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The Dedekind eta-function is an automorphic form in the complex plane.

In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group G to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup of the topological group. Automorphic forms are a generalization of the idea of periodic functions in Euclidean space to general topological groups.

Modular forms are holomorphic automorphic forms defined over the groups SL(2, R) or PSL(2, R) with the discrete subgroup being the modular group, or one of its congruence subgroups; in this sense the theory of automorphic forms is an extension of the theory of modular forms. More generally, one can use the adelic approach as a way of dealing with the whole family of congruence subgroups at once. From this point of view, an automorphic form over the group G(AF), for an algebraic group G and an algebraic number field F, is a complex-valued function on G(AF) that is left invariant under G(F) and satisfies certain smoothness and growth conditions.

Henri Poincaré first discovered automorphic forms as generalizations of trigonometric and elliptic functions. Through the Langlands conjectures, automorphic forms play an important role in modern number theory.[1]

Definition

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In mathematics, the notion of factor of automorphy arises for a group acting on a complex-analytic manifold. Suppose a group acts on a complex-analytic manifold . Then, also acts on the space of holomorphic functions from to the complex numbers. A function is termed an automorphic form if the following holds:

where is an everywhere nonzero holomorphic function. Equivalently, an automorphic form is a function whose divisor is invariant under the action of .

The factor of automorphy for the automorphic form is the function . An automorphic function is an automorphic form for which is the identity.

An automorphic form is a function F on G (with values in some fixed finite-dimensional vector space V, in the vector-valued case), subject to three kinds of conditions:

  1. to transform under translation by elements according to the given factor of automorphy j;
  2. to be an eigenfunction of certain Casimir operators on G; and
  3. to satisfy a "moderate growth" asymptotic condition a height function.

It is the first of these that makes F automorphic, that is, satisfy an interesting functional equation relating F(g) with F(γg) for . In the vector-valued case the specification can involve a finite-dimensional group representation ρ acting on the components to 'twist' them. The Casimir operator condition says that some Laplacians[citation needed] have F as eigenfunction; this ensures that F has excellent analytic properties, but whether it is actually a complex-analytic function depends on the particular case. The third condition is to handle the case where G/Γ is not compact but has cusps.

The formulation requires the general notion of factor of automorphy j for Γ, which is a type of 1-cocycle in the language of group cohomology. The values of j may be complex numbers, or in fact complex square matrices, corresponding to the possibility of vector-valued automorphic forms. The cocycle condition imposed on the factor of automorphy is something that can be routinely checked, when j is derived from a Jacobian matrix, by means of the chain rule.

A more straightforward but technically advanced definition using class field theory, constructs automorphic forms and their correspondent functions as embeddings of Galois groups to their underlying global field extensions. In this formulation, automorphic forms are certain finite invariants, mapping from the idele class group under the Artin reciprocity law. Herein, the analytical structure of its L-function allows for generalizations with various algebro-geometric properties; and the resultant Langlands program. To oversimplify, automorphic forms in this general perspective, are analytic functionals quantifying the invariance of number fields in a most abstract sense, therefore indicating the 'primitivity' of their fundamental structure. Allowing a powerful mathematical tool for analyzing the invariant constructs of virtually any numerical structure.

Examples of automorphic forms in an explicit unabstracted state are difficult to obtain, though some have directly analytical properties:

- The Eisenstein series (which is a prototypical modular form) over certain field extensions as Abelian groups.

- Specific generalizations of Dirichlet L-functions as class field-theoretic objects.

- Generally any harmonic analytic object as a functor over Galois groups which is invariant on its ideal class group (or idele).

As a general principle, automorphic forms can be thought of as analytic functions on abstract structures, which are invariant with respect to a generalized analogue of their prime ideal (or an abstracted irreducible fundamental representation). As mentioned, automorphic functions can be seen as generalizations of modular forms (as therefore elliptic curves), constructed by some zeta function analogue on an automorphic structure. In the simplest sense, automorphic forms are modular forms defined on general Lie groups; because of their symmetry properties. Therefore, in simpler terms, a general function which analyzes the invariance of a structure with respect to its prime 'morphology'.

History

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Before this very general setting was proposed (around 1960), there had already been substantial developments of automorphic forms other than modular forms. The case of Γ a Fuchsian group had already received attention before 1900 (see below). The Hilbert modular forms (also called Hilbert-Blumenthal forms) were proposed not long after that, though a full theory was long in coming. The Siegel modular forms, for which G is a symplectic group, arose naturally from considering moduli spaces and theta functions. The post-war interest in several complex variables made it natural to pursue the idea of automorphic form in the cases where the forms are indeed complex-analytic. Much work was done, in particular by Ilya Piatetski-Shapiro, in the years around 1960, in creating such a theory. The theory of the Selberg trace formula, as applied by others, showed the considerable depth of the theory. Robert Langlands showed how (in generality, many particular cases being known) the Riemann–Roch theorem could be applied to the calculation of dimensions of automorphic forms; this is a kind of post hoc check on the validity of the notion. He also produced the general theory of Eisenstein series, which corresponds to what in spectral theory terms would be the 'continuous spectrum' for this problem, leaving the cusp form or discrete part to investigate. From the point of view of number theory, the cusp forms had been recognised, since Srinivasa Ramanujan, as the heart of the matter.

Automorphic representations

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See also: Cuspidal representation

The subsequent notion of an "automorphic representation" has proved of great technical value when dealing with G an algebraic group, treated as an adelic algebraic group. It does not completely include the automorphic form idea introduced above, in that the adelic approach is a way of dealing with the whole family of congruence subgroups at once. Inside an L2 space for a quotient of the adelic form of G, an automorphic representation is a representation that is an infinite tensor product of representations of p-adic groups, with specific enveloping algebra representations for the infinite prime(s). One way to express the shift in emphasis is that the Hecke operators are here in effect put on the same level as the Casimir operators; which is natural from the point of view of functional analysis[citation needed], though not so obviously for the number theory. It is this concept that is basic to the formulation of the Langlands philosophy.

Poincaré on discovery and his work on automorphic functions

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One of Poincaré's first discoveries in mathematics, dating to the 1880s, was automorphic forms. He named them Fuchsian functions, after the mathematician Lazarus Fuchs, because Fuchs was known for being a good teacher and had researched on differential equations and the theory of functions. Poincaré actually developed the concept of these functions as part of his doctoral thesis. Under Poincaré's definition, an automorphic function is one which is analytic in its domain and is invariant under a discrete infinite group of linear fractional transformations. Automorphic functions then generalize both trigonometric and elliptic functions.

Poincaré explains how he discovered Fuchsian functions:

For fifteen days I strove to prove that there could not be any functions like those I have since called Fuchsian functions. I was then very ignorant; every day I seated myself at my work table, stayed an hour or two, tried a great number of combinations and reached no results. One evening, contrary to my custom, I drank black coffee and could not sleep. Ideas rose in crowds; I felt them collide until pairs interlocked, so to speak, making a stable combination. By the next morning I had established the existence of a class of Fuchsian functions, those which come from the hypergeometric series; I had only to write out the results, which took but a few hours.

See also

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Notes

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References

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Automorphic form

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In number theory and representation theory, an automorphic form is a smooth function ϕ:G(AF)→C\phi: G(\mathbb{A}_F) \to \mathbb{C} on the adelic points of a reductive algebraic group GG over a number field FF, which is invariant under the action of G(F)G(F), right-invariant under a maximal compact subgroup KK, Z(gC)Z(\mathfrak{g}_\mathbb{C})-finite, with a continuous central character, and of moderate growth at infinity. These functions generalize classical modular forms, which are holomorphic automorphic forms of weight kk on GL2GL_2 over Q\mathbb{Q}, transforming under the modular group SL2(Z)SL_2(\mathbb{Z}) via the slash operator f∣kγ(z)=j(γ,z)−kf(γz)f|_k \gamma(z) = j(\gamma, z)^{-k} f(\gamma z) for γ∈SL2(Z)\gamma \in SL_2(\mathbb{Z}), where j(γ,z)=cz+dj(\gamma, z) = cz + d. Automorphic forms trace their origins to the late , when introduced them as analytic functions invariant under discontinuous group actions on the , extending the concept of periodic functions to Fuchsian groups and laying foundational work in the theory of automorphic functions. This early development, detailed in Poincaré's papers from 1881–1883, connected such forms to Riemann surfaces and uniformization, influencing subsequent advances in and . The modern adelic framework emerged in the mid-20th century through the works of and , who reformulated automorphic forms in terms of representations of adelic groups to unify global and local aspects. Automorphic forms are central to the , which conjectures deep correspondences between automorphic representations—irreducible constituents of the space of automorphic forms—and Galois representations of the of the number field, facilitating the study of L-functions and arithmetic objects like elliptic curves. Key examples include , which generate non-cuspidal automorphic forms via summation over cosets, and cusp forms like the modular form Δ(z)=q∏n=1∞(1−qn)24\Delta(z) = q \prod_{n=1}^\infty (1 - q^n)^{24} that vanish at the cusps of the fundamental domain. Their associated L-functions encode arithmetic data, such as prime distributions, and have applications in proving results like the linking elliptic curves to s.

Definition and Fundamentals

Formal Definition

Automorphic forms are defined as certain smooth functions on quotient spaces such as G(Q)\G(A)G(\mathbb{Q}) \backslash G(\mathbb{A}), where GG is a reductive algebraic group over Q\mathbb{Q} and A\mathbb{A} denotes the adele ring of Q\mathbb{Q}, that satisfy specific transformation properties under the left action of G(Q)G(\mathbb{Q}). These functions ϕ:G(A)→C\phi: G(\mathbb{A}) \to \mathbb{C} are required to be left-invariant under G(Q)G(\mathbb{Q}), meaning ϕ(γg)=ϕ(g)\phi(\gamma g) = \phi(g) for all γ∈G(Q)\gamma \in G(\mathbb{Q}) and g∈G(A)g \in G(\mathbb{A}), and smooth with respect to the adelic topology, while also being right KK-finite for a maximal compact subgroup K⊂G(A)K \subset G(\mathbb{A}) and of moderate growth, satisfying ∣ϕ(g)∣≤C∥g∥N|\phi(g)| \leq C \|g\|^N for some constants C,N>0C, N > 0 and all g∈G(A)g \in G(\mathbb{A}). Additionally, they are Z(g)Z(\mathfrak{g})-finite at the archimedean places, where Z(g)Z(\mathfrak{g}) is the center of the universal enveloping algebra, ensuring the functions generate finite-dimensional spaces under differential operators. In the classical setting for G=SL2G = \mathrm{SL}_2, automorphic forms on the upper half-plane H\mathbb{H} are functions f:H→Cf: \mathbb{H} \to \mathbb{C} that transform under the action of SL2(Z)\mathrm{SL}_2(\mathbb{Z}) via f(az+bcz+d)=(cz+d)kf(z)f\left( \frac{az + b}{cz + d} \right) = (cz + d)^k f(z) for all γ=(abcd)∈SL2(Z)\gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathrm{SL}_2(\mathbb{Z}) and integer weight kk, with the automorphy factor j(γ,z)=cz+dj(\gamma, z) = cz + d. They exhibit moderate growth at the cusps, such as ∣f(z)∣≤C(1+y)k|f(z)| \leq C (1 + y)^k where y=Im(z)y = \mathrm{Im}(z), and admit a Fourier expansion f(z)=∑n∈Zane2πinzf(z) = \sum_{n \in \mathbb{Z}} a_n e^{2\pi i n z} with only finitely many nonzero coefficients ana_n for n<0n < 0. Holomorphic automorphic forms are those that are holomorphic on H\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 Δ=−y2(∂2∂x2+∂2∂y2)\Delta = -y^2 \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} \right) with eigenvalue λ=s(1−s)\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(A)G(\mathbb{A}).

Classical vs. Adelic Perspectives

The classical perspective on automorphic forms originates in the study of functions on the upper half-plane H\mathbb{H}, which is invariant under the action of discrete subgroups such as SL(2,Z)\mathrm{SL}(2, \mathbb{Z}). Specifically, a classical automorphic form of weight kk and level NN is a holomorphic function f:H→Cf: \mathbb{H} \to \mathbb{C} satisfying f(γz)=(cz+d)kf(z)f(\gamma z) = (cz + d)^k f(z) for all γ=(abcd)∈Γ0(N)\gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \Gamma_0(N), where Γ0(N)\Gamma_0(N) is the congruence subgroup consisting of matrices with c≡0(modN)c \equiv 0 \pmod{N}, along with moderate growth conditions at the cusps. This setup emphasizes the geometric and analytic properties tied to the hyperbolic plane and arithmetic subgroups. In contrast, the adelic formulation generalizes this to functions on the adelic quotient G(A)/G(Q)G(\mathbb{A})/G(\mathbb{Q}), where G=GL2G = \mathrm{GL}_2 and A\mathbb{A} denotes the adele ring over Q\mathbb{Q}, comprising the real numbers R\mathbb{R} at the infinite place and the finite adeles Af=∏p′Qp\mathbb{A}_f = \prod'_p \mathbb{Q}_p. An adelic automorphic form ϕ:G(A)→C\phi: G(\mathbb{A}) \to \mathbb{C} is smooth, satisfying ϕ(γg)=ϕ(g)\phi(\gamma g) = \phi(g) for γ∈G(Q)\gamma \in G(\mathbb{Q}) and g∈G(A)g \in G(\mathbb{A}), with right invariance under a compact open subgroup Kf⊆G(Af)K_f \subseteq G(\mathbb{A}_f) (such as K0(N)={g∈G(Z^)∣g≡(∗∗0∗)(modN)}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. The two perspectives are equivalent through an explicit embedding that leverages strong approximation for GL2(Q)\mathrm{GL}_2(\mathbb{Q}). Classical forms of level NN and weight kk correspond bijectively to adelic forms via the map sending f(z)f(z) to ϕf(g)=F(g∞)λ(kf)\phi_f(g) = F(g_\infty) \lambda(k_f), where g=γg∞kfg = \gamma g_\infty k_f decomposes using strong approximation GL2(A)=GL2(Q)⋅GL2+(R)⋅K0(N)\mathrm{GL}_2(\mathbb{A}) = \mathrm{GL}_2(\mathbb{Q}) \cdot \mathrm{GL}_2^+(\mathbb{R}) \cdot K_0(N), FF is the automorphy factor on the infinite component, and λ\lambda encodes the nebentypus character locally. This induces an isomorphism Γ0(N)\GL2+(R)≅GL2(Q)\GL2(A)/K0(N)\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 kk at infinity. 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 , by allowing automorphic forms to decompose into local components ϕv\phi_v at each prime pp (including p-adic completions Qp\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 GL2\mathrm{GL}_2.

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 Λ=Zω1+Zω2\Lambda = \mathbb{Z} \omega_1 + \mathbb{Z} \omega_2. 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. 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. 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. 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 ϑ(τ)=∑n=−∞∞exp⁡(πin2τ)\vartheta(\tau) = \sum_{n=-\infty}^{\infty} \exp(\pi i n^2 \tau) for τ\tau in the upper half-plane H\mathbb{H}, this entire function satisfies the modular transformation law ϑ(−1τ)=−iτ ϑ(τ),\vartheta\left(-\frac{1}{\tau}\right) = \sqrt{-i \tau} \, \vartheta(\tau),
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