^2 + 4^2\), \(29= 2^2 + 5^2\); however, the primes \(3\), \(7\), \(11\), \(19\), \(23\), cannot be written as a sum of two squares.
We can relate this problem to the problem of finding solutions modulo \(p\) of equations with integer coefficients: if there exist integers \(a\) and \(b\) such that \(a^2 + b^2=p\), then the equation \(x^2 + y^2\equiv 0\) modulo \(p\) has a solution (which, moreover, is not the trivial solution \(x\equiv y\equiv 0 \pmod{p}\)). Conversely, it can be shown, using Fermat’s infinite descent technique, that if there exists a solution modulo \(p\) of the equation \(x^2 + y^2\equiv 0\), which is not the trivial solution, then \(p\) can be written as a sum of two squares. Therefore, the original problem can be reformulated as follows: characterize the prime numbers \(p\) such that the equation \(x^2 + y^2\equiv 0\pmod p\) has a non-trivial solution. Using that every integer that is not a multiple of \(p\) has an inverse modulo \(p\), we can conclude that the above equation has a non-trivial solution if and only if the equation \(x^2 + 1\equiv 0 \pmod p\) has a solution.
Generalisations
This reformulation of the problem allows us to generalise it. For example, we can ask ourselves which primes verify that the equation \(x^2 -2\equiv 0 \pmod{p}\) has a solution. In this case, we find that the equation has a solution if and only if \(2\) is a square modulo \(p\), or, equivalently, if and only if \(p=2\) or \(p\equiv 1, 7 \pmod{8}\). In general, if we consider the problem of characterizing the set of primes \(p\) such that an equation of the form \(ax^2 + bx + c\equiv 0 \pmod{p}\) has a solution, we will find a condition given in terms of congruence relations that must satisfy the prime \(p\), modulo a certain number \(N\) which depends on the coefficients \(a\), \(b\), \(c\) of the original equation. To prove this result, one can use Gauss’s law of quadratic reciprocity: When \(p\) and \(q\) are two odd primes and either of them is congruent to \(1\) modulo \(4\), then \(p\) is a square modulo \(q\) if and only if \(q\) is a square modulo \(p\). When \(p\) and \(q\) are two odd primes congruent to \(3\) modulo \(4\), then \(p\) is a square modulo \(q\) if and only if \(q\) is not a square modulo \(p\).
A more complicated problem and the emergence of modular forms
Let’s consider a polynomial of higher degree, say \(P(x)= x^4 + x -1\), and we pose the problem of characterizing those primes \(p\) such that \(P(x)\equiv 0 \pmod{p}\) has 4 different solutions (in general, we can ask for which primes \(p\) a prefixed factorization of \(P(x)\) modulo \(p\) occurs; for example, \(P(x)=(x-a)(x-b)(x-c)(x-d)\) corresponds to the case where there are four solutions). The first primes satisfying this condition are \(83\), \(643\), \(773\), \(859\), \(1193\), \(1301\), \(1307\), \(\dots\) It can be shown that there is no set of congruence relations characterising the primes \(p\) for which the above condition is satisfied. However, there is an analytic object that gives us the solution to the problem. In this case, it is a modular form. Modular forms are functions of a complex variable \(f:\mathcal{H}\rightarrow \mathbb{C}\), defined on the upper half-plane \(\mathcal{H}\), which satisfy a symmetry with respect to a congruence subgroup (defined by a pair of integers, weight and level) and regularity conditions at infinity. The study of modular forms begins in the 19th century, in connection with doubly periodic functions, also called elliptic because of their relation to the calculation of elliptic integrals.
In this case, the function \(f\) characterising the set of primes \(p\) such that \(P(x)\equiv 0 \pmod{p}\) has 4 different solutions is a modular form of level \(283\) and weight \(1\). Denoting by \(q=e^{2 \pi i z}\), we can write the first terms of the Fourier development of \(f\):
$$f(z)=q – q^2 – q^3 + q^6 + q^8 – q^{13} – q^{16} + q^{23} – q^{24}
+ q^{25} + q^{26} + q^{27} – q^{29} – q^{31} + q^{39} – q^{41} + \cdots$$
This function verifies that those primes \(p\) such that the coefficient of \(q^p\) is exactly \(-2\) are those primes such that \(P(x)\equiv 0 \pmod{p}\) has 4 different solutions. Why would a complex variable function have to determine the factorisation of the polynomial \(P(x)\) modulo the different prime numbers? This curious relationship is part of the network connecting Galois representations and automorphic representations, conjectured by Langlands and embodied in a letter to the French mathematician André Weil in 1967. This letter, written at Weil’s own request, is preceded by the following lines: “In response to your invitation to come and talk, I wrote the enclosed letter. After writing it, I have realized there was hardly a statement in it of which I was certain. If you are willing to read it as pure speculation, I would appreciate that; if not – I’m sure you have a wastebasket handy”. Fortunately, the letter did not end up in the wastepaper basket, but was typed and circulated in the mathematical community.
In this letter, Langlands defines a key object, the Artin-Hecke \(L\)-series, which generalises both Artin’s \(L\)-series (associated with Galois group representations of a number field) and Hecke’s \(L\)-series (associated with modular forms). The \(L\)-series are complex variable functions, defined by means of an infinite sum
$$\sum_{n=1}^{\infty} \frac{a_n}{n^s}$$ for a family of complex coefficients \(\{a_n\}_{n\in \mathbb{N}}\). Let’s see some examples of \(L\)-series.
Riemann zeta function
The first example of \(L\)-series is the Riemann zeta function,
$$\zeta(s)=\sum_{n=1}^{\infty} \frac{1}{n^s}.$$
For every complex number \(s\) whose real part is greater than 1, this expression converges absolutely and defines a complex number. Riemann proved in 1859 that the function thus defined extends to a meromorphic function in the whole complex plane, with a single simple pole at \(s=1\). Thanks to the fundamental theorem of arithmetic, which states that every positive integer can be uniquely written as a product of prime numbers (up to the order of the factors), an expression can be given for the Riemann zeta function as an infinite product. This development is known as Euler’s product: $$\zeta(s)=\prod_{p\text{ prime}} \frac{1}{1-p^{-s}}.$$
Moreover, \(\zeta(s)\) satisfies a functional equation, which is a relation between \(\zeta(s)\) and \(\zeta(1-s)\).
The relationship between the Riemann zeta function and the prime numbers allows us to prove properties about the distribution of the latter: for example, Hadamard and de la Valleé Poussin proved that the function \(\pi(x)\), defined as the number of primes less than or equal to \(x\), grows as \(x/log(x)\).
Dirichlet series and representations of the groups \((\mathbb{Z}/n\mathbb{Z})^{\times}\)
The Riemann zeta function can be modified, giving rise to other \(L\)-series. For example, consider a Dirichlet character \(\chi:\mathbb{Z}/N\mathbb{Z}\rightarrow \mathbb{C}\), i.e., a morphism of multiplicative groups \((\mathbb{Z}/N\mathbb{Z})^{\times}\rightarrow \mathbb{C}^{\times}\) which extends to the set \(\mathbb{Z}/N\mathbb{Z}\) as \(\chi(a)=0\) if \(a\) is not prime with \(N\). We define the Dirichlet \(L\)-series as
$$L(\chi, s)=\sum_{n=1}^{\infty} \frac{\chi(n)}{n^s}.$$
This sum, as in the previous case, converges absolutely if the real part of \(s\) is greater than \(1\), and can be extended to a meromorphic function on the whole complex plane. Moreover, it also admits an expression like an infinite product
$$L(\chi, s)=\prod_{p \text{ prime}} \frac{1}{1-\chi(p)p^{-s}}$$
and a functional equation. Dirichlet (1837) used these \(L\)-series to show that, in every arithmetic progression \(\{an+ b: n\in \mathbb{N}\}\) there are infinitely many prime numbers, provided that \(a\) and \(b\) are prime to each other.
Other classes of \(L\)-series, associated with various mathematical objects, were defined more or less simultaneously.
Hecke’s \(L\)-series
In 1918, Hecke (1887–1947) defined the \(L\)-series attached to a type of characters, the so-called Grössencharaktere, and also the \(L\)-series attached to weight one modular forms. Each modular form \(f\) of level \(N\) and weight \(k\) has a Fourier expansion \(\sum_{n=0}^{\infty} a_n e^{2\pi i nz}\). We can use the family of coefficients \(\{a_n\}_n\) to define a series:
$$L(f, s)=\sum_{n=1}^{\infty} \frac{a_n}{n^s},$$ which converges in a complex half-plane. This function extends to a meromorphic function on the whole complex plane (holomorphic if \(f\) is cuspidal), and satisfies a functional equation. When \(f\) also satisfies the condition of being an autoform (or eigenform) for a family of operators, defined by Hecke, this function can also be expressed as an infinite Euler product
$$L(f, s)=\prod_{p\text{ prime}} \frac{1}{1-a_p p^{-s} + \varepsilon(p)p^{k-1-2s}}$$ for a certain Dirichlet character \(\varepsilon:\mathbb{Z}/N\mathbb{Z}\rightarrow \mathbb{C}\) associated to \(f\).
Artin’s \(L\)-series and the theory of class fields
Independently, Artin introduced in 1923 another kind of \(L\)-series, which are related to the question we discussed at the beginning about the factorisation of polynomials modulo different prime numbers, and Galois theory. If \(P(x)\) is a polynomial with rational coefficients of degree \(n\), the fundamental theorem of algebra tells us that it has exactly \(n\) complex roots (counting multiplicities). Galois (1811-1832), in his studies on the possibility of giving a formula for the roots of \(P(x)\) in terms of the coefficients of \(P(x)\) by elementary operations (addition, multiplication and division), and extraction of \(k\)-th roots, introduced the Galois group of the polynomial, which we can understand intuitively as the group of symmetries of the roots of the polynomial. If we call \(F\) the smallest subfield of \(\mathbb{C}\) containing the roots, we will say that the extension of fields \(F/\mathbb{Q}\) is Galois, and its Galois group is denoted as \(\mathrm{Gal}(F/\mathbb{Q})\). One of the great advances in late 19th and early 20th century mathematics is the theory of class fields. By replacing the field \(\mathbb{Q}\) by another field of numbers \(K\), the Galois group \(\mathrm{Gal}(F/K)\) can be defined in the same way. Class field theory classifies all Galois extensions \(F/K\) of a number field \(K\), whose Galois group \(\mathrm{Gal}(F/K)\) is abelian, in terms of arithmetic properties of the field \(K\) (namely, the group of ideals of \(K\)).
The Kronecker-Weber Theorem, one of the fundamental results of the theory, states that for every abelian extension \(F/\mathbb{Q}\) there exists an integer \(N\) such that \(F\) is contained in \(\mathbb{Q}(\zeta_N)\), the extension of \(\mathbb{Q}\) obtained by adding the \(N\)-th roots of unity. Now, these extensions of fields are particularly simple. For example, it is easy to study the decomposition modulo \(p\) of the polynomials defining them (which are the cyclotomic polynomials) in terms of the class of \(p\) modulo \(N\). It can be shown that the cyclotomic polynomial corresponding to the extension \(\mathbb{Q}(\zeta_N)\) decomposes into distinct linear factors if and only if \(p\equiv 1 \pmod{N}\). This is why, in the case of polynomials of degree \(2\), one can characterise the set of primes \(p\) for which the polynomial has different roots modulo \(p\) in terms of congruences modulo a natural number \(N\); in general, this will be true when the Galois group of the extension of fields \(F/\mathbb{Q}\) is abelian.
This theorem can be reformulated in terms of the \(L\)-series associated with Dirichlet characters: Let \(F/\mathbb{Q}\) be an abelian Galois extension. For each morphism of groups \(\rho:\mathrm{Gal}(F/\mathbb{Q})\rightarrow \mathbb{C}^{\times}\) we can define an \(L\)-function as follows: To every prime (except for a finite number which depends on the extension \(F/\mathbb{Q})\), we can associate an element \(\mathrm{Fr}_p\in\mathrm{Gal}(F/\mathbb{Q})\), the Frobenius element (actually, a conjugacy class), and thus consider its image through \(\rho\), which gives us a complex number. We can use this information to define a \(L\)-function .
$$L(\rho, s)=\prod_{p\text{ prime}}\frac{1}{1- \rho(\mathrm{Fr}_p)p^{-s}}.$$
The Kronecker-Weber Theorem states that there exists a Dirichlet character \(\chi:\mathbb{Z}/N\mathbb{Z}\rightarrow \mathbb{C}\) such that
$$L(\rho, s)=L(\chi, s).$$
When we replace the character \(\rho\) by a representation of dimension \(n\) (i.e., a group morphism of \(\mathrm{Gal}(F/\mathbb{Q})\) in the group \(\mathrm{GL}_n(\mathbb{C})\) of invertible matrices of size \(n\times n\)), one can also define a \(L\)-series, using the characteristic polynomial of \(\rho(\mathrm{Fr}_p)\). A natural question is whether there is a generalisation of the Kronecker-Weber Theorem. Given a representation \(\rho\) of dimension \(n\), is there an object \(\chi\) such that \(L(\rho, s)=L(\chi, s)\)? This is one of the questions Langlands considers in his letter: in particular, Langlands conjectures that the appropriate object to replace \(\chi\) is an automorphic representation.
Hasse-Weil \(L\)-series and Fermat’s theorem
We cannot fail to mention a third source of \(L\)-series: the algebraic varieties, which give rise to the Hasse-Weil \(L\)-series. In particular, the \(L\)-series associated with elliptic curves play an interesting role in this story. An elliptic curve is defined by an equation of the form \(y^2=x^3 + ax + b\); if we denote by \(N_p(E)\) the number of points of the curve modulo \(p\) (including the point at infinity), we can define the \(L\)-series associated to \(E\) as
$$L(E, s)= \prod_p \frac{1}{1- a_p(E) p^{-s} + \chi(p) p^{1-2s}},$$ where \(a_p(E)= p+1-N_p(E)\) and \(\chi\) is a Dirichlet character associated to \(E\) (precisely: \(\chi(p)=1\) if \(p\) is prime with the conductor of the curve, and \(0\) otherwise). The conjecture of Shimura and Taniyama, appearing around 1955, states that if \(E\) is an elliptic curve defined over \(\mathbb{Q}\), then there exists a modular form \(f\) of weight \(2\) such that \(L(E, s)=L(f, s)\). This conjecture attracted the attention of Andrew Wiles when, thanks to the work of Gerhard Frey, Jean-Pierre Serre and Kenneth Ribet, it was shown that a positive answer for semi-stable elliptic curves would imply Fermat’s Last Theorem. Wiles, together with Richard Taylor, proved this case in 1995; the general case was proved shortly afterwards.
Langlands’ \(L\)-series
Let us now return to the letter written by Langlands to the mathematician André Weil. In it, Langlands defines the Artin-Hecke \(L\)-series \(L(\Pi, \rho, s)\), which generalises both Artin’s \(L\)-series and Hecke’s \(L\)-series. These \(L\)-series depend on two objects: an automorphic representation \(\Pi\) of a reductive group \(G\) and a representation of the Galois group \(\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\) in the dual group \({}^LG\).
Langlands asks two questions in his letter. The first is whether Artin-Hecke \(L\)-series can be extended to meromorphic functions in the whole complex plane, and whether they satisfy a functional equation. The second question concerns the functoriality principle, central to Langlands’ programme. Intuitively, this principle can be formulated as follows: There are a number of natural operations we can perform on Galois representations (scalar restriction, composition with a morphism of groups \({}^LG\rightarrow ^L\hskip-0.15cm G’\), tensor product of representations, symmetric power, etc), and we get another Galois representation, in a different group (in general). The functoriality principle states that there are operations on automorphic representations which correspond to these operations on Galois representations. In particular, there must be an automorphic representation that corresponds to the new Galois representation.
Thus, we obtain a correspondence between the Galois representations \(\rho:G_{K}\rightarrow ^{L}\hskip-0.1cm G\) and the automorphic representations of the group \(G\). In particular, this correspondence provides a generalisation of class field theory: instead of Dirichlet characters, the objects whose \(L\)-series correspond to the complex Galois representations of dimension \(n\) are the automorphic representations of dimension \(n\).
Development of Langlands’ programme
Since its formulation, numerous mathematicians have devoted themselves to partially proving some of Langlands’ conjectures; experts such as Drinfeld, Lafforgue and Ngô have received the Fields Medal for their contributions to Langlands’ programme. In 1980, Langlands himself proved an important case of functoriality, the solvable base change for the group \(\mathrm{GL}(2)\) of \(2\times 2\) matrices. Using this result, Tunnells shows that, given an odd, irreducible Galois representation \(\rho:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\rightarrow \mathrm{GL}_2(\mathbb{C})\), whose image is a solvable group, then there exists a modular form \(f\) such that \(L(f, s)=L(\rho, s)\). In turn, this result is crucial in Wiles’ proof of Shimura and Taniyama’s conjecture, since it is used as the base case, from which the property of being modular is propagated.
However, despite spectacular progress, we are still a long way from having a proof of the lattice of connections established in Langlands’ visionary programme.
Learn more
The letter from Langlands to Weil
The database of \(L\) functions, modular forms and related objects is an open project where data and calculations concerning different mathematical objects and their \(L\)-series are collected.
An excellent start to the Langlands programme: Gelbart, Stephen.
An elementary introduction to the Langlands program. Bull. Amer. Math. Soc. (N.S.) 10 (1984), no. 2, 177–219.
A survey on the connection between Galois representations and various mathematical objects, including modular forms, automorphic representations and Shimura varieties.: Weinstein, J., Reciprocity laws and Galois representations: recent breakthroughs. Bull. Amer. Math. Soc. (N.S.) 53 (2016), no. 1, 1-3.
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