A MOTIVATED INTRODUCTION TO THE LANGLANDS PROGRAM

M. Ram Murty*

Dedicated to the memory of Professor R. Sitaramachandra Rao

§1. Motivation.

This paper is partially expository and is intended as an introduction to the Langlands program for number theorists. The new theorem in the paper is that automorphic induction map for Hecke characters implies both the Artin conjecture and the Langlands reciprocity law. (See below for definitions.) In the last section, we describe some recent work with K. Murty [18] that applies the theory of base change to elliptic curves. The paper is not an exhaustive survey. We have tried to use some classical problems of number theory as motivation for discussion. For instance, we concentrate on GL_{n} though the functoriality conjecture predicts that this is not a limitation. Nevertheless, our discussion is sufficiently motivated from the number theoretic point of view that a non-specialist in the field can appreciate the depth and profundity of these ideas.

We begin by considering two open problems confronting number theory: Fermat’s last theorem and the Sato-Tate conjecture.

First, we must understand the notion of a modular form. Let SL_{2}(ℤ) denote the full modular group. That is,

SL_{2}(ℤ) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) : a, b, c, d ∈ ℤ; ad − bc = 1\} .

If 𝔥 denotes the upper half-plane, a holomorphic function f : 𝔥→ℂ is called a modular form for SL_{2}(ℤ) of weight k if

f(\frac{az + b}{cz + d}) = (cz + d)^{k}f(z) ∀ ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(ℤ)

and f is “holomorphic at infinity”. Since f(z + 1) = f(z), such a function f has a Fourier expansion and the condition “holomorphic at infinity” can be stated by saying that f has a Fourier expansion of the form

f(z) = ∑_{n=0}^{∞} a_{n}e^{2πinz}.

* Research partially supported by an NSERC grant.

More generally, we may consider for each natural number N, the subgroup Γ_{0}(N) defined as the subgroup of matrices

γ = ( \begin{matrix}a & b \\ c & d \end{matrix} )

of SL_{2}(ℤ) satisfying c ≡ 0( mod N). A modular form of weight k for Γ_{0}(N) is defined analogously. (All rational numbers are SL_{2}(ℤ)-equivalent to i∞. The Γ_{0}(N) equivalence classes of rational numbers are called cusps for Γ_{0}(N) and sometimes we refer to a representative of the class as a cusp. We require that the modular form for Γ_{0}(N) be holomorphic at each of these cusps. )

Now let A, B, C ∈ ℤ be coprime integers such that A + B + C = 0 with 32|B and 4|(A + 1). Let E be the curve

E : y^{2} = x(x − A)(x + B).

For each prime p not dividing ABC let N_{p} be the number of solutions of E mod p. Define

a_{p} = p − N_{p}.

Then a classical theorem of Hasse (conjectured by E. Artin in his doctoral thesis) states that |a_{p}| ≤ 2\sqrt{p}. Set

L_{E}(s) = F (s) ∏_{p|/ABC} (1 − \frac{a_{p}}{p^{s}} + \frac{1}{p^{2s−1}} )^{−1}

where

F (s) = ∏_{p|A} (1 − (\frac{B}{p} ) \frac{1}{p^{s}} )^{−1} ∏_{p|B} (1 − (\frac{A}{p}) \frac{1}{p^{s}} )^{−1} ∏_{p|C} (1 − (\frac{−B}{p} ) \frac{1}{p^{s}} )^{−1} .

By virtue of Hasse’s inequality, this infinite product converges for Re (s) > 3/2 and so in this half-plane, we can write L_{E}(s) as a Dirichlet series

L_{E}(s) = ∑_{n=1}^{∞} \frac{a_{n}}{n^{s}} .

This defines the a_{n} (which coincides with a_{p} when n is prime, so the notation is consistent). Define

N = ∏_{p|ABC} p, f_{E}(z) = ∑_{n=1}^{∞} a_{n}e^{2πinz}.

In 1955, Taniyama [32] made the astounding conjecture that f_{E}(z) is a modular form of weight 2 for Γ_{0}(N). In 1985, Frey [7] noticed that there may be a link between Taniyama’s conjecture and Fermat’s last theorem. This led Serre [28] to formulate more precise conjectures concerning the ramification of modular Galois representations which culminated in K. Ribet [22] proving in 1989 the following remarkable theorem:

THEOREM 1. (Ribet, 1989) Taniyama’s conjecture implies Fermat’s last theorem.

Taniyama’s conjecture reduces to the problem of determining when a given sequence of numbers {a_{n}}_{n=1}^{∞} is the sequence of Fourier coefficients of a modular form. In 1967, A. Weil [34] answered this question in the following way. Let

L(s) = ∑_{n=1}^{∞} \frac{a_{n}}{n^{s}}

and for each Dirichlet character χ mod c define

L(s, χ) = ∑_{n=1}^{∞} \frac{a_{n}χ(n)}{n^{s}}.

Suppose that the a_{n}’s are of polynomial growth and for each primitive character χ mod c and (c, N) = 1, L(s, χ) extends to an entire function and satisfies the functional equation

(c\sqrt{N}/2π)^{s}Γ(s)L(s, χ) = w_{χ}(c\sqrt{N}/2π)^{k−s}χ(−N)L(k − s, χ̄)

where w_{χ} is a complex number of absolute value 1. Then

f(z) = ∑_{n=1}^{∞} a_{n}e^{2πinz}

is a modular form of weight k for Γ_{0}(N).

In view of Weil’s theorem, Theorem 1 reduces Fermat’s last theorem to an assertion about analytic continuation of certain Dirichlet series. This is not the first time that such an event has taken place in number theory. In retrospect, we see that the introduction of the zeta function to solve problems of the distribution of prime numbers or the use of Dirichlet L-functions to describe the behaviour of primes in arithmetic progressions foreshadowed this event.

As we shall see, the Taniyama conjecture is a special case of the Langlands program which seeks to unify representation theory, number theory and arithmetic algebraic geometry. The binding link between all these diverse disciplines is the notion of an L-function of an automorphic representation and the relation between its analytic properties and the underlying algebraic structures. The L-functions of automorphic forms and automorphic representations generalise the classical zeta and L-functions of Riemann, Dirichlet and Hecke.

To further motivate our understanding, I would like to describe the Sato-Tate conjecture. Let E be an elliptic curve defined over ℚ. By Hasse’s inequality, we know |a_{p}| ≤ 2\sqrt{p}. Let us write

a_{p} = \sqrt{p}(e^{iθ_{p}} + e^{−iθ_{p}}) = 2\sqrt{p} cos θ_{p}.

Sato and Tate independently asked the question how does θ_{p} vary as p varies and were led to conjecture that if the elliptic curve is not of CM type, then the θ_{p}’s are uniformly distributed with respect to the measure

\frac{2}{π} sin^{2} θdθ.

In his McGill lectures given in 1967, Serre [27] reformulated this conjecture as follows. Let α_{p} = e^{iθ_{p}} and β_{p} = e^{−iθ_{p}}. For each m, define the L-series

L_{m}(s) = ∏_{p} ∏_{j=0}^{m} (1 − \frac{α_{p}^{m−j}β_{p}^{j}}{p^{s}} )^{−1} .

Each L_{m}(s) converges for Re (s) > 1. Suppose that each L_{m}(s) extends to an entire function for all s ∈ ℂ and L_{m}(1 + it) ≠ 0 for all real values of t. Then, Serre [27] showed that the θ_{p}’s are uniformly distributed with respect to the (Sato-Tate) measure 2 sin^{2} θ/π. In 1979, Kumar Murty [17] showed that analytic continuation of each L_{m}(s) to Re (s) = 1 alone suffices to imply the Sato-Tate conjecture.

Since the a_{p}’s behave like Fourier coefficients of cusp forms of weight 2, it is reasonable to expect the same type of behaviour from Fourier coefficients of cusp forms which are eigenfunctions of Hecke operators. To illustrate, consider the Ramanujan τ function defined by the power series

q ∏_{n=1}^{∞}(1 − q^{n})^{24} = ∑_{n=1}^{∞} τ(n)q^{n}.

If we set q = e^{2πiz}, then the series defines a cusp form of weight 12 for the full modular group. In 1916, Ramanujan [20] conjectured that τ satisfies

(1) τ(nm) = τ(n)τ(m) if (n, m) = 1 and

(2) |τ(p)| ≤ 2p^{11/2} whenever p is prime.

(1) was proved by Mordell in 1928, but he overlooked the depth of the ideas that went into his proof. Hecke [12] saw in it the theory of certain operators acting on the space of cusp forms. But (2) defied many attempts until Deligne [4] in 1974 proved it as a consequence of his proof of the Weil conjectures. We can therefore write, in analogy with elliptic curves,

τ(p) = p^{11/2}(e^{iθ_{p}} + e^{−iθ_{p}}) = p^{11/2}(α_{p} + β_{p}).

With this notation, the series L_{m}(s) can be analogously defined. Serre [27] conjectured that the θ_{p}’s are uniformly distributed with respect to the Sato-Tate measure. In terms of L-functions and Kumar Murty’s theorem, this means we must continue each L_{m}(s) to the line Re (s) = 1. If m = 1, L_{1}(s) is the Mellin transform of a cusp form and so has analytic continuation by the work of Ramanujan and more generally by the work of Hecke. If m = 2, Rankin [21] and Selberg [26] (independently) showed in the 1940’s that ζ(s)L_{2}(s) has an analytic continuation and satisfies a functional equation. Since the ζ-function does not vanish on the line Re (s) = 1, it follows that L_{2}(s) extends to an entire function to Re (s) = 1. That in fact L_{2}(s) extends to an entire function for all values of s was proved by Shimura [30] in 1975 by a slight modification of the Rankin-Selberg method. In 1982, Shahidi [29] showed that L_{3}(s) and L_{4}(s) have analytic continuation up to Re (s) = 1. In 1985, Garrett [8] has also obtained results which imply these by another method. Their results imply

Theorem. If L_{1}(s) has no real zeroes in (1/2, 1), then L_{3}(s) is entire.

In 1952, Gelfand and Fomin [9] showed how a modular form gives rise to a representation of SL_{2}(ℝ). Langlands’ idea is to look at representations of GL_{2} of the adele ring of the rational numbers. More generally, he attaches L-functions to representations of adele groups. These L-functions play a central role in the Langlands program.

2. Artin L-series and Hecke’s L-series.

One can view the construction of L-series in a purely formal way. This is described by Serre [27] in his McGill lectures alluded to earlier. Let G be a compact group and μ its normalised Haar measure. Let X be the space of conjugacy classes. Suppose S is a finite set of primes and for each prime p ∉ S, let X_{p} be a conjugacy class. We can ask the question of how the X_{p}’s are distributed as p varies.

Let ρ be an irreducible representation of G and define

L(s, ρ) = ∏_{p∉S} det (1 − ρ(X_{p})p^{−s})^{−1}.

This defines an analytic function for Re (s) > 1. Suppose that each L(s, ρ) extends to an analytic function on Re (s) = 1 and does not vanish there. Then, the X_{p}’s are uniformly distributed with respect to the image of the Haar measure in X. This theorem is proved using the standard Tauberian theory.

Examples.

1. Let τ(p) = 2p^{11/2} cos θ_{p} or a_{p} = 2p^{1/2} cos θ_{p}. For each prime p, we can associate

( \begin{matrix}e^{iθ_{p}} \\ & e^{−iθ_{p}} \end{matrix} ) ∈ SU(2, ℂ).

Let ρ be the standard representation of SU(2, ℂ). Then, all irreducible representations of SU(2, ℂ) are Sym^{m}(ρ) and L(s, Sym^{m}(ρ)) is L_{m}(s) defined in the previous lecture. The image of the Haar measure in the space of conjugacy classes of SU(2, ℂ) is 2 sin^{2} θ/π.

2. Artin L-series. Let F be an algebraic number field and K/F be a Galois extension. For each prime ideal 𝔭 of F, we have the factorization

𝔭O_{K} = P_{1}^{e} ⋯ P_{r}^{e},

where P_{1}, ..., P_{r} are prime ideals of K. There is a finite set of primes S (namely the ramified primes of K) such that for p ∉ S, the decomposition group

D_{P} = {σ ∈ G : P^{σ} = P}

is cyclic. One can show that this group is canonically isomorphic to the Galois group of the extension of finite fields O_{K}/P over O_{F} /𝔭 which is generated by x ↦ x^{N_{F/ℚ}(𝔭)}. We denote by σ_{P} ∈ D_{P} which corresponds to this element. That is,

σ_{P}(x) ≡ x^{N_{F/ℚ}(𝔭)}( mod P)

for all x ∈ O_{K}. If P|𝔭, then the σ_{P} are conjugate and we denote by σ_{𝔭} the conjugacy class to which the σ_{P} belong. σ_{𝔭} is called the Artin symbol of 𝔭. It is a generalisation of the familiar Legendre symbol which distinguishes quadratic residues from nonresidues. For each irreducible complex representation of G, L(s, ρ) is called the Artin L-series attached to ρ.

Artin’s conjecture. If ρ is irreducible and ≠ 1, then L(s, ρ) extends to an entire function of s.

This is one of the major unsolved problems in number theory. If ρ has degree 1, then Artin showed that his conjecture is true by showing that L(s, ρ) = L(s, ψ) where ψ is a Hecke character. Hecke had already shown that such L-series L(s, ψ) arise as Mellin transforms of generalised theta functions and so established their analytic continuation and functional equation. This equality is a deep statement which is called the Artin reciprocity law. In the simplest case when F = ℚ and K is a quadratic extension, the identity is equivalent to the law of quadratic reciprocity. In the general case, Brauer showed that each L(s, ρ) has a meromorphic continuation for all s ∈ ℂ. This was a consequence of his induction theorem which says that every character of a finite group G can be written as an integral linear combination of monomial characters. Thus, every non-abelian L-series can be written as a quotient of two entire functions each of which is a product of abelian L-series. Since these L-series do not vanish on the line Re (s) = 1, we deduce immediately that the Artin symbols are uniformly distributed in G. This is the famous Tchebotarev density theorem, which plays a central role in many problems of number theory. For instance, if F = ℚ and K = ℚ(ζ_{k}), then Gal (K/ℚ) ≃ (ℤ/kℤ)^{∗} and the Artin symbol of p is p mod k. Thus, we recover Dirichlet’s theorem on primes in arithmetic progressions.

Brauer’s theorem is essentially group theoretic in nature. By a similar argument, one can show that Artin’s conjecture is true for all supersolvable groups where every irreducible character is monomial. But these are group theoretic arguments which have limitations. The natural question is what can we say about representations of degree 2. As we shall see, the Langlands program will make a precise conjecture as to what L(s, ρ) should be in general. In 1975, Langlands made significant progress by proving Artin’s conjecture in the “tetrahedral case” by using ideas of representation theory. More precisely, if

ρ : G→GL_{2}(ℂ)→^{nat}P GL_{2}(ℂ)

is a 2 dimensional representation, then the image of G in P GL_{2}(ℂ) is one of five possibilities: cyclic, dihedral, A_{4}, S_{4} or S_{5}. It was the case of A_{4} (tetrahedral) that Langlands [14] dealt with after developing the theory of base change for GL_{2}. Tunnell [33] in 1982 showed the same ideas work for S_{4} (the octahedral case). In his doctoral thesis, Buhler [2] showed that Artin’s conjecture is true in the icosahedral case for certain cases. The general case is still open.

Artin’s conjecture suggests the following question: given a sequence {a_{n}}_{n=1}^{∞}, define

L(s) = ∑_{n=1}^{∞} \frac{a_{n}}{n^{s}} .

When does L(s) have analytic continuation and functional equation. Hecke obtained such L-series as Mellin transforms of theta functions of several variables. These are L-series attached to grossencharactere. He also considered Mellin transforms of modular forms and showed that these series also enjoyed analytic continuation and functional equations. Let me illustrate with a concrete example. Let

Δ(z) = ∑_{n=1}^{∞} τ(n)e^{2πinz}.

Let SL_{2}(ℤ) be the group of matrices

( \begin{matrix}a & b \\ c & d \end{matrix} ) , a, b, c, d ∈ ℤ, ad − bc = 1.

Then,

Δ(\frac{az + b}{cz + d}) = (cz + d)^{12}Δ(z)

and in particular Δ(−1/z) = z^{12}Δ(z). Consider the Mellin transform

∫_{0}^{∞} Δ(iy)y^{s} \frac{dy}{y} = ∫_{0}^{∞} ∑_{n=1}^{∞} τ(n)e^{−2πny}y^{s} \frac{dy}{y}

= (2π)^{−s}Γ(s) ∑_{n=1}^{∞} \frac{τ(n)}{n^{s}}

= ∫_{0}^{1} Δ(iy)y^{s} \frac{dy}{y} + ∫_{1}^{∞} Δ(iy)y^{s} \frac{dy}{y}

= ∫_{1}^{∞} Δ(i/y)y^{−s} \frac{dy}{y} + ∫_{1}^{∞} Δ(iy)y^{s} \frac{dy}{y}

from which we get analytic continuation and functional equation upon using the modular relation.

To describe Hecke theory in some more detail, let 𝔥 denote the upper half plane. Let GL_{2}^{+}(ℝ) denote the non-singular 2 × 2 matrices with positive determinant. GL_{2}^{+}(ℝ) acts on 𝔥. Consider Γ, a congruence subgroup of SL_{2}(ℤ). That is, Γ ⊇ Γ(N) where

Γ(N) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ≡ I( mod N)\}

for some natural number N. For example, Hecke’s group

Γ_{0}(N) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) c ≡ 0( mod N)\}

a congruence subgroup. Define j(g, z) = (cz + d)( det g)^{−1/2} for all g ∈ GL_{2}^{+}(ℝ) and

g = ( \begin{matrix}∗ & ∗ \\ c & d \end{matrix} ) .

Set

(f|_{k}σ)(z) = j(σ, z)^{−k}f(σz)

for all positive integers k. A fundamental domain 𝔉 for Γ is any connected set in 𝔥 such that no two interior points of 𝔉 are Γ equivalent and every point in 𝔥 is equivalent to some point in 𝔉. For example, for Γ = SL_{2}(ℤ), it is easy to see that a fundamental domain is 𝔉 = {z = x + iy : −1/2 ≤ x ≤ 1/2, |z| ≥ 1} because SL_{2}(ℤ) is generated by

( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) and ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) .

For an arbitrary Γ, a fundamental domain is easily derived from this one as follows. Let γ_{1}, γ_{2}... be the left coset representatives of Γ in SL_{2}(ℤ). Then a fundamental domain for Γ is ∪_{i} γ_{i}^{−1}𝔉.

A cusp for Γ is an element of ℚ ∪ {i∞} which is fixed by an element of trace ±2. In the case of SL_{2}(ℤ), i∞ is the only cusp up to equivalence. Let 𝔥^{∗} = 𝔥 ∪ cusps of Γ. Then, Γ\𝔥^{∗} has a complex analytic structure of a Riemann surface. We also identify this with a fundamental domain of Γ.

An automorphic form for Γ of weight k is a complex valued function f on 𝔥 such that (1) f|_{k}γ = f for all γ ∈ Γ,

(2) f is holomorphic on 𝔥^{∗}.

A cusp form for Γ is an automorphic form which vanishes at all the cusps of Γ. Let M_{k}(Γ) denote the space of modular forms for Γ and S_{k}(Γ) the space of cusp forms. For SL_{2}(ℤ), dim M_{k}(Γ) = dim S_{k}(Γ) + 1 where the term +1 arises from the subspace spanned by the Eisenstein series:

E_{k}(z) = ∑_{(m,n)≠(0,0)}(mz + n)^{−k}.

In the general case for k even and k > 2, dim M_{k}(Γ) - dim S_{k}(Γ) is equal to the number of inequivalent cusps of Γ so that there is an Eisenstein series attached to each cusp. The space of cusp forms is an inner product space where the inner product is defined by

(f, g) = ∫ ∫_{Γ\𝔥^{∗}} f(z)\overline{g(z)}y^{k} \frac{dxdy}{y^{2}} .

Hecke’s First theorem. Let f ∈ S_{k}(Γ_{0}(N)) and write

f(z) = ∑_{n=1}^{∞} a_{n}e^{2πinz}

for its Fourier expansion at i∞. Then,

L(s, f) = ∑_{n=1}^{∞} \frac{a_{n}}{n^{s}}

has an analytic continuation and functional equation relating L(s, f) and L(s, f|w_{N}) where

w_{N} = ( \begin{matrix}0 & 1 \\ −N & 0 \end{matrix} ).

The Artin conjecture for 2-dimensional Galois representations can now be reformulated and made more precise. Every 2-dimensional Artin L-series is the Mellin transform of a cusp form of weight 1 on Γ_{0}(N) for some natural number N. This can be viewed as the “2-dimensional reciprocity law”.

Hecke was led to ask which L(s, f) have Euler product expansions because only these will correspond to Artin L-series. He defined certain linear operators (called Hecke operators) on the space S_{k}(Γ) and showed that if f is an eigenfunction of these operators, then L(s, f) has an Euler product. Such f’s are called eigenforms. In 1976, Deligne and Serre showed that for each eigenform of weight one on Γ_{0}(N) of “odd type”, there is a 2-dimensional Galois representation ρ_{f} over ℚ such that L(s, ρ_{f}) = L(s, f).

3. Hecke operators.

Let Γ = Γ_{0}(N) and let S_{k}(Γ) be the space of cusp forms of Γ of weight k. Define operators

T_{p}(f) = p^{k−1}f(pz) + \frac{1}{p} ∑_{b mod p} f(\frac{z + b}{p})

where p |̸N is a prime. Then, T_{p} is a linear operator on S_{k}(Γ).

Hecke’s Second Theorem. Let

f(z) = ∑_{n=1}^{∞} a_{n}e^{2πinz}, a_{1} = 1.

The a_{n}’s are multiplicative if and only if f is an eigenfunction of all the T_{p}’s. In this case, L(s, f) has an Euler product of the form

L(s, f) = ∏_{p|/N} (1 − \frac{a_{p}}{p^{s}} + \frac{1}{p^{2s+1−k}} )^{−1} ∏_{p|N} (1 − \frac{a_{p}}{p^{s}} )^{−1} .

Another way of thinking about the T_{p}’s is via double cosets. Let

Γ ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ) Γ = ∪Γγ_{i}.

The γ_{i} can be taken to be

( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ) ( \begin{matrix}1 & b \\ 0 & p \end{matrix} ) b mod p.

It is then easily verified that

T_{p}(f) = ∑_{i} f|γ_{i}.

This interpretation will be useful in later generalizations.

4. Modular forms as representations of SL_{2}(ℝ).

In 1952, Gelfand and Fomin [9] noticed that cusp form of S_{k}(Γ) define representations of SL_{2}(ℝ) in the following way. In SL_{2}(ℝ), let

N = \{( \begin{matrix}1 & n \\ 0 & 1 \end{matrix} )\}, A = \{( \begin{matrix}a \\ & a^{−1} \end{matrix} )\}, K = \{( \begin{matrix}cos θ & − sin θ \\ sin θ & cos θ \end{matrix} ) = R_{θ} : 0 ≤ θ ≤ 2π\}.

Then, SL_{2}(ℝ) = NAK (Iwasawa decomposition).

If G is any group acting transitively on a set S, then S can be identified with the coset space G/Γ_{x} where Γ_{x} is the stabilizer of any x ∈ S. Since SL_{2}(ℝ) acts transitively on 𝔥, and Γ_{i} = K, we can identify the upper half plane with SL_{2}(ℝ)/K. Thus, elements of SL_{2}(ℝ) can be thought of as pairs (z, θ) with z ∈ 𝔥. Let G = SL_{2}(ℝ). Define for g ∈ G and f ∈ S_{k}(Γ),

ϕ_{f}(g) = j(g, i)^{−k}f(g · i).

This satisfies

(1) ϕ(γg) = ϕ(g) for all γ ∈ Γ,

(2) ϕ(gR_{θ}) = e^{−ikθ}ϕ(g),

(3) ∫_{Γ\G} |ϕ(g)|^{2}dg < ∞,

(4) for all ρ ∈ SL_{2}(ℤ), and all g ∈ G,

∫_{ℤ\ℝ} ϕ (ρ ( \begin{matrix}1 & xh \\ 0 & 1 \end{matrix} ) g) dx = 0

where h is the “width” of the cusp ρ(i∞).

Define the Laplace operator on G

Δ = −y^{2} (\frac{∂^{2}}{∂x^{2}} + \frac{∂^{2}}{∂y^{2}} ) − y \frac{∂^{2}}{∂x∂θ} .

Then ϕ_{f} satisfies (5’) Δϕ_{f} = −k(k − 2)ϕ_{f}/4.

Theorem. Any function ϕ ∈ L^{2}(Γ\G) satisfying (1) - (4) and (5’) is necessarily a ϕ_{f} for some f ∈ S_{k}(Γ).

If we replace (5’) by

(5) ϕ is an eigenfunction of Δ

we get real analytic forms in addition to holomorphic forms. Such forms were first studied by Maass and are called Maass wave forms. They have a Fourier expansion of the type

f(z) = ∑_{n} a_{n}e^{2πinx}\sqrt{y}K_{ir}(2π|n|y)

where r^{2} + 1/4 is an eigenvalue of

−y^{2} (\frac{∂^{2}}{∂x^{2}} + \frac{∂^{2}}{∂y^{2}} )

and

K_{v}(z) = ∫_{0}^{∞} e^{−z cosh t} cosh vt dt

is the Bessel function. Connected with these eigenvalues is the famous conjecture of Selberg that predicts that if λ = r^{2} + 1/4 is an eigenvalue for Γ_{0}(N), then λ ≥ 1/4. This is equivalent to saying that r cannot be purely imaginary. This conjecture is known if Γ = SL_{2}(ℤ). Selberg [26] showed that λ ≥ 3/16 for Γ_{0}(N). Iwaniec and Selberg have noted that there are arithmetic applications of this conjecture.

Let L^{2}(Γ\G/K) = V be the vector space of square integrable, left-invariant by Γ and right K-invariant functions on G. Define the right regular representation of G by

(R(g) · ϕ)(h) = ϕ(hg).

Then, ΔR = RΔ. Thus, irreducible representations occurring in R coincide with eigenfunctions of Δ. (We are ignoring certain growth conditions.) These ideas can be generalised to GL_{n}(ℝ).

5. GL_{n}(ℝ).

Let K = O_{n}(ℝ) be the orthogonal group, Z the group of scalar matrices. Let Γ = GL_{n}(ℤ). The analog of the upper half plane is the symmetric space

H = Γ\GL_{n}(ℝ)/KZ.

Every element of G/KZ can be written as a product

g = ( \begin{matrix}1 & x_{12} & ⋯ & ⋯ & x_{1n} \\ & 1 & x_{23} & ⋯ & x_{2n} \\ & ⋯ & ⋯ & ⋯ \\ & & & & 1 \end{matrix} ) ( \begin{matrix}y_{1}y_{2} ⋯ y_{n} \\ & y_{2} ⋯ y_{n} \\ & ⋯ & ⋯ \\ & & y_{n−1} \\ & & & 1 \end{matrix} ) .

For example, if n = 2,

( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} ) ( \begin{matrix}y & 0 \\ 0 & 1 \end{matrix} ) = ( \begin{matrix}y & x \\ 0 & 1 \end{matrix} )

corresponds to x + iy ∈ 𝔥.

We will consider functions satisfying

(1) ϕ(γg) = ϕ(g) for all γ ∈ Γ.

(2) ϕ is square integrable,

(3) ϕ should be an eigenfunction of n − 1 “Laplacians”. More precisely, if D is algebra of G-invariant differential operators, it is isomorphic to a polynomial ring of rank n − 1. We require Dϕ = ρ(D)ϕ for all D ∈ D. If these conditions are satisfied by ϕ in addition to the following growth condition

(4) |ϕ(g)| ≤ C(y_{1} ⋯ y_{n−1})^{A} then we say that ϕ is an automorphic form for G. An automorphic form is called cuspidal if

(5) ∫ ϕ (( \begin{matrix}I & X \\ 0 & I \end{matrix} ) g) dx = 0 for all g ∈ G and X is an n_{1} × n_{2} matrix with n_{1} + n_{2} = n. (This definition is due to Gelfand.) One can equally well develop the theory of Hecke operators. Let

ξ(p, i) = ( \begin{matrix}pI_{i} \\ & I_{n−i} \end{matrix} ) .

Let

Γξ(p, i)Γ = ∪Γγ_{p,i,v}

be a double coset decomposition. Define the Hecke operator T_{p,i} by

(T_{p,i}ϕ)(g) = ∑_{v} ϕ(gγ_{p,i,v}).

If ϕ is an eigenfunction for all Hecke operators T_{p,i}, we will write

T_{p,i}ϕ = λ_{p,i}p^{i(n−i)}ϕ, 1 ≤ i ≤ n.

Define for such a ϕ,

L(s, ϕ) = ∏_{p} (1 − λ_{p,1}p^{−s} + λ_{p,2}p^{−2s} + ⋯ ± λ_{p,n}p^{−ns})^{−1} .

One has the contragredient form ϕ̃ defined by ϕ̃(g) = ϕ(^{ι}g) where

ιg = w^{t}g^{−1}w

and

w = ( \begin{matrix}& & & −1 \\ & & −1 \\ & ⋯ \\ −1 \end{matrix} ) .

Then, it is known that:

Theorem. L(s, ϕ) has an analytic continuation and functional equation relating L(s, ϕ) to L(s, ϕ̃) where ϕ̃ is the contragredient automorphic form.

These L-functions are related to the ones described previously in the case n = 1 (Hecke L-series attached to grossencharactere) and n = 2 (Hecke L-series attached to modular forms or L-series attached to Maass wave forms).

Langlands has formulated a strong reciprocity conjecture: every Artin L-series L(s, ρ) is an L(s, ϕ) for some automorphic form ϕ on GL_{r}(ℝ), where r = deg ρ. The truth of this conjecture implies Artin’s conjecture in view of the Theorem above. With reference to the Sato-Tate conjecture, it is also conjectured that L_{m}(s) should be L(s, ϕ) for some automorphic form ϕ on GL_{m+1}(ℝ). Again, such a reciprocity conjecture would imply the Sato-Tate conjecture. One can also show that Selberg’s eigenvalue conjecture and Ramanujan conjectures also follow from this reciprocity. Thus, reciprocity is the problem of converse theory (that is generalisation of Weil’s theorem) to higher GL_{n}.

6. Adeles and ideles.

For x ∈ ℚ, put v_{p}(x) = ord_{p}(x). Then,

(1) v_{p}(x + y) ≥ min (v_{p}(x), v_{p}(y)),

(2) v_{p}(xy) = v_{p}(x) + v_{p}(y),

(3) v_{p}(0) = ∞.

Define a metric on ℚ by |x − y|_{p} = e^{−v_{p}(x−y)}. Denote by ℚ_{p} the completion of ℚ with respect to this p-adic metric. We call ℚ_{p} the p-adic number field. Let ℤ_{p} be the subring of p-adic integers. Every p-adic number x can be written uniquely as

x = ∑_{i=−N}^{∞} a_{i}p^{i}, a_{i} ∈ {0, 1, ..., p − 1}.

We will say that two metrics are equivalent if they induce the same topology on ℚ. We have the famous:

Theorem. (Ostrowski) Upto equivalence, the only metrics on ℚ are | · |_{p} and | · |_{∞} (the usual absolute value).

A similar result holds for any algebraic number field. Metrics which do not correspond to a finite prime (ideal) are called Archimedean valuations (or infinite primes). Thus, for a number field K, we can consider analogously K_{v} the field of v-adic numbers and the ring O_{v} of v-adic integers.

Consider the following problem: what are the characters of the additive group of rational numbers ℚ ?

We can write down some obvious ones: given α ∈ ℝ, let χ_{α}(x) = e^{2πiαx}. If β ∈ ℚ_{p}, then β = ∑_{i=−N}^{∞} a_{i}p^{i}. Let β̃ = ∑_{i=−N}^{−1} a_{i}p^{i} be the “fractional part” of β. Define χ_{β}(x) = e^{2πiβ̃x} is also a character of ℚ. It turns out that all characters of ℚ are finite products of such characters. To state this neatly, we can say every character of ℚ is given by a sequence

a = (a_{∞}, a_{2}, a_{3}, a_{5}, ...)

where a_{∞} ∈ ℝ and a_{p} ∈ ℚ_{p} and a_{p} ∈ ℤ_{p} for all p sufficiently large. Such a sequence is called an adele. Defining componentwise addition and multiplication, we see that the set of all adeles is a ring, called the adele ring of ℚ and denoted A_{ℚ} or simply A. Hence, for a ∈ A_{ℚ} we set

χ_{a}(x) = e^{2πi(a_{∞}x+a_{2}x+···)}

then every character of ℚ is of this form. Is this a one-one correspondence ? It turns out to be not. It is a simple exercise to show that two characters χ_{a} and χ_{b} are equal if and only if a − b = (x, x, x, ...) for some rational number x. Thus, we can view ℚ as a subgroup of A_{ℚ} embedded diagonally in this way. Then we have:

Theorem. The character group of ℚ is isomorphic to A_{ℚ}/ℚ as an abstract group.

Since ℚ has a natural topology on it, one can ask whether this isomorphism can be made as topological groups. For this purpose, it is necessary to define a topology on A_{ℚ} so as to have this. To this end, we begin by putting the product topology on

A^{0} = ℝ × ∏_{p} ℤ_{p}

and declaring that A^{0} is an open neighbourhood of 0. Thus, a sequence of adeles a^{(n)} = (a_{∞}^{(n)}, a_{2}^{(n)}, ⋯) tends to zero if and only if a_{p}^{(n)} tends to zero and a_{p} ∈ ℤ_{p} for all n sufficiently large. Thus, this is stronger than the product topology. Thus, the adele ring with this topology is the restricted direct product

A = ∏_{p}(ℚ_{p} : ℤ_{p}).

In general, if K is a number field, the adele ring A_{K} is the restricted direct product

A_{K} = ∏_{v}(K_{v} : O_{v}).

If G is a linear algebraic group, we can define topological group of its adelic points by the restricted direct product

G(A_{K}) = ∏_{v}(G(K_{v}) : G(O_{v})).

A_{K} becomes locally compact topological group and as such has a Haar measure on it. A similar result holds for G(A). In the course of this paper, we will be interested in G = GL_{n}. If G = GL_{1}, then G(K)\G(A_{K}) is called the idele class group of K.

7. Automorphic representations of adele groups.

Let Z be the group of scalar adeles and consider the vector space

V = L^{2}(ZG(ℚ)\G(A_{ℚ}).

We will be interested in irreducible unitary representations of G(A). Every such representation π factors as a estricted tensor product:

π = ⊗_{p}π_{p}

where π_{p} is an irreducible unitary representation of G(ℚ_{p}). In case G = GL_{1}, these are Hecke’s grossencharactere. In case G = GL_{2}, every irreducible unitary representation arises from either a modular form or a Maass wave form. Our goal will be to attach an Lfunction to such a representation π. There is a finite set S of primes (namely the ramified primes of π ) such that for each prime p ∉ S, we can associate a conjugacy class A_{p} in GL_{n}(ℂ). The exact assignment needs more background (see [15]) which we will not discuss here. Suffice it to say that in case f is a Hecke eigenform, and π_{f} is the corresponding automorphic representation, then π_{f,p} is associated to ( \begin{matrix}α_{p} \\ & β_{p} \end{matrix} ) with notation as in §1.

G(A) acts on L^{2}(ZG(ℚ)\G(A)) by right translation:
(R(g) · ϕ)(x) = ϕ(xg).

An automorphic representation is an irreducible constituent of R. (Cuspidal automorphic representations form a subspace corresponding to cusp forms.)

Let K be a number field and π an automorphic representation of G(A_{K}). Let A_{v} be the conjugacy class attached to π_{v}. Define

L(s, π_{v}) = det (1 − A_{v}Nv^{−s})^{−1}

for non-Archimedean valuations. Set

L_{S}(s, π) = ∏_{v∉S} L(s, π_{v}).

Theorem. L_{S}(s, π) has an analytic continuation and at the places v ∈ S, L(s, π_{v}) can be defined so that the global L-function L(s, π) has a functional equation.

This theorem, for n = 1 is the theory of Hecke’s L-series attached to grossencharacter and Tate, who formulated it adelically in his doctoral thesis. For n = 2, it is due to Hecke, Maass and the final adelic formulation is the work of Jacquet and Langlands [13]. The general case for GL_{n} is due to Godement and Jacquet. A good write up for the Jacquet-Langlands theory is found in Robert [23].

In this adelic framework, Langlands conjectures that every Artin L-series L(s, ρ) is and L(s, π) for some automorphic representation π on GL_{n}(A_{K}) where n = deg ρ. Thus, to each ρ there should be an automorphic representation π(ρ). In case deg ρ = 1, this is Artin’s reciprocity law since Hecke’s grossencharactere are automorphic representations of GL_{1}(A). We have already discussed the situation if deg ρ = 2.

To make further progress, we must clarify the converse problem and resolve it. Namely, given a Dirichlet series

L(s) = ∑_{n=1}^{∞} \frac{a_{n}}{n^{s}}

when is L(s) = L(s, π) for some automorphic representation π of GL_{n}(A). Weil’s theorem is an incomplete converse theorem for GL_{2} in that it applies to only modular forms. Jacquet-Langlands [13] filled this gap to cover Maass forms as well and generalised to an arbitrary number field. Thus the analog of Weil’s theorem holds for GL_{2}. For GL_{3}, Gelbart, Jacquet and Piatetski-Shapiro proved an analogous result. However, it was already shown by Piatetski-Shapiro [19] that for GL_{4}, one needs to twist by not just abelian characters but all 2-dimensional representations and establish analytic continuation and functional equations of the right type before one can conclude that L(s) = L(s, π) for some automorphic representation π on GL_{4}(A). In general, one expects to twist by all representations of degree d ≤ m − 2 before we can conlcude that π lives on GL_{m}(A). To establish continuation and functional equations, one needs the Rankin-Selberg method.

8. The Rankin-Selberg method.

Recall that the problem of the Sato-Tate conjecture was equivalent to the problem of analytic continuation of L_{m}(s) to the line Re (s) = 1. L_{1}(s) was treated by Hecke. We have mentioned that ζ(s)L_{2}(s) was shown to extend to a meromorphic function with only a simple pole at s = 1 by Rankin and Selberg (independently) in the 1940’s. Their idea is fundamental in understanding the mechanism of converse theory so we will describe this in some detail. I will illustrate it for the full modular group SL_{2}(ℤ).

Suppose that f(z) = ∑_{n=1}^{∞} a_{n}e^{2πinz} and g(z) = ∑_{n=1}^{∞} b_{n}e^{2πinz} are two cusp forms of weight k for the full modular group. (Weaker assumptions on f and g still lead to the same results but then, our exposition will be bogged down by unnecessary technical details.) Define the Eisenstein series

E(z, s) = ∑_{γ∈Γ_{∞}\Γ} Im(γz)^{s}

where

Γ_{∞} = \{( \begin{matrix}1 & n \\ 0 & 1 \end{matrix} ) : n ∈ ℤ\} .

Note that

y^{k}f(z)\overline{g(z)}

is SL_{2}(ℤ)-invariant. Consider

∫ ∫_{Γ\𝔥} y^{k}f(z)\overline{g(z)}E(z, s)\frac{dxdy}{y^{2}} .

Inserting the definition of E(z, s) as a sum and interchanging the sum and integration we get

∑_{γ∈Γ_{∞}\Γ} ∫ ∫_{Γ\𝔥}( Im (γy))^{k}f(γz)\overline{g(γz)}( Im (γz))^{s} \frac{dxdy}{y^{2}}

= ∑_{γ∈Γ_{∞}\Γ} ∫ ∫_{Γ\𝔥} y^{k}f(z)\overline{g(z)}y^{s} \frac{dxdy}{y^{2}}

= ∫ ∫_{Γ_{∞}\𝔥} y^{k}f(z)\overline{g(z)}y^{s} \frac{dxdy}{y^{2}}

because

Γ_{∞}\𝔥 = ∪_{γ∈Γ_{∞}\Γ}γ(Γ\𝔥)

which can be taken to be the region −1/2 ≤ x ≤ 1/2, y > 0. Thus, the above is

= ∫_{0}^{∞} ∫_{−1/2}^{1/2} y^{k}f(z)\overline{g(z)}y^{s} \frac{dxdy}{y^{2}}

= ∫_{0}^{∞} y^{k+s} ∑_{n,m} a_{n}\overline{b_{m}}e^{−2π(n+m)y} ∫_{−1/2}^{1/2} e^{2πi(n−m)x}dx\frac{dy}{y^{2}}

== ∫_{0}^{∞} ∑_{n=1}^{∞} a_{n}\overline{b_{n}}y^{k+s}e^{−4πny} \frac{dy}{y^{2}}

= (4π)^{k+s+1}Γ(k + s + 1) ∑_{n=1}^{∞} \frac{a_{n}\overline{b_{n}}}{n^{s}} .

It is known that

E^{∗}(z, s) = ζ(2s)E(z, s)

has an analytic continuation except for a simple pole at s = 1 and satisfies a functional equation relating the value at s to the value at 1 − s. Therefore, from this integral representation, we obtain the analytic continuation of the series

∑_{n=1}^{∞} \frac{a_{n}\overline{b_{n}}}{n^{s}} .

Now suppose that f and g are Hecke eigenforms. That is, a_{n} and b_{n} are multiplicative. Our goal is to write the above Dirichlet series as an Euler product. Let us write

a_{p} = 2p^{(k−1)/2}(α_{p} + β_{p}), b_{p} = 2p^{(k−1)/2}(γ_{p} + δ_{p}).

Let ã_{n} = a_{n}/n^{(k−1)/2} and b̃_{n} = b_{n}/n^{(k−1)/2}. It is not difficult to show that

˜a_{p^{n}} = \frac{α_{p}^{n+1} − β_{p}^{n+1}}{α_{p} − β_{p}}

and a similar formula holds for b̃_{p^{n}} whenever p is prime. Then,

∑_{n=1}^{∞} \frac{ã_{n}b̃_{n}}{n^{s}} = ∏_{p} (∑_{n=1}^{∞} (\frac{α_{p}^{n+1} − β_{p}^{n+1}}{α_{p} − β_{p}} ) (\frac{γ_{p}^{n+1} − δ_{p}^{n+1}}{γ_{p} − δ_{p}} ) p^{−ns}) .

Now we invoke Ramanujan’s identity:

∑_{n=1}^{∞} (\frac{α^{n+1} − β^{n+1}}{α − β} ) (\frac{γ^{n+1} − δ^{n+1}}{γ − δ} ) T^{n} = \frac{1 − αβγδT^{2}}{(1 − αγT)(1 − αδT)(1 − βγT)(1 − βδT)}.

Thus, we get the product equal to

ζ(2s)^{−1} ∏_{p} (1 − \frac{α_{p}γ_{p}}{p^{s}} )^{−1} (1 − \frac{α_{p}δ_{p}}{p^{s}} )^{−1} (1 − \frac{β_{p}γ_{p}}{p^{s}} )^{−1} (1 − \frac{β_{p}δ_{p}}{p^{s}} )^{−1} .

This is precisely L(s, π_{f} ⊗ π_{g}) in the notation of the Langlands’ L-function. We have therefore

L(s, π_{f} ⊗ π_{g}) = ∫_{Γ\𝔥} y^{k}f(z)g(z)E^{∗}(z, s)\frac{dxdy}{y^{2}} .

If we let f = g, we get

L(s, π_{f} ⊗ π_{f}) = ζ(s)L_{2}(s)

since π ⊗ π = Alt (π) ⊕ Sym^{2}(π). Since the zeta function does not vanish on the line Re (s) = 1, this gives the analytic continuation of L_{2}(s) until this line.

Though we have not mentioned it explicitly, the above construction is a special case of a more general construction of Langlands of L-functions attached to the tensor product of two automorphic representations. Indeed, Gelbart, Jacquet and Piatetski-Shapiro [11] showed that L(s, π_{1} ⊗ π_{2}) has analytic continuation and functional equation except when π_{2} is the contragredient of π_{1} in which case it has a simple pole at s = 1.

The method however does not give the continuation of L_{2}(s) for all values of s. This was done by Shimura [30] by making an ingenious observation. He noticed that L_{2}(s) is essentially

∑_{n=1}^{∞} \frac{a_{n^{2}}}{n^{s}}

and so this should be obtained by taking the Rankin-Selberg convolution of f with the classical theta function (which is a modular form of half integral weight). He was able to show that the theory extends to such a situation and obtained an integral representation

∑_{n=1}^{∞} \frac{a_{n^{2}}}{n^{s}} = (fudge factor) ∫_{Γ_{0}(4)\𝔥} y^{k+1/2}f(z)\overline{θ(z)}E_{Γ_{0}(4)}(z, s)\frac{dxdy}{y^{2}}

where E_{Γ_{0}(4)}(z, s) is a half-integral weight Eisenstein series belonging to Γ_{0}(4). Therefore, we obtain the analyticity of L_{2}(s).

But can we establish Langlands’ conjecture for L_{2}(s), namely that it is an L-series attached to an automorphic representation of GL_{3} ? As mentioned earlier, Gelbart and Jacquet proved a converse theorem for GL_{3} which requires twisting our original series by Dirichlet characters. The method of Shimura extends to give analytic continuation and functional equations for all these twists and so Gelbart and Jacquet were able to prove Langlands’ conjecture for L_{2}(s). We state their theorem more precisely:

Theorem. If π is an irreducible unitary automorphic representation of GL_{2}(A_{K}), then the map π ↦ Sym^{2}(π) (Gelbart-Jacquet lift) is a map into irreducible unitary automorphic representations on GL_{3}(A_{K}).

For further discussion, let us introduce the notation A(GL_{n}) to denote the space of irreducible unitary automorphic representations of GL_{n}(A_{K}). With the information at hand, we can take Rankin-Selberg convolutions:

π ⊗ Sym^{2}(π) = π ⊕ Sym^{3}(π)
Sym^{2}(π) ⊗ Sym^{2}(π) = 1 ⊕ Sym^{2}(π) ⊕ Sym^{4}(π).

From the first equation, we deduce that L_{3}(s) is analytic for Re (s) ≥ 1 upon showing that L(s, π) does not vanish on this line. This is a minor modification of the classical argument of Hadamard and de la Vallée Poussin. The second equation shows that L_{4}(s) is analytic for Re (s) ≥ 1 by a similar process. If we can consider Sym^{2}(π)⊗ Sym^{3}(π) then we would obtain information on L_{5}(s). This however we cannot do because we do not know that Sym^{3}(π) ∈ A(GL_{4}). We need a converse theorem for GL_{4}.

Piatetski-Shapiro showed that the analog of Weil’s theorem does not hold for GL_{4} and we need the analyticity of additonal twists. It turns out that these additional twists are representations of GL_{2}. Thus, the converse theory for GL_{4} requires analytic continuations of L(s, ϕ⊗π) as π ranges over A(GL_{2}). It is in this context that the result of Paul Garrett [8] in 1985 on triple convolutions was intriguing. (We shall see later other reasons for the intrigue.) Namely, Garrett [8] showed that if f, g and h are modular cusp forms of weight k, with Fourier coefficients a_{n}, b_{n} and c_{n} respectively, then

∑_{n=1}^{∞} \frac{a_{n}b_{n}c_{n}}{n^{s}}

can be written as an integral transform of Rankin-Selberg type and an analytic continuation can be obtained for it. In the notation of Langlands, this corresponds to L(s, π_{f} ⊗π_{g} ⊗π_{h}), an L-function attached to GL_{2} × GL_{2} × GL_{2}. In the context of converse theory, this result implies that we will have a map

A(GL_{2}) × A(GL_{2})→A(GL_{4}).

From this, we would obtain the analytic continuation of L_{5}(s), L_{6}(s), L_{7}(s) and L_{8}(s) up to Re (s) = 1. At the end of this paper, we will see another fundamental reason for the interest in this map.

For the general converse theory on GL_{n}, one needs to twist by representations of GL_{n−2} though this has not been formally written up. There are informal notes of Piatetski-Shapiro which prove this theorem for function fields.

9. Base change and automorphic induction.

Let K/k be a Galois extension and let G = Gal (K/k). If ρ is an irreducible representation of G, then L(s, ρ, K/k) was conjectured by Artin to be entire if ρ ≠ 1. We can extend the definition of Artin L-series to an arbitrary representation of G by additivity:

L(s, ρ_{1} ⊕ ρ_{2}, K/k) = L(s, ρ_{1}, K/k)L(s, ρ_{2}, K/k).

If now ψ is a representation of a subgroup H of G, then L(s, ψ, K/K^{H}) is the Artin Lseries belonging to the extension K/K^{H} where K^{H} denotes the field fixed by H. A simple calculation shows that Artin L-series are invariant under induction:

L(s, Ind _{H}^{G}ψ, K/k) = L(s, ψ, K/K^{H}).

This last property implies a corresponding property for L-series attached to automorphic representations. Recall the Langlands’ reciprocity conjecture: for each ρ, there is π(ρ) ∈ A(GL_{n}(A_{K})) so that

L(s, ρ, K/k) = L(s, π).

The natural question is, how does the map ρ ↦ π(ρ) behave under restriction to a subgroup? By the reciprocity conjecture, we should have ρ|_{H} ↦ π(ρ|_{H}) ∈ A(GL_{n}(A_{K^{H}})). What is L(s, ρ|_{H}, K/K^{H}) ? A standard group theoretic result is

Ind _{H}^{G}(ρ|_{H} ⊗ ψ) = ρ ⊗ Ind _{H}^{G}ψ.

Thus,

L(s, Ind _{H}^{G}(ρ|H ⊗ ψ), K/k) = L(s, ρ ⊗ Ind _{H}^{G}ψ, K/k).

But the former L-function is L(s, ρ|_{H} ⊗ ψ, K/K^{H}) by the invariance of Artin L-series. Therefore,

L(s, ρ|_{H}, K/K^{H}) = L(s, ρ ⊗ Ind _{H}^{G}1, K/k).

But Ind _{H}^{G}1 = reg_{G/H} is the permutation representation on the cosets of H. This suggests that we make the following definition. Let π ∈ A(GL_{n}(A_{K})). For each π_{v}, we associated an A_{v} ∈ GL_{n}(ℂ). Define,

L_{v}(s, B(π)) = det (1 − A_{v} ⊗ reg_{G/H}(σ_{v})Nv^{−s})^{−1},

where σ_{v} is the Artin symbol of v. Set

L(s, B(π)) = ∏_{v} L_{v}(s, B(π)).

B(π) should correspond to an element of A(GL_{n}(A_{M})) where M = K^{H}. The problem of base change is to determine when this map exists.

For n = 2, this was done by Langlands [14] when M/k is cyclic. He then used these ideas to deal with the tetrahedral case of Artin’s conjecture. For arbitrary n, it is the recent work of Arthur and Clozel [1] Again, the situation is for M/k cyclic.

Now suppose the ψ is a representation of H. Corresponding to ψ there should be a π ∈ A(GL_{n}(A_{M})) where n = deg ψ. But the invariance of Artin L-series under induction implies that there should be an I(π) ∈ A(GL_{nr}(A_{k})) (r = [G : H]) so that

L(s, I(π)) = L(s, Ind _{H}^{G}ψ, K/k).

This map π ↦ I(π), called the automorphic induction map, is conjectured to exist. Again, Arthur and Clozel showed this exists when M/k is cyclic and arbitrary n. Thus, if M/k is contained in a solvable extension of k, the base change maps and automorphic induction maps exist.

Main Theorem. Let K be an algebraic number field of degree n over ℚ, and ψ a Hecke character of K. If there exists an automorphic representation I(ψ) of A(GL_{n}(A_{ℚ})) such that

L(s, ψ) = L(s, I(ψ))

for every Hecke character ψ of K, then both Artin’s conjecture and the Langlands reciprocity conjecture are true.

Proof. Let L/ℚ be a Galois extension of finite degree and Galois group G. If ρ is an irreducible representation of G, and χ is its character, we can write by the Brauer induction theorem,

χ = ∑_{i} m_{i} Ind _{H_{i}}^{G}χ_{i},

where m_{i} are integers, χ_{i} are abelian characters of certain subgroups H_{i} of G. By the abelian reciprocity law, L(s, χ_{i}, L/L^{H_{i}}) is equal to a Hecke L-series L(s, ψ_{i}) where ψ_{i} is a Hecke character of L^{H_{i}}. As such, we can therefore write

L(s, ρ, L/ℚ) = ∏_{i} L(s, π_{i})^{m_{i}}

where π_{i} are automorphic representations of ℚ. By a theorem of Langlands [15], we can further decompose the product so that each constituent L-function corresponds to an irreducible automorphic representation of ℚ. Thus, without loss, in the above product, we can suppose that each π_{i} is irreducible and distinct from π_{j} when i ≠ j. The identity reveals that

χ(p) = ∑_{i} m_{i}a_{p}^{(i)}

where a_{p}^{(i)} denotes the p-th coefficient in the Dirichlet series of L(s, π_{i}). On one hand, the Chebotarev density theorem shows that

∑_{p} \frac{|χ(p)|^{2}}{p^{s}}

has a simple pole at s = 1. On the other hand, the series

∑_{p} | ∑_{i} m_{i}a_{p}^{(i)}|^{2}/p^{s}

is on expanding

∑_{i,j} m_{i}m_{j} ∑_{p} a_{p}^{(i)}\overline{a_{p}^{(j)}}p^{−s}.

By a result of Jacquet, Piatetski-Shapiro and Shalika [11], the inner sum has a simple pole if and only if i = j. Thus, the order of the pole is

∑_{i} m_{i}^{2}

which must equal 1. This forces m_{1} (say) to be ±1 and the remaining ones to be zero. The possiblity m_{1} = −1 implies that L(s, χ) has no trivial zeroes which is not the case. Hence, m_{1} = 1 and L(s, χ, L/ℚ) = L(s, π_{1}) as desired.

Unfortunately, the hypothesis of the theorem is known to be satisfied only if K is a solvable extension of ℚ by the theorem of Arthur and Clozel [1]. In the next section, we apply these ideas to the theory of elliptic curves.

10. Application to elliptic curves.

Let E/k be an elliptic curve defined over a number field k. One can think of this as the set of solutions of an equation of the form

E :  y^{2} = x^{3} + ax + b,  a, b ∈ k.

Let E(k) be the set of k-rational points of E. Then, E(k) can be given the structure of an additive abelian group. We have the classical:

Mordell-Weil theorem. E(k) is finitely generated.

Thus,

E(k) ≃ ℤ^{r} ⊕ E(k)_{tors}.

There is a folklore conjecture that |E(k)_{tors}| ≤ C_{k} where C_{k} is a constant depending only on the field. For k = ℚ, Mazur proved that C_{ℚ} = 16. Recently, Kamienny established this for quadratic fields. Given any curve, there is a finite algorithm which enables us to find E(k)_{tors}.

More intriguing is to find r = r_{k}, called the rank of E over k. It is unknown at present whether the rank of E over ℚ is unbounded though this is known in the function field case. Birch and Swinnerton-Dyer made an analytic conjecture about the rank in the following way: for each prime ideal 𝔭 of k, let E(O_{k}/𝔭) have cardinality N𝔭 + 1 − a_{𝔭}. We know, |a_{𝔭}| ≤ (N𝔭)^{1/2}. Define

L(E/k, s) = ∏_{𝔭} (1 − \frac{a_{𝔭}}{N𝔭^{s}} + \frac{1}{N𝔭^{2s−1}} )^{−1} ,

where the product is defined over primes where E has good reduction. This defines an analytic function for Re (s) > 3/2. We have the famous:

Birch and Swinnerton-Dyer conjecture: L(E/k, s) extends to an analytic function for all s ∈ ℂ and

ord_{s=1}L(E/k, s) = rank E(k).

If K/k is an arbitrary extension, then we can consider E over K and it is easy to see that rank E(K) ≥ rank E(k). This raises two questions. First, does L(E/K, s) have analytic continuation given that L(E/k, s) does. Second, if it does, then can we prove that

ord_{s=1}L(E/K, s) ≥ ord_{s=1}L(E/k, s).

If we consider the ring of endomorphisms of E, then it is not difficult to show that there are two possibilities for this. Either it is isomorphic to ℤ or an order in an imaginary quadratic field. In the latter case, we say E has CM (complex multiplication). The latter case is easier to deal with. For instance, we shall see that the answer to the first question is affirmative in this case.

Deuring [5] showed that if E has CM by an order in F, then L(E/k, s) is a product of two Hecke L-series of k if k ⊃ F or equal to a Hecke L-series of kF which is a quadratic extension of k.

Thus if E has CM, then L(E/k, s) extends to an entire function of s. But recall Taniyama’s conjecture, namely that L(E/k, s) should be an L(s, π) for some π ∈ A(GL_{2}(A_{k})). We can verify this conjecture for E with CM by using automorphic induction.

Suppose that Taniyama’s conjecture is true. Let K/k be any Galois extension with group G. Then, the theory of ℓ-adic representations shows that

L(E/K, s) = ∏_{ρ irred} L(E/k, s, ρ)^{ρ(1)},

where L(E/k, s, ρ) is the “twist” by ρ. On Taniyama’s conjecture, this corresponds to L(s, π ⊗ reg_{G}) where reg_{G} is the regular representation of G. By what we have seen in the previous section, this is the base change of π, B(π). We can prove the following theorem:

Theorem. (joint with Kumar Murty) ([18])

(1) If E over k has CM then, L(E/k, s) extends to an entire function for all values of s.

(2) Suppose that E/k satisfies Taniyama’s conjecture and K is a finite extension of k. If K is a solvable extension of k, then, L(E/K, s) extends to an entire function of s.

(3) In both cases, if K is a solvable extension, then L(E/K, s)/L(E/k, s) is entire. In

particular,

ord_{s=1}L(E/K, s) ≥ ord_{s=1}L(E/k, s).

If the base change map always exists, one would not have to make the assumption that K is a solvable extension of k for part (3) of the theorem. In the CM case, one could strengthen the result by allowing K to be contained in a solvable extension.

This is highly reminiscent of the Brauer-Aramata theorem that states that if K/k is Galois, then ζ_{K}(s)/ζ_{k}(s) is entire, where ζ_{k}(s) denotes the zeta function of k. There is a classical conjecture of Dedekind that predicts that this should be true even if K/k is not Galois. This is of course implied by Artin’s conjecture. All these would follow from the development of the theory of base change.

Our method of proof deviates from the method of Brauer or Aramata. Instead, we use a device exploited by Kumar Murty in his exceedingly simple proof of the Brauer-Aramata theorem (see [6]).

Fix s_{0} ∈ ℂ. Let K/k be Galois with group G. For each character χ of G, let

n_{χ} = ord_{s=s_{0}}L(s, χ, K/k).

Theorem. (Kumar Murty) ∑_{χ} n_{χ}^{2} ≤ (n_{reg})^{2}.

Corollary. ζ_{K}(s)/ζ_{k}(s) is entire.

Proof. From the inequality, we get

n_{1}^{2} ≤ n_{reg}^{2}

from which the result follows.

Proof of the theorem. Let H ≤ G. For each character ψ of H define

n_{ψ} = ord_{s=s_{0}}L(s, ψ, K/K^{H}).

Define

θ_{H} = ∑_{ψ∈Ĥ} n_{ψ}ψ,

where Ĥ denotes the set of irreducible characters of H. Then,

θ_{G}|_{H} = ∑_{χ∈Ĝ} n_{χ}χ|_{H} = ∑_{χ∈Ĝ} n_{χ} ∑_{ψ∈Ĥ}(χ|_{H}, ψ)ψ.

The inner sum is by Frobenius reciprocity

= ∑_{ψ∈Ĥ}(χ, Ind _{H}^{G}ψ)ψ.

Inserting this and interchanging summation we obtain

θ_{G}|_{H} = ∑_{ψ∈Ĥ} (∑_{χ∈Ĝ} n_{χ}(χ, Ind _{H}^{G}ψ)) ψ.

We recognise the inner sum as n_{ψ} because Artin L-series are invariant under induction. Thus, θ_{G}|_{H} = θ_{H}. Now,

(θ_{G}, θ_{G}) = ∑_{χ∈Ĝ} n_{χ}^{2} = \frac{1}{|G|} ∑_{g∈G} |θ_{G}(g)|^{2}.

But by what was shown above, θ_{G}(g) = θ_{⟨g⟩}(g). If H is cyclic,

|θ_{H}(g)| ≤ ∑_{ψ∈Ĥ} n_{ψ} = n_{reg}

by the abelian reciprocity law. This gives

(θ_{G}, θ_{G}) ≤ (n_{reg})^{2}.

This same proof carries over with a few modifications to show that if ρ is an arbitrary representation, then

L(s, ρ ⊗ reg_{G}, K/k)/L(s, ρ, K/k)

is entire. By a similar method, one can show that if π is an automorphic cuspidal representation and B(π) is the base change of it, then L(s, B(π))/L(s, π) is entire.

11. Concluding remarks.

We therefore see that Artin Galois representations should correspond to automorphic representations. The Taniyama conjecture predicts the same should happen for elliptic curves. But there are more automorphic representations than Galois representations or elliptic curves. This led Langlands to conjecture a super-reciprocity law in the following form.

If K is a field and G is a group, we can consider the category of all K representations of G. This is a category with tensor products and fibre functor (namely the map that takes each representation to its underlying vector space). Langlands drew attention to the fact that there is a result of Rivano Saavedra [24] that the converse of this is true. That is, if C is a category with tensor products and fibre functors (together with a few other minor compatibility assumptions), then C is the category of representations of a group G. Such a category with tensor products is called a Tannakian category. Is the category of automorphic representations Tannakian ? If so, then there is a giant (or monstrous) group which we can call the automorphic Galois group from which all L-functions arise. Such a situation is not outlandish. In the context of converse theory, it requires us to show that the Rankin-Selberg convolution is the L-series of an automorphic representation. More precisely, there should be a map

A(GL_{n}) × A(GL_{m})→A(GL_{mn}).

By the predicted converse theory of GL_{mn}, we must twist the Rankin-Selberg convolution by automorphic representations of GL_{r} for all r ≤ mn − 2, and so we are in the case of triple convolutions. That is why it is tantalizing to understand Garrett’s work and extend it (if possible) to higher GL_{n}. Such a step would be a major one in establishing a superreciprocity law and which would resolve a huge chunk of the classical unsolved problems of number theory.

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M. Ram Murty Department of Mathematics McGill University Montreal H3A 2K6, Canada e-mail: murty@gauss.math.mcgill.ca