PMATH 842: Automorphic Forms and the Langlands Program

These notes synthesize material from D. Bump’s Automorphic Forms and Representations, S.S. Gelbart’s Automorphic Forms on Adele Groups, F. Diamond and J. Shurman’s A First Course in Modular Forms, and D. Goldfeld’s Automorphic Forms and L-Functions for GL(n,R), enriched with S.S. Kudla’s expository article “From Modular Forms to Automorphic Representations” and J. Bernstein and S. Gelbart’s An Introduction to the Langlands Program.

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Chapter 1: Classical Modular Forms

1.1 The Upper Half-Plane and Its Symmetries

The theory of modular forms begins with one of the most natural objects in complex analysis: the upper half-plane

\[ \mathcal{H} = \{ z \in \mathbb{C} : \operatorname{Im}(z) > 0 \}. \]

This is a connected, simply connected open subset of \(\mathbb{C}\), and it carries a natural hyperbolic metric

\[ ds^2 = \frac{dx^2 + dy^2}{y^2}, \qquad z = x + iy, \]

making it a model of the hyperbolic plane with constant curvature \(-1\). The corresponding volume form is \(d\mu = \frac{dx\, dy}{y^2}\).

The group \(\mathrm{SL}(2, \mathbb{R})\) acts on \(\mathcal{H}\) by Möbius transformations: for \(\gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathrm{SL}(2, \mathbb{R})\), we set

\[ \gamma \cdot z = \frac{az + b}{cz + d}. \]

A direct computation shows that \(\operatorname{Im}(\gamma \cdot z) = \frac{\operatorname{Im}(z)}{|cz + d|^2}\), confirming that this action preserves \(\mathcal{H}\). Moreover, the hyperbolic metric and the measure \(d\mu\) are both invariant under this action. The kernel of the action is \(\{\pm I\}\), so the effective group of isometries is \(\mathrm{PSL}(2, \mathbb{R}) = \mathrm{SL}(2, \mathbb{R}) / \{\pm I\}\), which in fact equals the full group of orientation-preserving isometries of \(\mathcal{H}\).

The stabiliser of the point \(i \in \mathcal{H}\) is the special orthogonal group \(\mathrm{SO}(2)\), giving the identification \(\mathcal{H} \cong \mathrm{SL}(2, \mathbb{R}) / \mathrm{SO}(2)\) as a Riemannian symmetric space.

1.2 The Modular Group and Congruence Subgroups

The modular group is the discrete subgroup \(\mathrm{SL}(2, \mathbb{Z}) \subset \mathrm{SL}(2, \mathbb{R})\). Its importance stems from its role as the automorphism group of the lattice \(\mathbb{Z}^2 \subset \mathbb{R}^2\), or equivalently, from the fact that the quotient \(\mathrm{SL}(2, \mathbb{Z}) \backslash \mathcal{H}\) parametrises isomorphism classes of elliptic curves over \(\mathbb{C}\).

A fundamental domain for the action of \(\mathrm{SL}(2, \mathbb{Z})\) on \(\mathcal{H}\) is given by

\[ \mathcal{F} = \left\{ z \in \mathcal{H} : |z| \geq 1, \; |\operatorname{Re}(z)| \leq \tfrac{1}{2} \right\}, \]

with appropriate boundary identifications. The quotient \(\mathrm{SL}(2, \mathbb{Z}) \backslash \mathcal{H}\) is a non-compact Riemann surface of genus zero with one cusp (at \(i\infty\)), one elliptic point of order 2 (at \(i\)), and one elliptic point of order 3 (at \(e^{2\pi i/3}\)).

For a positive integer \(N\), the principal congruence subgroup of level \(N\) is

\[ \Gamma(N) = \ker\bigl(\mathrm{SL}(2, \mathbb{Z}) \to \mathrm{SL}(2, \mathbb{Z}/N\mathbb{Z})\bigr) = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathrm{SL}(2, \mathbb{Z}) : a \equiv d \equiv 1, \; b \equiv c \equiv 0 \pmod{N} \right\}. \]

A subgroup \(\Gamma \leq \mathrm{SL}(2, \mathbb{Z})\) is called a congruence subgroup if it contains \(\Gamma(N)\) for some \(N\). The two most important families are:

\[ \Gamma_0(N) = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathrm{SL}(2, \mathbb{Z}) : c \equiv 0 \pmod{N} \right\}, \]\[ \Gamma_1(N) = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathrm{SL}(2, \mathbb{Z}) : a \equiv d \equiv 1, \; c \equiv 0 \pmod{N} \right\}. \]

These satisfy \(\Gamma(N) \trianglelefteq \Gamma_1(N) \trianglelefteq \Gamma_0(N) \trianglelefteq \mathrm{SL}(2, \mathbb{Z})\), with \(\Gamma_0(N)/\Gamma_1(N) \cong (\mathbb{Z}/N\mathbb{Z})^\times\). The quotients \(\Gamma_0(N) \backslash \mathcal{H}\) are the modular curves \(Y_0(N)\), which parametrise isogenies of elliptic curves of degree \(N\).

1.3 Modular Forms and Cusp Forms

The central objects of this chapter are holomorphic functions on \(\mathcal{H}\) that transform in a prescribed way under a congruence subgroup.

Since \(\Gamma\) always contains a translation \(T^h = \begin{pmatrix} 1 & h \\ 0 & 1 \end{pmatrix}\) for some minimal positive integer \(h\) (the width of the cusp at infinity), any \(f \in M_k(\Gamma)\) satisfies \(f(z + h) = f(z)\), and therefore admits a Fourier expansion

\[ f(z) = \sum_{n=0}^{\infty} a_n q_h^n, \qquad q_h = e^{2\pi i z / h}. \]

For \(\Gamma = \mathrm{SL}(2, \mathbb{Z})\) we have \(h = 1\), and we write \(q = e^{2\pi i z}\). The condition of being a cusp form is then \(a_0 = 0\) (and similarly at all other cusps).

1.4 Eisenstein Series

The simplest and most explicit examples of modular forms are the Eisenstein series.

The sum converges absolutely and uniformly on compact subsets of \(\mathcal{H}\) when \(k \geq 4\). Rearranging the sum and using the Lipschitz summation formula, one computes the Fourier expansion:

\[ G_k(z) = 2\zeta(k) + \frac{2(2\pi i)^k}{(k-1)!} \sum_{n=1}^{\infty} \sigma_{k-1}(n) q^n, \]

where \(\sigma_{k-1}(n) = \sum_{d \mid n} d^{k-1}\). The normalised Eisenstein series is

\[ E_k(z) = \frac{G_k(z)}{2\zeta(k)} = 1 - \frac{2k}{B_k} \sum_{n=1}^{\infty} \sigma_{k-1}(n) q^n, \]

where \(B_k\) is the \(k\)-th Bernoulli number.

1.5 The Discriminant Function and the \(j\)-Invariant

The product formula is the celebrated identity of Jacobi. The first few values are \(\tau(1) = 1\), \(\tau(2) = -24\), \(\tau(3) = 252\), \(\tau(4) = -1472\). The fact that \(\Delta\) never vanishes on \(\mathcal{H}\) is essential: it implies that \(S_k(\mathrm{SL}(2, \mathbb{Z})) = \Delta \cdot M_{k-12}(\mathrm{SL}(2, \mathbb{Z}))\), giving an inductive computation of dimensions.

The coefficient 196884 famously equals \(196883 + 1\), the dimension of the smallest non-trivial representation of the Monster group plus 1, a connection explained by Borcherds’ proof of the Monstrous Moonshine conjecture.

1.6 Hecke Operators

The theory of Hecke operators is one of the great achievements of early 20th-century number theory. Erich Hecke (1937) discovered that the spaces \(M_k(\mathrm{SL}(2, \mathbb{Z}))\) carry a commutative algebra of operators that simultaneously diagonalise the space, and whose eigenvalues encode arithmetic information.

1.7 The Petersson Inner Product

The space of cusp forms carries a natural Hermitian inner product.

1.8 Atkin-Lehner Theory: Newforms and Oldforms

For general level \(N\), the spaces \(S_k(\Gamma_0(N))\) contain forms that “come from” lower levels. The theory of Atkin and Lehner (1970), refined by W. Li, provides a canonical decomposition.

If \(M | N\) and \(d | (N/M)\), then the map \(f(z) \mapsto f(dz)\) embeds \(S_k(\Gamma_0(M))\) into \(S_k(\Gamma_0(N))\). The oldspace \(S_k^{\mathrm{old}}(\Gamma_0(N))\) is the subspace spanned by all such images with \(M < N\). Its orthogonal complement with respect to the Petersson inner product is the newspace \(S_k^{\mathrm{new}}(\Gamma_0(N))\).

This theorem is the classical precursor of the strong multiplicity one theorem for automorphic representations, which we will encounter in Chapter 5.

1.9 \(L\)-functions of Modular Forms

To a normalised Hecke eigenform \(f = \sum a_n q^n \in S_k(\Gamma_0(N))\), one associates the \(L\)-function

\[ L(s, f) = \sum_{n=1}^{\infty} \frac{a_n}{n^s} = \prod_{p \nmid N} \frac{1}{1 - a_p p^{-s} + p^{k-1-2s}} \prod_{p | N} \frac{1}{1 - a_p p^{-s}}. \]

The Euler product converges absolutely for \(\operatorname{Re}(s) > \frac{k+1}{2}\) (using the Ramanujan-Petersson bound \(|a_p| \leq 2p^{(k-1)/2}\) for \(p \nmid N\), proved by Deligne in 1974).

This construction gives the first bridge between modular forms and number theory through \(L\)-functions. In subsequent chapters, we will see how this picture generalises to automorphic representations and how the Langlands program organises these \(L\)-functions into a vast web.

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Chapter 2: Representation Theory of \(\mathrm{GL}(2)\) over Local Fields

The passage from classical modular forms to automorphic representations requires a deep understanding of the representation theory of \(\mathrm{GL}(2)\) over local fields. This chapter develops the representation theory over \(p\)-adic fields \(\mathbb{Q}_p\), with parallel remarks about the archimedean place \(\mathbb{R}\).

2.1 Smooth and Admissible Representations

Let \(F\) be a non-archimedean local field (e.g., \(\mathbb{Q}_p\)) with ring of integers \(\mathcal{O}\), uniformiser \(\varpi\), and residue field \(\mathbb{F}_q\). We write \(G = \mathrm{GL}(2, F)\), \(K = \mathrm{GL}(2, \mathcal{O})\), and \(B = \left\{ \begin{pmatrix} a & b \\ 0 & d \end{pmatrix} \right\}\) for the standard Borel subgroup.

The condition of smoothness replaces the continuity condition in the theory of representations of Lie groups. Since \(G\) is a totally disconnected locally compact group, its open compact subgroups form a neighbourhood basis of the identity, and the smooth representations are precisely those for which the action map \(G \times V \to V\) is continuous when \(V\) is given the discrete topology.

2.2 Parabolic Induction and Principal Series Representations

The most fundamental construction of representations of \(G\) is via parabolic induction from the Borel subgroup \(B\).

Let \(\chi_1, \chi_2 : F^\times \to \mathbb{C}^\times\) be smooth (i.e., locally constant) characters. They define a character of \(B\) via \(\begin{pmatrix} a & b \\ 0 & d \end{pmatrix} \mapsto \chi_1(a) \chi_2(d)\). The (normalised) principal series representation is the induced representation

\[ I(\chi_1, \chi_2) = \operatorname{Ind}_B^G(\chi_1 \otimes \chi_2) = \left\{ f : G \to \mathbb{C} \;\middle|\; f\left(\begin{pmatrix} a & b \\ 0 & d \end{pmatrix} g\right) = \chi_1(a)\chi_2(d) \left|\frac{a}{d}\right|^{1/2} f(g), \; f \text{ smooth} \right\}. \]

The normalising factor \(|a/d|^{1/2}\) is the square root of the modulus character of \(B\), included so that the induction preserves unitarity.

When \(\chi_1 \chi_2^{-1} = |\cdot|\), the representation \(I(\chi_1, \chi_2)\) has a unique irreducible subrepresentation (the special representation) and a one-dimensional quotient; when \(\chi_1\chi_2^{-1} = |\cdot|^{-1}\), the roles of subrepresentation and quotient are reversed.

2.3 The Special (Steinberg) Representation

The Steinberg representation plays a fundamental role: it is the representation associated to modular forms with square-free level at primes dividing the level. Geometrically, \(\mathrm{St}\) can be realised as the space of locally constant functions on \(\mathbb{P}^1(F)\) modulo constants, with \(G\) acting by Möbius transformations.

2.4 Supercuspidal Representations

Supercuspidal representations are the “building blocks” of the representation theory that do not arise from lower-dimensional groups. They can be constructed via compact induction from maximal compact-mod-centre subgroups. For \(\mathrm{GL}(2, F)\), the supercuspidal representations are parametrised by characters of quadratic extensions \(E/F\) that do not factor through the norm map \(N_{E/F}\), via a construction of Jacquet-Langlands.

2.5 The Jacquet Module

The Jacquet module provides a systematic tool for studying the relationship between representations of \(G\) and characters of the torus \(T \cong F^\times \times F^\times\).

The vanishing of the Jacquet module is thus the defining property of supercuspidal representations; they cannot be “seen” from the Borel subgroup.

2.6 Whittaker Models and the Kirillov Model

Whittaker models are fundamental to the theory of automorphic forms, as they provide the local counterparts of Fourier expansions.

This multiplicity-one result is a theorem of Gelfand and Kazhdan, valid more generally for \(\mathrm{GL}(n)\). It is the key local ingredient behind the uniqueness of Fourier-Whittaker expansions of automorphic forms.

2.7 Local \(L\)-factors and \(\epsilon\)-factors

To each irreducible admissible representation \(\pi\) of \(\mathrm{GL}(2, F)\) and each additive character \(\psi\), one associates local \(L\)-factors and \(\epsilon\)-factors. These are the local building blocks of global \(L\)-functions.

2.8 The Local Langlands Correspondence for \(\mathrm{GL}(2)\)

The local Langlands correspondence, proved for \(\mathrm{GL}(2)\) by Kutzko (supercuspidal case) and Jacquet-Langlands, establishes a canonical bijection between irreducible admissible representations of \(\mathrm{GL}(2, F)\) and two-dimensional Frobenius-semisimple Weil-Deligne representations of the Weil group \(W_F\).

This correspondence is the foundation of the Langlands program at the local level: it translates between harmonic analysis on reductive groups and Galois theory.

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Chapter 3: Automorphic Forms on Adele Groups

The adelic viewpoint, pioneered by Tate in his thesis (1950) for \(\mathrm{GL}(1)\) and extended to \(\mathrm{GL}(2)\) by Jacquet and Langlands (1970), reformulates the theory of modular forms in the language of representation theory. This reformulation is essential for generalisation to higher-rank groups and for the Langlands program.

3.1 The Adele Ring and the Idele Group

We recall the basic constructions from algebraic number theory.

The rational numbers embed diagonally into \(\mathbb{A}\) via \(q \mapsto (q, q, q, \ldots)\), and the image is a discrete, cocompact subgroup of \(\mathbb{A}\). Similarly, \(\mathbb{Q}^\times\) embeds into \(\mathbb{A}^\times\) as a discrete subgroup, and the quotient \(\mathbb{Q}^\times \backslash \mathbb{A}^\times / \mathbb{R}_{>0}\) is compact — this is the adelic restatement of the finiteness of the class number.

3.2 \(\mathrm{GL}(2)\) over the Adeles and Strong Approximation

For \(\mathrm{GL}(2)\), strong approximation fails (because the determinant map detects the class group), but a modified version holds:

\[ \mathrm{GL}(2, \mathbb{A}) = \mathrm{GL}(2, \mathbb{Q}) \cdot \mathrm{GL}(2, \mathbb{R})^+ \cdot K_f, \]

where \(\mathrm{GL}(2, \mathbb{R})^+ = \{g \in \mathrm{GL}(2, \mathbb{R}) : \det g > 0\}\). This is a consequence of the fact that \(\mathbb{Q}\) has class number one.

3.3 Automorphic Forms on \(\mathrm{GL}(2, \mathbb{A})\)

3.4 The Dictionary: Classical Forms ↔ Adelic Forms

The passage from classical modular forms to adelic automorphic forms is one of the most important translations in modern number theory. It was first made explicit by Gelbart and Jacquet-Langlands.

Under this dictionary, the Fourier expansion of \(f\) at the cusp \(\infty\) corresponds to the Whittaker expansion of the adelic form:

\[ \varphi_f(g) = \sum_{\alpha \in \mathbb{Q}^\times} W_\varphi\left(\begin{pmatrix} \alpha & 0 \\ 0 & 1 \end{pmatrix} g\right), \]

where \(W_\varphi(g) = \int_{\mathbb{Q} \backslash \mathbb{A}} \varphi_f\left(\begin{pmatrix} 1 & x \\ 0 & 1 \end{pmatrix} g\right) \psi(-x)\, dx\) is the global Whittaker function, and \(\psi\) is the standard additive character of \(\mathbb{Q} \backslash \mathbb{A}\).

3.5 Automorphic Representations

The space of cusp forms decomposes as a direct sum (Hilbert space completion with respect to the Petersson inner product):

\[ L^2_0(\mathrm{GL}(2, \mathbb{Q}) \backslash \mathrm{GL}(2, \mathbb{A}), \omega) = \widehat{\bigoplus}_\pi m(\pi) \cdot \pi, \]

where the sum runs over cuspidal automorphic representations with central character \(\omega\), and \(m(\pi)\) is the multiplicity.

3.6 Flath’s Theorem: The Tensor Product Decomposition

One of the most powerful structural results is that automorphic representations factor as restricted tensor products of local representations.

3.7 Cuspidal Automorphic Representations and the Spectral Decomposition

The space of all automorphic forms decomposes into the cuspidal part and the Eisenstein part. The cuspidal spectrum is discrete (as shown by the multiplicity-one theorem above), while the Eisenstein spectrum is continuous.

This spectral decomposition is the starting point for the trace formula and for understanding the full automorphic spectrum.

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Chapter 4: \(L\)-functions and the Langlands Program for \(\mathrm{GL}(2)\)

Having established the local and global theory of automorphic representations on \(\mathrm{GL}(2)\), we now turn to the \(L\)-functions that encode their arithmetic content and the first incarnation of the Langlands conjectures.

4.1 Global \(L\)-functions

Let \(\pi = \bigotimes_v' \pi_v\) be a cuspidal automorphic representation of \(\mathrm{GL}(2, \mathbb{A})\). Using the local \(L\)-factors defined in Chapter 2, we form the global \(L\)-function.

For an unramified prime \(p\), the local component \(\pi_p\) is the unramified principal series \(I(\chi_1, \chi_2)\) with \(\chi_i\) unramified, and the local \(L\)-factor is

\[ L(s, \pi_p) = \frac{1}{(1 - \alpha_p p^{-s})(1 - \beta_p p^{-s})}, \]

where \(\alpha_p = \chi_1(\varpi_p)\) and \(\beta_p = \chi_2(\varpi_p)\) are the Satake parameters. The relationship to classical modular forms is: if \(\pi = \pi_f\) for a normalised newform \(f\), then \(\alpha_p + \beta_p = a_p(f) / p^{(k-1)/2}\) and \(\alpha_p \beta_p = \chi(p)\), where \(\chi\) is the nebentypus.

4.2 Analytic Continuation and Functional Equation

4.3 Rankin-Selberg \(L\)-functions

One of the most powerful tools in automorphic forms is the Rankin-Selberg method, which constructs \(L\)-functions for pairs of representations.

The Rankin-Selberg \(L\)-function is constructed via the integral

\[ I(s, \varphi, \varphi') = \int_{\mathrm{GL}(2, \mathbb{Q}) Z(\mathbb{A}) \backslash \mathrm{GL}(2, \mathbb{A})} \varphi(g) \varphi'(g) E(g, s) \, dg, \]

where \(E(g, s)\) is a real-analytic Eisenstein series. This “unfolding” technique reduces the integral to a product of local zeta integrals.

4.4 The Langlands Dual Group

To state the Langlands conjectures, one needs the notion of the dual group.

4.5 The Local and Global Langlands Conjectures for \(\mathrm{GL}(2)\)

We can now state the Langlands conjectures in their simplest non-trivial case.

4.6 Connections to Artin \(L\)-functions

The Langlands program provides a conjectural framework for understanding Artin \(L\)-functions through automorphic representations.

This theorem, due to Langlands (dihedral and tetrahedral cases) and Tunnell (octahedral case), is a cornerstone of the proof of Fermat’s Last Theorem: it provides the automorphicity needed to start the modularity argument for residual representations. The remaining case — when the image of \(\rho\) modulo scalars is \(A_5\) (the icosahedral case) — was established by Khare-Wintenberger as a consequence of Serre’s modularity conjecture.

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Chapter 5: Automorphic Forms on \(\mathrm{GL}(n)\)

We now generalise the theory from \(\mathrm{GL}(2)\) to \(\mathrm{GL}(n)\) for arbitrary \(n\). This generalisation, developed primarily by Jacquet, Piatetski-Shapiro, and Shalika in the 1970s-80s, reveals new phenomena and deeper structures.

5.1 Cuspidal Representations of \(\mathrm{GL}(n, \mathbb{A})\)

For \(\mathrm{GL}(2)\), there is only one proper parabolic (the Borel), so the cuspidality condition is a single vanishing. For \(\mathrm{GL}(n)\), the standard parabolics correspond to partitions of \(n\), and all constant terms must vanish. In practice, by a theorem of Jacquet, it suffices to check the vanishing for the Borel subgroup alone.

5.2 Parabolic Induction and the Jacquet Functor

The constructions of Chapter 2 generalise to \(\mathrm{GL}(n)\) in a natural way.

Parabolic induction and the Jacquet functor are adjoint functors — this is the Frobenius reciprocity in this context:

\[ \operatorname{Hom}_G(\pi, \mathrm{Ind}_P^G \sigma) \cong \operatorname{Hom}_M(r_P^G(\pi), \sigma). \]

5.3 The Langlands Classification

Every irreducible admissible representation of \(\mathrm{GL}(n, F)\) can be built from supercuspidal representations via parabolic induction and a specific procedure involving “twists.”

For \(\mathrm{GL}(n)\), the tempered representations are themselves understood: every irreducible tempered representation is a full induced representation from a parabolic, with all inducing data being unitary supercuspidal representations twisted by unitary characters. This means the classification ultimately reduces to supercuspidal representations.

5.4 Eisenstein Series and Spectral Decomposition

The continuous spectrum of \(L^2(\mathrm{GL}(n, \mathbb{Q}) \backslash \mathrm{GL}(n, \mathbb{A})^1)\) is generated by Eisenstein series.

For \(\mathrm{GL}(n)\), the discrete spectrum equals the cuspidal spectrum (by the theorem of Mœglin and Waldspurger for the residual spectrum): the residual spectrum consists of the generalised Speh representations.

5.5 Rankin-Selberg Method for \(\mathrm{GL}(n) \times \mathrm{GL}(m)\)

The Rankin-Selberg method generalises beautifully to products of general linear groups.

The construction uses the integral representation

\[ I(s, \varphi, \varphi') = \begin{cases} \displaystyle\int_{\mathrm{GL}(m, \mathbb{Q}) \backslash \mathrm{GL}(m, \mathbb{A})} \varphi\begin{pmatrix} g & \\ & I_{n-m} \end{pmatrix} \varphi'(g) |\det g|^{s - (n-m)/2} \, dg & \text{if } n > m, \\[8pt] \displaystyle\int_{\mathrm{GL}(n, \mathbb{Q}) Z(\mathbb{A}) \backslash \mathrm{GL}(n, \mathbb{A})} \varphi(g) \varphi'(g) E(g, s) \, dg & \text{if } n = m. \end{cases} \]

5.6 The Strong Multiplicity One Theorem

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Chapter 6: The Langlands Program: General Framework

We now describe the Langlands program in its full generality, extending beyond \(\mathrm{GL}(n)\) to arbitrary reductive groups. This is the most ambitious framework in modern number theory, proposing deep connections between automorphic representations and Galois representations.

6.1 Reductive Groups over Number Fields

The structure theory of reductive groups is encoded in their root datum \(\Psi(G, T) = (X^*(T), \Phi, X_*(T), \Phi^\vee)\), where \(T\) is a maximal torus, \(X^*(T)\) and \(X_*(T)\) are its character and cocharacter lattices, and \(\Phi, \Phi^\vee\) are the roots and coroots.

6.2 The \(L\)-group

The formation of the \(L\)-group is a contravariant functor: a homomorphism \(G \to H\) does not generally induce a map \({}^LG \to {}^LH\), but rather depends on the relationship between the root data.

6.3 Langlands Parameters

6.4 The Langlands Functoriality Conjecture

The central conjecture of the Langlands program predicts that homomorphisms of \(L\)-groups give rise to correspondences between automorphic representations.

6.5 The Satake Isomorphism and Unramified Representations

The Satake isomorphism provides the crucial link between unramified representations and the \(L\)-group.

6.6 Examples of Functoriality: Symmetric Power \(L\)-functions

One of the most important applications of functoriality concerns the symmetric power \(L\)-functions of a cuspidal representation \(\pi\) of \(\mathrm{GL}(2)\).

The Langlands functoriality conjecture predicts that there exists an automorphic representation \(\mathrm{Sym}^n(\pi)\) of \(\mathrm{GL}(n+1, \mathbb{A})\) whose standard \(L\)-function equals \(L(s, \pi, \mathrm{Sym}^n)\). The following cases are known:

6.7 Base Change and Automorphic Induction

Two further important instances of functoriality are base change and automorphic induction.

These operations are fundamental tools for constructing automorphic representations and proving cases of functoriality. Base change for \(\mathrm{GL}(2)\) over cyclic extensions was proved by Langlands (1980) using the trace formula, and the general case for \(\mathrm{GL}(n)\) was proved by Arthur and Clozel (1989).

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Chapter 7: Applications and Modern Developments

The Langlands program has far-reaching consequences throughout number theory, algebraic geometry, and mathematical physics. This final chapter surveys some of the most spectacular applications and indicates directions of current research.

7.1 Modularity of Elliptic Curves

The most celebrated application of the Langlands program is the modularity theorem for elliptic curves over \(\mathbb{Q}\), which includes Fermat’s Last Theorem as a consequence.

The strategy of proof is a landmark in mathematics. We outline the key ideas.

Step 1: The Galois representation. For an elliptic curve \(E/\mathbb{Q}\) and a prime \(\ell\), the Tate module \(T_\ell(E) = \varprojlim E[\ell^n]\) gives a continuous representation

\[ \rho_{E, \ell} : \mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q}) \to \mathrm{GL}(2, \mathbb{Z}_\ell). \]

The modularity conjecture asserts that this representation is modular, i.e., arises from a weight-2 newform.

Step 2: The residual representation. One first considers the mod-\(\ell\) reduction \(\bar{\rho}_{E, \ell} : \mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q}) \to \mathrm{GL}(2, \mathbb{F}_\ell)\). For \(\ell = 3\), the Langlands-Tunnell theorem (which handles the solvable case) shows that \(\bar{\rho}_{E,3}\) is modular (when it is irreducible). For \(\ell = 5\), a 3-5 switching argument handles the case when \(\bar{\rho}_{E,3}\) is reducible.

Step 3: Modularity lifting. The Taylor-Wiles method shows that if \(\bar{\rho}_{E, \ell}\) is modular and satisfies suitable conditions, then \(\rho_{E, \ell}\) itself is modular. This uses deformation theory of Galois representations and the numerical coincidence between the dimensions of deformation rings and Hecke algebras.

7.2 The Sato-Tate Conjecture

7.3 The Ramanujan Conjecture

For \(\mathrm{GL}(2)\) over \(\mathbb{Q}\) and holomorphic cusp forms of weight \(k\), the Ramanujan conjecture states that \(|a_p| \leq 2p^{(k-1)/2}\) for primes \(p\) not dividing the level. This was proved by Deligne (1974) as a consequence of his proof of the Weil conjectures (specifically, the Riemann hypothesis for varieties over finite fields).

For \(\mathrm{GL}(n)\) with \(n \geq 3\), the generalised Ramanujan conjecture remains open in general. The best known bounds are:

7.4 The Arthur-Selberg Trace Formula

The trace formula is the primary tool for proving cases of functoriality. It originates in Selberg’s work on spectral theory of locally symmetric spaces and was vastly generalised by Arthur.

The trace formula equates spectral information (representations) with geometric information (conjugacy classes), and by comparing trace formulas for different groups, one can establish instances of functoriality.

7.5 Endoscopy

Endoscopy arises naturally from the trace formula when one attempts to compare trace formulas for different groups.

The comparison of trace formulas requires matching orbital integrals on \(G\) with “stable” orbital integrals on \(H\). This transfer of orbital integrals was conjectured by Langlands and Shelstad, and the required identities constitute the Fundamental Lemma.

The Fundamental Lemma, awarded the Fields Medal in 2010, was one of the most important technical advances in the Langlands program, as it cleared the path for the endoscopic classification of automorphic representations.

7.6 The Langlands-Shahidi Method

An alternative approach to proving analytic properties of \(L\)-functions is the Langlands-Shahidi method, which uses the constant terms of Eisenstein series.

This method has been remarkably successful: it applies to a large class of \(L\)-functions that cannot be accessed by the Rankin-Selberg method, and it was a key ingredient in the Kim-Shahidi proof of the symmetric cube and fourth power functoriality.

7.7 \(p\)-adic and mod-\(p\) Langlands (Preview)

The classical Langlands correspondence relates complex representations. A major direction of modern research extends this to \(p\)-adic and mod-\(p\) coefficients.

This correspondence is much richer than the classical one: the Banach space representations are infinite-dimensional and carry a wealth of additional structure. Extending it to \(\mathrm{GL}(n)\) with \(n \geq 3\) is a major open problem.

The mod-\(p\) Langlands correspondence similarly seeks to relate representations over \(\overline{\mathbb{F}}_p\):

7.8 The Geometric Langlands Program (Preview)

A remarkable parallel development replaces number fields with function fields of curves over algebraically closed fields.

7.9 Summary and Outlook

The Langlands program, initiated by Robert Langlands in a famous letter to André Weil in 1967, has grown into one of the most far-reaching frameworks in mathematics. Let us summarise the main threads:

1. The local Langlands correspondence for \(\mathrm{GL}(n)\) has been fully established by Harris-Taylor and Henniart (2001), relating irreducible admissible representations of \(\mathrm{GL}(n, F)\) to \(n\)-dimensional Weil-Deligne representations.

2. The global Langlands correspondence for \(\mathrm{GL}(n)\) over function fields was proved by Drinfeld (\(n = 2\)) and L. Lafforgue (\(n\) general). Over number fields, it remains conjectural in full generality, though the Taylor-Wiles method and its extensions have established many cases.

3. Functoriality remains the deepest conjecture. The trace formula approach, particularly Arthur’s endoscopic classification, has established functoriality for classical groups. The recent proof of symmetric power functoriality by Newton-Thorne opens new vistas.

4. Beyond \(\mathrm{GL}(n)\): Arthur’s work classifies the automorphic spectrum of symplectic and orthogonal groups; Mok and Kaletha-Minguez-Shin-White handle unitary groups; the exceptional groups remain largely open.

5. \(p\)-adic and geometric Langlands represent new frontiers that interact deeply with algebraic geometry, \(p\)-adic Hodge theory, and mathematical physics.

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Appendix: Key Notation and Conventions

Symbol	Meaning
\(\mathcal{H}\)	Upper half-plane \(\{z \in \mathbb{C} : \operatorname{Im}(z) > 0\}\)
\(\Gamma_0(N), \Gamma_1(N), \Gamma(N)\)	Standard congruence subgroups of \(\mathrm{SL}(2, \mathbb{Z})\)
\(M_k(\Gamma), S_k(\Gamma)\)	Modular forms, cusp forms of weight \(k\) for \(\Gamma\)
\(q = e^{2\pi i z}\)	Standard local parameter at the cusp \(\infty\)
\(\mathbb{A} = \mathbb{A}_\mathbb{Q}\)	Ring of adeles of \(\mathbb{Q}\)
\(\mathbb{A}_f\)	Finite adeles
\(K_v\)	Maximal compact subgroup of \(\mathrm{GL}(n, \mathbb{Q}_v)\)
\({}^LG = \hat{G} \rtimes W_F\)	\(L\)-group of \(G\)
\(W_F\)	Weil group of \(F\)
\(W_F' = W_F \times \mathrm{SL}(2, \mathbb{C})\)	Weil-Deligne group
\(\tilde{\pi}\)	Contragredient representation of \(\pi\)
\(L(s, \pi)\)	Standard \(L\)-function of \(\pi\)
\(\epsilon(s, \pi, \psi)\)	Local \(\epsilon\)-factor
\(\Lambda(s, \pi)\)	Completed \(L\)-function