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Automorphic form

In number theory and representation theory, an automorphic form is a smooth function \phi: G(\mathbb{A}_F) \to \mathbb{C} on the adelic points of a reductive algebraic group G over a number field F, which is invariant under the action of G(F), right-invariant under a maximal compact subgroup K, Z(\mathfrak{g}_\mathbb{C})-finite, with a continuous central character, and of moderate growth at infinity.^[1] These functions generalize classical modular forms, which are holomorphic automorphic forms of weight k on GL_2 over \mathbb{Q}, transforming under the modular group SL_2(\mathbb{Z}) via the slash operator f|_k \gamma(z) = j(\gamma, z)^{-k} f(\gamma z) for \gamma \in SL_2(\mathbb{Z}), where j(\gamma, z) = cz + d.^[2] Automorphic forms trace their origins to the late 19th century, when Henri Poincaré introduced them as analytic functions invariant under discontinuous group actions on the complex plane, extending the concept of periodic functions to Fuchsian groups and laying foundational work in the theory of automorphic functions.^[3] This early development, detailed in Poincaré's papers from 1881–1883, connected such forms to Riemann surfaces and uniformization, influencing subsequent advances in complex analysis and geometry.^[4] The modern adelic framework emerged in the mid-20th century through the works of André Weil and Harish-Chandra, who reformulated automorphic forms in terms of representations of adelic groups to unify global and local aspects.^[5] Automorphic forms are central to the Langlands program, which conjectures deep correspondences between automorphic representations—irreducible constituents of the space of automorphic forms—and Galois representations of the absolute Galois group of the number field, facilitating the study of L-functions and arithmetic objects like elliptic curves.^[1] Key examples include Eisenstein series, which generate non-cuspidal automorphic forms via summation over cosets, and cusp forms like the discriminant modular form \Delta(z) = q \prod_{n=1}^\infty (1 - q^n)^{24} that vanish at the cusps of the fundamental domain.^[2]^[6] Their associated L-functions encode arithmetic data, such as prime distributions, and have applications in proving results like the modularity theorem linking elliptic curves to modular forms.^[7]

Definition and Fundamentals

Formal Definition

Automorphic forms are defined as certain smooth functions on quotient spaces such as G(\mathbb{Q}) \backslash G(\mathbb{A}), where G is a reductive algebraic group over \mathbb{Q} and \mathbb{A} denotes the adele ring of \mathbb{Q}, that satisfy specific transformation properties under the left action of G(\mathbb{Q}).^[8] These functions \phi: G(\mathbb{A}) \to \mathbb{C} are required to be left-invariant under G(\mathbb{Q}), meaning \phi(\gamma g) = \phi(g) for all \gamma \in G(\mathbb{Q}) and g \in G(\mathbb{A}), and smooth with respect to the adelic topology, while also being right K-finite for a maximal compact subgroup K \subset G(\mathbb{A}) and of moderate growth, satisfying |\phi(g)| \leq C \|g\|^N for some constants C, N > 0 and all g \in G(\mathbb{A}).^[9] Additionally, they are Z(\mathfrak{g})-finite at the archimedean places, where Z(\mathfrak{g}) is the center of the universal enveloping algebra, ensuring the functions generate finite-dimensional spaces under differential operators.^[8] In the classical setting for G = \mathrm{SL}_2, automorphic forms on the upper half-plane \mathbb{H} are functions f: \mathbb{H} \to \mathbb{C} that transform under the action of \mathrm{SL}_2(\mathbb{Z}) via f\left( \frac{az + b}{cz + d} \right) = (cz + d)^k f(z) for all \gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathrm{SL}_2(\mathbb{Z}) and integer weight k, with the automorphy factor j(\gamma, z) = cz + d.^[9] They exhibit moderate growth at the cusps, such as |f(z)| \leq C (1 + y)^k where y = \mathrm{Im}(z), and admit a Fourier expansion f(z) = \sum_{n \in \mathbb{Z}} a_n e^{2\pi i n z} with only finitely many nonzero coefficients a_n for n < 0.^[9] Holomorphic automorphic forms are those that are holomorphic on \mathbb{H} and at the cusps (after suitable transformations), while non-holomorphic ones, such as Maass forms, are real-analytic eigenfunctions of the hyperbolic Laplacian \Delta = -y^2 \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} \right) with eigenvalue \lambda = s(1-s), satisfying the same transformation and growth conditions but without holomorphy. This framework of automorphic forms is intimately connected to the abstract notion of automorphic representations, which provide a representation-theoretic perspective on these functions as matrix coefficients of irreducible unitary representations of G(\mathbb{A}).^[8]

Classical vs. Adelic Perspectives

The classical perspective on automorphic forms originates in the study of functions on the upper half-plane \mathbb{H}, which is invariant under the action of discrete subgroups such as \mathrm{SL}(2, \mathbb{Z}). Specifically, a classical automorphic form of weight k and level N is a holomorphic function f: \mathbb{H} \to \mathbb{C} satisfying f(\gamma z) = (cz + d)^k f(z) for all \gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \Gamma_0(N), where \Gamma_0(N) is the congruence subgroup consisting of matrices with c \equiv 0 \pmod{N}, along with moderate growth conditions at the cusps.^[10] This setup emphasizes the geometric and analytic properties tied to the hyperbolic plane and arithmetic subgroups.^[11] In contrast, the adelic formulation generalizes this to functions on the adelic quotient G(\mathbb{A})/G(\mathbb{Q}), where G = \mathrm{GL}_2 and \mathbb{A} denotes the adele ring over \mathbb{Q}, comprising the real numbers \mathbb{R} at the infinite place and the finite adeles \mathbb{A}_f = \prod'_p \mathbb{Q}_p. An adelic automorphic form \phi: G(\mathbb{A}) \to \mathbb{C} is smooth, satisfying \phi(\gamma g) = \phi(g) for \gamma \in G(\mathbb{Q}) and g \in G(\mathbb{A}), with right invariance under a compact open subgroup K_f \subseteq G(\mathbb{A}_f) (such as K_0(N) = \{ g \in G(\hat{\mathbb{Z}}) \mid g \equiv \begin{pmatrix} * & * \\ 0 & * \end{pmatrix} \pmod{N} \}) and suitable growth conditions at infinity.^[11]^[12] The two perspectives are equivalent through an explicit embedding that leverages strong approximation for \mathrm{GL}_2(\mathbb{Q}). Classical forms of level N and weight k correspond bijectively to adelic forms via the map sending f(z) to \phi_f(g) = F(g_\infty) \lambda(k_f), where g = \gamma g_\infty k_f decomposes using strong approximation \mathrm{GL}_2(\mathbb{A}) = \mathrm{GL}_2(\mathbb{Q}) \cdot \mathrm{GL}_2^+(\mathbb{R}) \cdot K_0(N), F is the automorphy factor on the infinite component, and \lambda encodes the nebentypus character locally. This induces an isomorphism \Gamma_0(N) \backslash \mathrm{GL}_2^+(\mathbb{R}) \cong \mathrm{GL}_2(\mathbb{Q}) \backslash \mathrm{GL}_2(\mathbb{A}) / K_0(N), preserving Hecke actions and ensuring classical holomorphic forms embed as those generating the discrete series representation of weight k at infinity.^[10]^[11]^[12] The adelic viewpoint offers significant advantages, particularly in unifying local and global aspects through the product structure over places. It facilitates the treatment of local-global principles, such as those arising in the Langlands program, by allowing automorphic forms to decompose into local components \phi_v at each prime p (including p-adic completions \mathbb{Q}_p), where unramified behavior at most places simplifies computations via Satake parameters. This contrasts with the classical approach, which is inherently tied to the archimedean place and requires separate handling of finite-level conditions, making the adelic framework more amenable to generalization over number fields and reductive groups beyond \mathrm{GL}_2.^[13]^[12]

Historical Development

Early Origins in Function Theory

The early conceptual foundations of automorphic forms trace back to the development of elliptic functions in the 1820s and 1830s by Niels Henrik Abel and Carl Gustav Jacob Jacobi. Abel's groundbreaking work, particularly in his 1827–1828 memoir Recherches sur les fonctions elliptiques published in Crelle's Journal, introduced the inversion of elliptic integrals to define doubly periodic meromorphic functions on the complex plane, characterized by two fundamental periods forming a lattice \Lambda = \mathbb{Z} \omega_1 + \mathbb{Z} \omega_2.^[14] These functions, such as the Weierstrass \wp-function, satisfied addition theorems and exhibited invariance under translations by lattice elements, laying the groundwork for understanding functions tied to discrete group symmetries.^[14] Jacobi advanced this theory significantly with his 1829 treatise Fundamenta Nova Theoriae Ellipticarum, the first systematic exposition of elliptic functions employing complex variable methods to simplify integrals involving square roots of cubic or quartic polynomials.^[15] Building on Abel's inversion techniques, Jacobi emphasized the double periodicity and developed expressions for elliptic functions in terms of trigonometric substitutions, highlighting their role in solving physical problems through differential equations.^[15] His work also explored transformations that preserved the functional form, connecting elliptic functions to broader invariance properties under group actions. A key early example illustrating these transformation properties is the Jacobi theta function, introduced in the 1829 treatise. Defined as

\vartheta(\tau) = \sum_{n=-\infty}^{\infty} \exp(\pi i n^2 \tau)

for \tau in the upper half-plane \mathbb{H}, this entire function satisfies the modular transformation law

\vartheta\left(-\frac{1}{\tau}\right) = \sqrt{-i \tau} \, \vartheta(\tau),

where the square root uses the principal branch of the logarithm, alongside the periodicity \vartheta(\tau + 2) = \vartheta(\tau).^[16] These relations demonstrate invariance under the generators of the modular group \mathrm{SL}(2, \mathbb{Z}), making the theta function an automorphic object of weight $1/2 on the congruence subgroup \Gamma_0(4).^[16] The shift from purely periodic functions to automorphic functions arose by generalizing discrete group actions: single-periodic functions like \exp(2\pi i z) are automorphic under the translation group \mathbb{Z}, while elliptic functions are automorphic under two-dimensional lattices as discrete subgroups of \mathbb{C}^\times.^[17] This perspective extended naturally to non-compact domains, where discontinuous actions of discrete groups produce quotient spaces with rich geometric structure, paving the way for functions invariant under such symmetries.^[17] Henri Poincaré advanced this framework decisively in his 1882 paper Théorie des groupes fuchsiennes, published in Acta Mathematica, where he defined Fuchsian groups as discrete subgroups of \mathrm{PSL}(2, \mathbb{R}) acting properly discontinuously on the upper half-plane or unit disk.^[18] Poincaré developed a combinatorial-geometric method to classify these groups and showed that they enable the uniformization of arbitrary Riemann surfaces, representing them as quotients \mathbb{H}/\Gamma for suitable Fuchsian \Gamma.^[19] His study of meromorphic functions invariant under Fuchsian actions—termed Fuchsian functions—directly generalized elliptic functions to hyperbolic geometries, establishing the core idea of automorphy through discrete group invariance.^[18]

Poincaré's Contributions

In the early 1880s, Henri Poincaré made groundbreaking contributions to the theory of automorphic functions by introducing Fuchsian functions, which generalize elliptic functions to settings involving non-Euclidean geometries. In his 1882 memoir "Mémoire sur les fonctions fuchsiennes," published in Acta Mathematica, Poincaré defined these functions as meromorphic functions on the upper half-plane that remain invariant under the action of a Fuchsian group—a discrete subgroup of the projective special linear group PSL(2, ℝ).^[20] This discovery arose from his realization that the linear fractional transformations preserving such functions coincide precisely with the isometries of hyperbolic geometry, thereby bridging complex analysis and non-Euclidean structures.^[21] Poincaré's insight, which struck him suddenly during a journey in 1881, extended the classical theory of elliptic functions—rooted in doubly periodic behaviors—to singly periodic or aperiodic cases adapted to hyperbolic planes.^[22] A key tool in Poincaré's framework was the Poincaré series, an infinite summation method for constructing automorphic forms from seed functions. These series take the form

\sum_{\gamma \in \Gamma} \chi(\gamma) \, j(\gamma, z)^{-k} \, f(\gamma z),

where \Gamma is a Fuchsian group, \chi is a character on \Gamma, j(\gamma, z) is the automorphic factor (typically cz + d for \gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix}), k is the weight, and f is a suitable holomorphic function ensuring convergence.^[20] Introduced in his 1880s papers, such as those in Acta Mathematica (1882–1884), the series provided a systematic way to generate Fuchsian functions by summing over group orbits, with convergence analyzed via estimates on the group's action.^[22] This construction not only facilitated explicit examples but also laid analytic foundations for studying the spectral theory of automorphic forms, influencing later developments in representation theory.^[20] Poincaré's work culminated in applications to the uniformization of Riemann surfaces, where automorphic functions play a central role in classifying conformal structures. In 1907, independently of Paul Koebe, Poincaré proved the uniformization theorem, asserting that every simply connected Riemann surface is conformally equivalent to one of three canonical domains: the Riemann sphere, the complex plane, or the unit disk.^[23] His proof, detailed in "Sur l'uniformisation des fonctions analytiques" (Acta Mathematica, 1907), employed potential theory and automorphic functions to construct the required mappings, showing how Fuchsian groups uniformize punctured surfaces via quotients of the disk.^[23] This result resolved a longstanding conjecture by Felix Klein and Henri Poincaré himself from the 1880s, demonstrating the profound geometric impact of automorphic functions on the classification of Riemann surfaces.^[22] As a specific example, Poincaré explored theta series associated with quadratic forms to generate automorphic functions, particularly linking arithmetic transformations of indefinite ternary quadratic forms to non-Euclidean geometries. In his 1880s investigations, he constructed such series by summing exponentials over lattice points defined by quadratic forms, yielding functions invariant under the corresponding orthogonal groups acting on hyperbolic space.^[21] This approach, inspired by a second sudden insight during a seaside walk, provided early examples of automorphic forms arising from number-theoretic data, foreshadowing connections between quadratic forms and modular phenomena.^[21]

Modern Evolution

In the 1930s, Erich Hecke introduced Hecke operators as linear operators acting on spaces of modular forms, establishing a profound correspondence between these forms and Dirichlet L-series, which generalized earlier analytical approaches and laid the foundation for modern arithmetic interpretations of automorphic forms. This development shifted the focus from purely analytical properties to algebraic structures, enabling the study of eigenvalues and multiplicities in the theory of modular forms. Hecke's operators became central tools for decomposing spaces of forms into eigenspaces, influencing subsequent advancements in number theory.^[24] The 1940s and 1950s saw Hans Maass extend the scope of automorphic forms beyond holomorphic examples by introducing non-holomorphic variants, now called Maass forms, constructed as eigenfunctions of the Laplace–Beltrami operator on the hyperbolic plane that remain invariant under the action of the modular group SL(2, ℤ). These forms, first detailed in Maass's 1949 paper, allowed for the association of non-holomorphic functions with L-functions, broadening the analytical framework to include real-analytic objects and providing new insights into spectral theory on arithmetic quotients.^[25] This innovation complemented the holomorphic case and facilitated connections to eigenvalue problems in hyperbolic geometry. By the 1960s, the adelic perspective revolutionized the field, with André Weil, Harish-Chandra, and John Tate developing the framework of adeles and ideles to reformulate automorphic forms in a global, non-archimedean setting that unified local and global behaviors. Weil's work emphasized the role of algebraic groups over adele rings, Harish-Chandra's contributions advanced the representation-theoretic foundations for automorphic forms on semisimple groups, while Tate's work linked this adelization to class field theory, enabling the precise description of automorphic representations via induced constructions and intertwining operators. This shift from classical upper half-plane models to adelic quotients G(ℚ)\G(𝔸) provided a uniform language for general reductive groups, bridging analytic number theory with algebraic geometry.^[26] The 1970s and 1980s marked a pivotal era with Robert Langlands's formulation of functoriality conjectures, which posit the existence of transfers (lifts) between automorphic representations on different algebraic groups, thereby connecting them to Galois representations and extending reciprocity laws beyond abelian cases. These conjectures, outlined in Langlands's 1970 monograph, aimed to establish a non-abelian class field theory through L-functions attached to automorphic forms.^[27] A landmark achievement within this context was Pierre Deligne's 1974 proof of the Ramanujan conjecture for holomorphic cusp forms, demonstrating that the Fourier coefficients satisfy |a_p| ≤ 2p^{(k-1)/2} for prime p, derived as a corollary to his resolution of the Weil conjectures on étale cohomology of varieties over finite fields.^[28] This result confirmed the boundedness of Hecke eigenvalues and solidified the arithmetic significance of automorphic forms up to the late 20th century.

Automorphic Representations

Core Concepts

Automorphic representations form the foundational framework in the representation-theoretic study of automorphic forms, viewing them as irreducible components in the unitary dual of the adelic group G(\mathbb{A}), where G is a reductive algebraic group over a number field F (often \mathbb{Q}) and \mathbb{A} denotes the adele ring of F. Specifically, an automorphic representation \pi is an irreducible admissible unitary representation of G(\mathbb{A}) that appears as a direct summand in the decomposition of the right regular representation on the Hilbert space L^2(G(F) \backslash G(\mathbb{A})), generated by the action on matrix coefficients of automorphic forms. These representations are realized on spaces of functions satisfying the automorphy condition under the left action of G(F).^[29] In the adelic perspective, this setup unifies classical and modern approaches by embedding the theory within the broader structure of representations of adelic groups.^[1] A key structural property is that every automorphic representation decomposes as a restricted tensor product \pi = \otimes'_v \pi_v over all places v of F, where each \pi_v is an irreducible admissible representation of the local group G(F_v), and \pi_v is unramified (i.e., has a nonzero vector fixed by a hyperspecial maximal compact subgroup) for all but finitely many finite places v.^[30] This local-global decomposition reflects the product structure of the adele ring and allows the global representation to be reconstructed from its local constituents. The space of smooth vectors \pi^\infty in \pi, consisting of vectors with open stabilizers under G(\mathbb{A}), carries a Fréchet topology and admits an action by the global Hecke algebra \mathcal{H}(G(\mathbb{A}), G(F)), generated by smooth, compactly supported, bi-G(F)-invariant functions. At non-archimedean places, this action is realized via convolution with Schwartz-Bruhat functions on G(F_v), ensuring admissibility by bounding the dimensions of fixed subspaces under compact open subgroups.^[1] The automorphy condition is encoded in the transformation properties of the matrix coefficients of \pi. For vectors \phi, \psi \in \pi, the matrix coefficient function \mu_{\phi, \psi}(g) = \int_{G(\mathbb{A})} \phi(h) \overline{\psi(h^{-1} g)} \, dh (or variants thereof) satisfies \mu_{\phi, \psi}(\gamma g) = \mu_{\phi, \psi}(g) for all \gamma \in G(F) and g \in G(\mathbb{A}), reflecting the G(F)-invariance essential to the quotient space.^[29] This ensures that the coefficients descend to well-defined functions on G(F) \backslash G(\mathbb{A}) and generate the representation as a subspace of automorphic forms.^[30] Associated to each irreducible automorphic representation \pi is a central character \chi_\pi, a continuous unitary multiplicative character of the center Z(\mathbb{A}) of G(\mathbb{A}) such that \pi(z) \xi = \chi_\pi(z) \xi for all z \in Z(\mathbb{A}) and smooth vectors \xi \in \pi^\infty. For compatibility with the automorphy under G(F), \chi_\pi satisfies \chi_\pi(z g) = \chi_\pi(z) for all z \in Z(\mathbb{A}) and g \in G(F), implying that \chi_\pi is trivial on the image of Z(F) in Z(\mathbb{A}). This character governs the action of the center and often corresponds to a grossencharacter or Hecke character in classical settings.^[1]

Types of Representations

Automorphic representations are classified based on their occurrence in the spectral decomposition of the space L^2(G(\mathbb{Q}) \backslash G(\mathbb{A})), where G is a reductive algebraic group over \mathbb{Q}. This space decomposes orthogonally into a discrete spectrum and a continuous spectrum. The discrete spectrum consists of the cuspidal spectrum and the residual spectrum, while the continuous spectrum is generated by induced representations from proper parabolic subgroups.^[31]^[32] Cuspidal representations form the core of the cuspidal spectrum. An irreducible unitary automorphic representation \pi of G(\mathbb{A}) is cuspidal if, for every vector \phi \in V_\pi (the underlying space of \pi) and every proper parabolic subgroup P = MN \subset G with unipotent radical N, the constant term

\int_{N(\mathbb{A})} \phi(ng) \, dn = 0

for all g \in G(\mathbb{A}), where dn is the Tamagawa measure on N(\mathbb{A}). This condition implies that the Fourier expansion of functions in V_\pi along any proper parabolic has no nonzero constant term, ensuring square-integrable matrix coefficients modulo the center and rapid decay at the cusps. Cuspidal representations correspond to cusp forms and embed densely into the cuspidal subspace L^2_{\mathrm{cusp}}(G(\mathbb{Q}) \backslash G(\mathbb{A})).^[31]^[32] The residual spectrum comprises the remaining irreducible unitary representations in the discrete spectrum that are not cuspidal. These arise as complementary factors in the discrete part after extracting the cuspidal spectrum and are characterized by nonzero constant terms along certain parabolic subgroups, though they still contribute to the square-integrable part of L^2. Unlike cuspidal representations, residual ones do not vanish at all cusps but form a finite-dimensional complement in the decomposition.^[31]^[32] Induced representations generate the continuous spectrum and are constructed via parabolic induction from proper parabolic subgroups. For a parabolic subgroup P = MN \subset G with Levi component M and unipotent radical N, an induced representation is given by

\pi = \mathrm{Ind}_{P(\mathbb{A})}^{G(\mathbb{A})} (\sigma \otimes \chi),

where \sigma is an automorphic representation of M(\mathbb{A}) (typically tempered and unitary) and \chi is a unitary character of N(\mathbb{A}). The induction is normalized by the modulus character \delta_P^{1/2} to preserve unitarity, and the resulting representation is irreducible under generic conditions on \sigma and \chi. These representations do not appear discretely but contribute to the direct integral decomposition of the continuous spectrum, reflecting non-square-integrable behavior.^[31]^[32]

Key Examples

Modular Forms

Modular forms represent a classical instance of holomorphic automorphic forms on the group GL(2) over the rationals, specifically arising as functions on the upper half-plane \mathbb{H} = \{ z \in \mathbb{C} : \operatorname{Im}(z) > 0 \} that satisfy certain transformation properties under the action of congruence subgroups of SL(2, \mathbb{Z}).^[33] A modular form of weight k \in 2\mathbb{Z}_{\geq 0} and level N \geq 1 with respect to the congruence subgroup \Gamma_0(N) = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{SL}(2, \mathbb{Z}) : c \equiv 0 \pmod{N} \right\} and a Dirichlet character \varepsilon: (\mathbb{Z}/N\mathbb{Z})^\times \to \mathbb{C}^\times is a holomorphic function f: \mathbb{H} \to \mathbb{C} such that for all \gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \Gamma_0(N),

f(\gamma z) = \varepsilon(\gamma) j(\gamma, z)^k f(z), \quad z \in \mathbb{H},

where j(\gamma, z) = cz + d is the standard automorphy factor.^[33] Additionally, f must extend holomorphically to the cusps of the modular curve associated to \Gamma_0(N), meaning its Fourier expansion at each cusp has no negative powers of the nome q = e^{2\pi i z / h} for some width h at that cusp.^[33] The space M_k(\Gamma_0(N), \varepsilon) of such modular forms forms a finite-dimensional complex vector space, with dimension given by formulas involving the genus of the modular curve X_0(N). The subspace S_k(\Gamma_0(N), \varepsilon) of cusp forms consists of those modular forms that vanish at all cusps, i.e., the constant term in their Fourier expansions at every cusp is zero; these are the holomorphic automorphic forms proper in this context.^[33] Hecke operators provide key linear endomorphisms on these spaces, acting multiplicatively on the Fourier coefficients. For a positive integer n, the Hecke operator T_n on M_k(\Gamma_0(N), \varepsilon) is defined via the double coset operator

(T_n f)(z) = \sum_{\gamma \in \Gamma_0(N) \backslash \Gamma_0(N) \alpha_n \Gamma_0(N)} f \big| _k \gamma (z),

where \alpha_n runs over matrices representing the double coset \Gamma_0(N) \begin{pmatrix} * & * \\ * & n \end{pmatrix} \Gamma_0(N) with determinant n, and the slash operator is (f \big| _k \gamma)(z) = j(\gamma, z)^{-k} f(\gamma z).^[34] These operators commute with each other and with the action of \Gamma_0(N), and they normalize the subspace of cusp forms; a normalized Hecke eigenform f satisfies T_n f = \lambda_n f for eigenvalues \lambda_n that are the nth Fourier coefficient of f.^[34] A prototypical example is the Ramanujan \Delta-function, the unique cusp form in S_{12}(\Gamma_0(1), \varepsilon_1) (where \varepsilon_1 is the trivial character and \Gamma_0(1) = \operatorname{SL}(2, \mathbb{Z})), given by the Fourier expansion

\Delta(z) = \sum_{n=1}^\infty \tau(n) q^n = q \prod_{m=1}^\infty (1 - q^m)^{24}, \quad q = e^{2\pi i z},

with Fourier coefficients \tau(n) known as the Ramanujan tau function; it generates the ring of modular forms for \operatorname{SL}(2, \mathbb{Z}) when adjoined to the Eisenstein series of weight 4 and 6.

Maass Forms

Maass forms are non-holomorphic automorphic forms on the upper half-plane \mathbb{H}, defined as real-analytic functions f: \mathbb{H} \to \mathbb{C} that are invariant under the action of a Fuchsian group \Gamma \subseteq \mathrm{SL}_2(\mathbb{Z}), satisfy the eigenvalue equation for the hyperbolic Laplacian \Delta f = s(1-s) f where \Delta = -y^2 \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} \right), exhibit moderate growth at the cusps, and vanish at the cusps for cusp forms (i.e., f(z) \to 0 exponentially fast as \mathrm{Im}(z) \to \infty along cusp directions).^[35] These forms generalize holomorphic modular forms to the weight-zero case, replacing holomorphy with Laplacian eigenfunction properties.^[25] In the spectral theory of automorphic forms, Maass cusp forms embed into the discrete spectrum of the unitary representation of \Gamma \backslash \mathbb{H} in L^2(\Gamma \backslash \mathbb{H}, d\mu) with hyperbolic measure d\mu = y^{-2} dx dy. The eigenvalues are \lambda = s(1-s) = \frac{1}{4} + r^2 for spectral parameter r \in \mathbb{R}, ensuring \lambda \geq \frac{1}{4} and distinguishing cusp forms from the continuous spectrum contributed by Eisenstein series. This spectral embedding allows Maass forms to form an orthonormal basis for the cusp subspace, facilitating applications in trace formulas and eigenvalue distribution studies.^[35]^[25] The Fourier expansion of a Maass form f(z) = f(x + iy) along the cusp at infinity takes the form f(z) = \sum_{n \in \mathbb{Z}} \rho_j(n) W\left( y^{1/2} e^{2\pi i x / j} \right) in its Whittaker model, where W denotes the Whittaker function (asymptotically related to modified Bessel functions K_{ir}(2\pi |n| y)) and \rho_j(n) are Fourier coefficients capturing arithmetic data; for the full modular group \Gamma = \mathrm{[SL](/page/SL)}_2(\mathbb{Z}), j=1. This expansion arises from the action of the unipotent radical and provides a decomposition into characters, enabling the study of Hecke operators and L-functions attached to the form.^[35] A representative example is the non-holomorphic Eisenstein series E(z, s) = \sum_{\gamma \in \Gamma_\infty \backslash \Gamma} \mathrm{Im}(\gamma z)^s, where \Gamma_\infty is the stabilizer of infinity; this satisfies \Delta E(\cdot, s) = s(1-s) E(\cdot, s) and generates the continuous spectrum, with its Fourier expansion involving terms y^s + \phi(s) y^{1-s} + 2 y^{1/2} \sum_{n=1}^\infty \sigma_{1-2s}(n) K_{s-1/2}(2\pi n y) \cos(2\pi n x), where \phi(s) incorporates zeta values and \sigma denotes the divisor function. Unlike cusp forms, it exhibits logarithmic growth at cusps but remains bounded away from poles.^[35]^[25]

Applications and Connections

Links to L-Functions

Automorphic forms give rise to associated L-functions through their Fourier coefficients. For a cuspidal automorphic representation \pi on \mathrm{[GL](/page/GL)}_n(\mathbb{A}) corresponding to an automorphic form f, the standard L-function is constructed as the Dirichlet series L(s, \pi) = \sum_{n=1}^\infty a_n n^{-s}, where a_n are the normalized Fourier coefficients of f, and this series converges absolutely in the half-plane \mathrm{Re}(s) > 1.^[36] This L-function admits an Euler product decomposition L(s, \pi) = \prod_p L_p(s, \pi), where the local factors L_p(s, \pi) at each prime p are defined via the Satake parameters of the unramified local component \pi_p, typically of the form L_p(s, \pi) = \prod_{j=1}^n (1 - \alpha_{p,j} p^{-s})^{-1} for unramified p.^[36] The L-function L(s, \pi) extends meromorphically to the entire complex plane and satisfies a functional equation of the form L(s, \pi) = \varepsilon(s, \pi) L(1-s, \tilde{\pi}), where \varepsilon(s, \pi) incorporates root number and Gamma factors such as products of \Gamma\left(\frac{s + \mu_j}{2}\right) with shifts \mu_j depending on the representation; for holomorphic forms of weight k on \mathrm{[GL](/page/GL)}_2, these are \Gamma\left( \frac{s + k - 1}{2} \right) \Gamma\left( \frac{s - k + 1}{2} \right). Automorphic L-functions are members of the Selberg class of L-functions, which axiomatizes Dirichlet series with Euler products, analytic continuation, functional equations, and bounded coefficients; specifically, they have degree n corresponding to \mathrm{GL}_n.^[37] The Ramanujan conjecture posits that the normalized coefficients satisfy |a_p| \leq n at primes p (for unramified p), ensuring the L-function belongs to the Selberg class under this bound, though it remains unproven in general.^[37]

Role in the Langlands Program

Automorphic forms occupy a pivotal position in the Langlands program, a vast conjectural framework linking representation theory, algebraic number theory, and geometry. At its heart lies the Langlands correspondence, which posits a bijection between irreducible automorphic representations of the adele group G(\mathbb{A}) for a reductive algebraic group G over \mathbb{Q}, and continuous homomorphisms \rho: \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to {}^L G, where {}^L G denotes the Langlands dual group. This bijection is anticipated to equate key arithmetic invariants, such as local parameters at unramified places and global L-functions, thereby unifying the spectral theory of automorphic forms with Galois representations.^[38]^[39] A foundational instance of this correspondence appears in the reciprocity conjecture for the general linear group \mathrm{GL}_n, where automorphic forms are conjectured to parametrize n-dimensional Galois representations. Specifically, it asserts a bijection between cuspidal automorphic representations \pi of \mathrm{GL}_n(\mathbb{A}) and irreducible n-dimensional representations \rho: \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \mathrm{GL}_n(\mathbb{C}), such that the local components match via the local Langlands correspondence and the associated L-functions coincide. This parametrization extends Artin's reciprocity to higher dimensions, providing a dictionary between the analytic properties of automorphic forms and the arithmetic of Galois groups.^[40] Complementing the correspondence is the principle of functoriality, which conjectures that every homomorphism {}^L H \to {}^L G between the L-groups of reductive groups H and G over \mathbb{Q} induces a transfer map on automorphic representations, sending irreducible representations of H(\mathbb{A}) to those of G(\mathbb{A}). This map preserves the unramified local parameters and enables the lifting of automorphic forms across groups, facilitating proofs of multiplicity-one theorems and stability of trace formulas. Functoriality thus serves as a mechanism for relating representations on different groups, underpinning many instances of the Langlands program.^[41]^[42] A landmark advancement in this framework occurred in 2008 with Ngô Bao Châu's proof of the fundamental lemma, a non-trivial identity comparing orbital integrals in the theory of endoscopy for reductive groups. This result, obtained via the geometry of the Hitchin fibration and affine Springer fibers, resolved a long-standing obstacle to stabilizing the trace formula and advancing the geometric Langlands program, thereby strengthening the links between automorphic representations and their Galois counterparts. Ngô's work earned him the Fields Medal in 2010 and has catalyzed further progress in establishing functoriality for classical groups.^[43]

Arithmetic Geometry Implications

Automorphic forms play a pivotal role in arithmetic geometry by establishing deep connections between analytic objects and geometric structures such as elliptic curves and higher-dimensional abelian varieties. A landmark result is the modularity theorem, which asserts that every elliptic curve over the rational numbers \mathbb{Q} corresponds to a cusp form of weight 2 on the modular group \mathrm{GL}_2(\mathbb{A}_\mathbb{Q}), where \mathbb{A}_\mathbb{Q} denotes the adele ring of \mathbb{Q}. This correspondence, proved by Breuil, Conrad, Diamond, and Taylor in 2001, not only resolves the Taniyama-Shimura conjecture but also implies key properties for the reduction behavior of elliptic curves at primes of bad reduction, facilitating the study of their semistable models in Diophantine contexts.^[44] Shimura varieties provide a geometric framework where automorphic forms manifest as cohomology classes parametrizing families of abelian varieties. These varieties are coarse moduli spaces that classify principally polarized abelian varieties of a given dimension equipped with additional automorphic level structures, such as actions by endomorphism rings or Hecke correspondences derived from automorphic representations on reductive groups over \mathbb{Q}. As detailed in foundational work by Shimura and further developed by Deligne, Shimura varieties of Hodge type embed into the moduli stack of abelian varieties with tensor structures, allowing automorphic forms to encode the étale cohomology of these spaces and thereby link number-theoretic invariants to geometric invariants like the Néron models of abelian varieties.^[45] The Langlands-Kottwitz conjecture posits that automorphic forms on general linear groups classify the motive structures underlying the cohomology of Shimura varieties, providing a bridge between automorphic representations and pure motives in the sense of Grothendieck. Specifically, it conjectures that the \ell-adic cohomology groups of a Shimura variety decompose into a direct sum of motives associated to cuspidal automorphic representations, with the Hecke eigenvalues of these representations governing the Frobenius action on the cohomology. This framework, originating from Langlands' visionary essay and refined by Kottwitz's trace formula applications, enables the geometric realization of automorphic forms as regulators of motivic cohomology, with implications for Diophantine problems such as the distribution of rational points on varieties.^[46] A striking application of these implications is the proof of the Sato-Tate conjecture for non-CM elliptic curves over \mathbb{Q}, achieved by Clozel, Harris, and Taylor in 2008 using automorphic lifting techniques. The conjecture predicts that the angles of the Frobenius traces on the Tate module of an elliptic curve, normalized appropriately, follow the distribution of the trace of a random element in \mathrm{SU}(2). By establishing the automorphy of symmetric powers of the associated Galois representation via cohomology computations on unitary Shimura varieties, the proof translates the Sato-Tate equidistribution into a consequence of the Weyl law for automorphic forms, thereby confirming the conjectured statistical behavior through geometric means.

Advanced Structures

Eisenstein Series

Eisenstein series constitute a primary class of non-cuspidal automorphic forms, obtained by parabolic induction from characters or representations on the Levi subgroup of a parabolic subgroup.^[47] In the adelic framework for a reductive algebraic group G over \mathbb{Q}, consider a maximal parabolic subgroup P = MN with unipotent radical N and Levi factor M. Given a section \phi: G(\mathbb{A}) \to \mathbb{C} in the space of the induced representation \mathrm{Ind}_{P(\mathbb{A})}^{G(\mathbb{A})} ( \delta_P^{1/2} \chi_M |\det_M|^{s} ), where \delta_P is the modular function of P and \chi_M a character of M(\mathbb{A}), the Eisenstein series is defined by

E(g, s) = \sum_{\gamma \in P(\mathbb{Q}) \backslash G(\mathbb{Q})} \phi(\gamma g)

for g \in G(\mathbb{A}); the sum converges absolutely for \mathrm{Re}(s) sufficiently large, depending on the group and parabolic.^[47] This construction yields functions on G(\mathbb{A}) that are left-invariant under G(\mathbb{Q}) and right-invariant under a maximal compact subgroup K, transforming under the center according to the given character, thus qualifying as automorphic forms.^[47] For the standard maximal parabolic in \mathrm{GL}_n, the section \phi is typically chosen to be unramified outside a finite set of places, factoring into local sections via the restricted tensor product.^[47] In the classical setting for G = \mathrm{GL}_2 over \mathbb{Q}, corresponding to modular forms on the upper half-plane \mathbb{H}, the Eisenstein series of weight k (a non-negative even integer) takes the form

E_k(z, s) = \sum_{(c,d)=1} (cz + d)^{-k} y^s ,

where z = x + iy \in \mathbb{H}, y = \Im(z), and the sum runs over coprime integers c, d \in \mathbb{Z} with d > 0 if c = 0; this converges for \mathrm{Re}(s) > 1.^[48] Here, the section \phi is constructed from the standard local factors, ensuring invariance under \mathrm{SL}_2(\mathbb{Z}), and the weight k arises from the automorphy factor j(\gamma, z) = cz + d.^[47] For s = k/2, this specializes to the holomorphic Eisenstein series E_k(z), a classical modular form of weight k on \Gamma = \mathrm{SL}_2(\mathbb{Z}).^[48] The Fourier expansion of E_k(z, s) reveals its structure, with the constant term involving the Riemann zeta function and the non-constant terms given by Bessel functions, facilitating analytic properties.^[48] The meromorphic continuation of E(g, s) to all s \in \mathbb{C} is established through the constant term along the unipotent radical N, which unfolds the sum and reduces the problem to intertwining operators on the Levi factor.^[49] Specifically, the functional equation relates the Eisenstein series at s to that at $1 - s via the normalized intertwining operator M(w, s), associated to the longest Weyl element w normalizing the parabolic P, defined by the integral

(M(w, s) \phi)(g) = \int_{N(\mathbb{A})} \phi\left( n w g \right) \psi^{-1}(n) \, dn

for \mathrm{Re}(s) > 1/2, where \psi is a non-trivial additive character on N(\mathbb{A}); this operator analytically continues to a meromorphic function on \mathbb{C} with possible poles, and satisfies E(g, M(w, s) \phi, 1 - s) = E(g, \phi, s) up to a global normalizing factor involving L-values.^[47]^[49] For \mathrm{GL}_2, the intertwining operator has a simple pole at s = 1/2, but the composition with the section yields the desired relation.^[47] Regarding poles, in the \mathrm{GL}_2 case, E(g, s) exhibits a simple pole at s = 1, independent of the choice of unramified section, with residue equal to the constant automorphic form $1 (normalized by the volume of G(\mathbb{Q}) \backslash G(\mathbb{A}) / K), which spans the trivial representation in the constant term of the spectral decomposition.^[48]^[47] No other poles occur in \mathrm{Re}(s) > 1/2 for the standard maximal parabolic Eisenstein series, ensuring the residue at s = 1 generates the invariant trace in the space of automorphic forms.^[49]

Global and Local Theories

Automorphic forms are studied through their decomposition into local components, where the local Langlands correspondence establishes a bijection between certain irreducible admissible representations \pi_v of the group G(\mathbb{Q}_v) and Frobenius-semisimple Weil–Deligne parameters over the local Weil group.^[50] This correspondence, conjectured by Langlands in the 1970s and proven for general linear groups GL_n over p-adic fields by Guy Henniart (2000), with a new geometric proof by Peter Scholze (2013) using perfectoid spaces, links the representation theory of reductive groups over local fields to Galois representations, providing a precise dictionary for the local factors of automorphic representations.^[51] Automorphic representations, as global objects, factor into these local representations \pi_v at each place v, ensuring compatibility across the adele group.^[1] Global base change lifts automorphic forms from the rationals \mathbb{Q} to automorphic forms on number fields K via the restriction of scalars functor, which embeds G_K into a form over \mathbb{Q}.^[52] For GL_2, Langlands developed this lifting in the 1980s, showing that cuspidal automorphic representations on GL_2(\mathbb{A}_\mathbb{Q}) transfer to representations on GL_2(\mathbb{A}_K) under compatible central characters, preserving L-functions and enabling the study of base change for motives.^[53] This process is functorial and compatible with the local Langlands correspondence at places of K, facilitating the global reciprocity conjectures by allowing descent and ascent between fields.^[52] Strong approximation theorems for algebraic groups over number fields imply that the image of G(\mathbb{Q}) is dense in the adeles G(\mathbb{A}_\mathbb{Q}^f) restricted to finite places, modulo compact open subgroups, which is crucial for approximating automorphic forms on adelic groups by classical ones.^[54] For simply connected groups like SL_n, this density holds fully, while for general reductive groups, it applies off a finite set of places, leading to computations on "thin sets" where the quotient space G(\mathbb{Q}) \backslash G(\mathbb{A}_\mathbb{Q}) is modified by compact factors to ensure convergence of integrals defining automorphic forms.^[55] These approximations underpin the transition from classical modular forms to adelic automorphic forms, allowing explicit calculations via Hecke operators on dense subgroups.^[54] The Jacquet–Langlands correspondence provides an equivalence between automorphic forms on GL_2(\mathbb{A}_\mathbb{Q}) and those on the multiplicative group of a quaternion algebra D over \mathbb{Q}, ramified at specified places.^[56] Established by Jacquet and Langlands in their 1970 monograph, this bijection maps cuspidal representations on GL_2 to representations on D^\times, preserving the associated L-functions up to finite factors and central characters, thus unifying the theory of modular forms with more general division algebra settings.^[57] It extends to global fields and plays a key role in endoscopic classifications, ensuring that every automorphic form on the inner form arises from one on the split form.^[56]

Recent Developments

A significant advancement in the local Langlands correspondence came with Peter Scholze's proof for general linear groups GL_n over p-adic fields, utilizing perfectoid spaces to establish a geometric framework that links Galois representations to automorphic representations. This work, building on foundational local theories, provides a novel geometric realization of the correspondence, confirming the bijection between irreducible smooth representations of GL_n and certain Galois representations.^[58] In the realm of p-adic automorphic forms, Eric Urban's construction of eigenvarieties for reductive groups has enabled the interpolation of classical automorphic forms into p-adic families, extending Hida's earlier theories to broader contexts. These eigenvarieties parameterize overconvergent eigenforms, allowing for the study of p-adic deformations and continuity properties that bridge classical and p-adic settings. This interpolation is central to the p-adic Langlands program, where overconvergent forms play a key role in establishing correspondences for potentially crystalline representations.^[59] James Arthur's 2013 endoscopic classification resolves key aspects of functoriality for classical groups, such as orthogonal and symplectic groups, by parameterizing automorphic representations in terms of global Arthur parameters and endoscopic transfers. This classification facilitates the lifting of representations between these groups and GL_N, providing a complete description of the discrete spectrum and advancing the understanding of functorial lifts in the Langlands program. Recent progress in the geometric Langlands program culminated in the 2024 proof of the categorical unramified conjecture by Dennis Gaitsgory and collaborators, establishing an equivalence between categories of automorphic sheaves on the moduli stack of bundles and categories of representations on the spectral side via the Hitchin fibration. This work incorporates categorical traces to handle multiplicities and confirms long-standing expectations for equivalences in characteristic zero, with implications for deeper connections between algebraic geometry and number theory. The p-adic Langlands program has similarly evolved, with overconvergent forms now integral to constructions of p-adic L-functions and Galois representations, addressing gaps in earlier treatments.^[60]

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