nLab automorphic form

Idea

Quite generally, automorphic forms are suitably well-behaved functions on a quotient space $K\backslash X$ where $K$ is typically a discrete group, hence suitable functions on $X$ which are invariant under the action of a discrete group. The precise definition has evolved a good bit through time.

Henri Poincaré considered analytic functions invariant under a discrete infinite group of fractional linear transformations and called them Fuchsian functions (after his advisor Lazarus Fuchs).

More generally, automorphic forms in the modern sense are suitable functions on a coset space $K \backslash G$, hence functions on groups $G$ which are invariant with respect to the action of the subgroup $K \hookrightarrow G$. The archetypical example here are modular forms regarded as functions on $K\backslash PSL(2,\mathbb{R})$ where $K$ is a congruence subgroup, and for some time the terms “modular form” and “automorphic form” were used essentially synonymously, see below. Based on the fact that a modular form is a section of some line bundle on the moduli stack of elliptic curves, Pierre Deligne defined an automorphic form to be a section of a line bundle on a Shimura variety.

By pullback of functions the linear space of such functions hence constitutes a representation of $G$ and such representations are then called automorphic representations (e.g. Martin 13, p. 9) , specifically so if $G = GL_n(\mathbb{A}_K)$ is the general linear group with coefficients in a ring of adeles of some global field and $K = GL_n(K)$. This is the subject of the Langlands program. There one also considers unramified such representations, which are constituted by functions that in addition are invariant under the action of $GL_n$ with coefficients in the integral adeles, see below.

Modular forms as classical automorphic forms on $PSL(2,\mathbb{R})$

By a standard definition, a modular form is a holomorphic function on the upper half plane $\mathfrak{H}$ satisfying a specified transformation property under the action of a given congruence subgroup $\Gamma$ of the modular group $G = PSL(2,\mathbb{Z})$ (e.g. Martin 13, definition 1, Litt, def. 1).

But the upper half plane is itself the coset of the projective linear group $G = PSL(\mathbb{R})$ by the subgroup $K = Stab_G(\{i\}) \simeq SO(2)/\{\pm I\}$

$$f\colon \mathfrak{H} \simeq PSL(2,\mathbb{R})/K \,.$$

In view of this, one finds that every modular function $f \colon \mathfrak{H} \to \mathbb{C}$ lifts to a function

$$\tilde f \colon \Gamma\backslash PSL(2,\mathbb{R}) \longrightarrow \mathbb{C} \,,$$

hence to a function on $G$ which is actually invariant with respect to the $\Gamma$-action (“automorphy”), but which instead now satisfies some transformation property with respect to the action of $K$, as well as some well-behavedness property

This $\tilde f$ is the incarnation as an automorphic function of the modular function $f$ (e.g. Martin 13, around def. 3, Litt, section 2). For emphasis these automorphic forms on $PSL(2,\mathbb{R})$ equivalent to modular forms are called classical modular forms.

This is where the concept of automorphic forms originates (for more on the history see e.g. this MO comment for the history of terminology) and this one.

Modular forms as adelic automorphic forms on $GL(2,\mathbb{A})$

Where by the above an ordinary modular form is equivalently a suitably periodic function on $SL(2,\mathbb{R})$, one may observe that the real numbers $\mathbb{R}$ appearing as coefficients in the latter are but one of many p-adic number completions of the rational numbers. Hence it is natural to consider suitably periodic functions on $SL(2,\mathbb{Q}_p)$ of all these completions at once. This means to consider functions on $SL(2,\mathbb{A})$, for $\mathbb{A}$ the ring of adeles. These are the adelic automorphic forms. They may be thought of as subsuming ordinary modular forms for all level structures. (e.g. Martin 13, p. 8, also Goldfeld-Hundley 11, lemma 5.5.10, Bump, section 3.6, Gelbhart 84, p. 22): we have

$$\Gamma \backslash PSL(2,\mathbb{R}) \simeq Z(\mathbb{A}) GL_2(\mathbb{Q})\backslash GL_2(\mathbb{A})/ GL_2(\mathbb{A}_{\mathbb{Z}}) \,,$$

where $\mathbb{A}_{\mathbb{Z}}$ are the integral adeles. (The double coset on the right is analogous to that which appears in the Weil uniformization theorem, see the discussion there and at geometric Langlands correspondence for more on this analogy.)

This leads to the more general concept of adelic automorphic forms below.

General adelic automorphic forms

More generally, for the general linear group $G = GL_n(\mathbb{A}_F)$, for any $n$ and with coefficients in a ring of adeles $\mathbb{A}_F$ of some number field $F$, and for the subgroup $GL_n(F)$, then sufficiently well-behaved functions on $GL_n(F)\backslash GL_n(\mathbb{A}_F)$ form representations of $GL_n(\mathbb{A}_{F})$ which are called automorphic representations. Here “well-behaved” typically means

1. finiteness – the functions invariant under the action of the maximal compact subgroup span a finite dimensional vector space;

2. central character – the action by the center is is controled by (…something…);

3. growth – the functions are bounded functions;

4. cuspidality – (…)

(e.g. Frenkel 05, section 1.6, Loeffler 11, page 4, Martin 13, definition 4, Litt, def.4).

(These conditions are not entirely set in stone, they are being varied according to application (see e.g. this MO comment)).

In particular one considers subspaces of “unramified” such functions, namely those which are in addition trivial on the subgroup of $GL_n$ of the integral adeles $\mathcal{O}_F$ (Goldfeld-Hundley 11, def. 2.1.12). This means that that unramified automorphic representations are spaces of functions on a double coset of the form

$$GL_n(F)\backslash GL_n(\mathbb{A}_F) / GL_n(\mathcal{O}_F) \,.$$

See at Langlands correspondence for more on this. Such double cosets are analogous to those appearing in the Weil uniformization theorem in complex analytic geometry, an analogy which leads to the conjecture of the geometric Langlands correspondence.

Dirichlet characters

For the special case of $n = 1$ in the discussion of adelic automorphic forms above, the group

$$GL_1(\mathbb{A}_F) = (\mathbb{A}_F)^\times = \mathbb{I}_F$$

is the group of ideles and the quotient

$$GL_1(F) \backslash GL_1(\mathbb{A}_F) = F^\times \backslash (\mathbb{A}_F)^\times$$

is the idele class group. Automorphic forms in this case are effectively Dirichlet characters in disguise… (Goldfeld-Hundley 11, theorem 2.1.9).

Properties

Function field analogy

function field analogy

	number fields (“function fields of curves over F1”)	function fields of curves over finite fields $\mathbb{F}_q$ (arithmetic curves)	Riemann surfaces/complex curves
affine and projective line
	$\mathbb{Z}$ (integers)	$\mathbb{F}_q[z]$ (polynomials, polynomial algebra on affine line $\mathbb{A}^1_{\mathbb{F}_q}$)	$\mathcal{O}_{\mathbb{C}}$ (holomorphic functions on complex plane)
	$\mathbb{Q}$ (rational numbers)	$\mathbb{F}_q(z)$ (rational fractions/rational function on affine line $\mathbb{A}^1_{\mathbb{F}_q}$)	meromorphic functions on complex plane
	$p$ (prime number/non-archimedean place)	$x \in \mathbb{F}_p$, where $z - x \in \mathbb{F}_q[z]$ is the irreducible monic polynomial of degree one	$x \in \mathbb{C}$, where $z - x \in \mathcal{O}_{\mathbb{C}}$ is the function which subtracts the complex number $x$ from the variable $z$
	$\infty$ (place at infinity)		$\infty$
	$Spec(\mathbb{Z})$ (Spec(Z))	$\mathbb{A}^1_{\mathbb{F}_q}$ (affine line)	complex plane
	$Spec(\mathbb{Z}) \cup place_{\infty}$	$\mathbb{P}_{\mathbb{F}_q}$ (projective line)	Riemann sphere
	$\partial_p \coloneqq \frac{(-)^p - (-)}{p}$ (Fermat quotient)	$\frac{\partial}{\partial z}$ (coordinate derivation)	“
	genus of the rational numbers = 0		genus of the Riemann sphere = 0
formal neighbourhoods
	$\mathbb{Z}/(p^n \mathbb{Z})$ (prime power local ring)	$\mathbb{F}_q [z]/((z-x)^n \mathbb{F}_q [z])$ ($n$-th order univariate local Artinian $\mathbb{F}_q$-algebra)	$\mathbb{C}[z]/((z-x)^n \mathbb{C}[z])$ ($n$-th order univariate Weil $\mathbb{C}$-algebra)
	$\mathbb{Z}_p$ (p-adic integers)	$\mathbb{F}_q[ [ z -x ] ]$ (power series around $x$)	$\mathbb{C}[ [z-x] ]$ (holomorphic functions on formal disk around $x$)
	$Spf(\mathbb{Z}_p)\underset{Spec(\mathbb{Z})}{\times} X$ (“$p$-arithmetic jet space” of $X$ at $p$)		formal disks in $X$
	$\mathbb{Q}_p$ (p-adic numbers)	$\mathbb{F}_q((z-x))$ (Laurent series around $x$)	$\mathbb{C}((z-x))$ (holomorphic functions on punctured formal disk around $x$)
	$\mathbb{A}_{\mathbb{Q}} = \underset{p\; place}{\prod^\prime}\mathbb{Q}_p$ (ring of adeles)	$\mathbb{A}_{\mathbb{F}_q((t))}$ ( adeles of function field )	$\underset{x \in \mathbb{C}}{\prod^\prime} \mathbb{C}((z-x))$ (restricted product of holomorphic functions on all punctured formal disks, finitely of which do not extend to the unpunctured disks)
	$\mathbb{I}_{\mathbb{Q}} = GL_1(\mathbb{A}_{\mathbb{Q}})$ (group of ideles)	$\mathbb{I}_{\mathbb{F}_q((t))}$ ( ideles of function field )	$\underset{x \in \mathbb{C}}{\prod^\prime} GL_1(\mathbb{C}((z-x)))$
theta functions
	Jacobi theta function
zeta functions
	Riemann zeta function	Goss zeta function
branched covering curves
	$K$ a number field ($\mathbb{Q} \hookrightarrow K$ a possibly ramified finite dimensional field extension)	$K$ a function field of an algebraic curve $\Sigma$ over $\mathbb{F}_p$	$K_\Sigma$ (sheaf of rational functions on complex curve $\Sigma$)
	$\mathcal{O}_K$ (ring of integers)		$\mathcal{O}_{\Sigma}$ (structure sheaf)
	$Spec_{an}(\mathcal{O}_K) \to Spec(\mathbb{Z})$ (spectrum with archimedean places)	$\Sigma$ (arithmetic curve)	$\Sigma \to \mathbb{C}P^1$ (complex curve being branched cover of Riemann sphere)
	$\frac{(-)^p - \Phi(-)}{p}$ (lift of Frobenius morphism/Lambda-ring structure)	$\frac{\partial}{\partial z}$	“
	genus of a number field	genus of an algebraic curve	genus of a surface
formal neighbourhoods
	$v$ prime ideal in ring of integers $\mathcal{O}_K$	$x \in \Sigma$	$x \in \Sigma$
	$K_v$ (formal completion at $v$)		$\mathbb{C}((z_x))$ (function algebra on punctured formal disk around $x$)
	$\mathcal{O}_{K_v}$ (ring of integers of formal completion)		$\mathbb{C}[ [ z_x ] ]$ (function algebra on formal disk around $x$)
	$\mathbb{A}_K$ (ring of adeles)		$\prod^\prime_{x\in \Sigma} \mathbb{C}((z_x))$ (restricted product of function rings on all punctured formal disks around all points in $\Sigma$)
	$\mathcal{O}$		$\prod_{x\in \Sigma} \mathbb{C}[ [z_x] ]$ (function ring on all formal disks around all points in $\Sigma$)
	$\mathbb{I}_K = GL_1(\mathbb{A}_K)$ (group of ideles)		$\prod^\prime_{x\in \Sigma} GL_1(\mathbb{C}((z_x)))$
Galois theory
	Galois group	“	$\pi_1(\Sigma)$ fundamental group
	Galois representation	“	flat connection (“local system”) on $\Sigma$
class field theory
	class field theory	“	geometric class field theory
	Hilbert reciprocity law	Artin reciprocity law	Weil reciprocity law
	$GL_1(K)\backslash GL_1(\mathbb{A}_K)$ (idele class group)	“
	$GL_1(K)\backslash GL_1(\mathbb{A}_K)/GL_1(\mathcal{O})$	“	$Bun_{GL_1}(\Sigma)$ (moduli stack of line bundles, by Weil uniformization theorem)
non-abelian class field theory and automorphy
	number field Langlands correspondence	function field Langlands correspondence	geometric Langlands correspondence
	$GL_n(K) \backslash GL_n(\mathbb{A}_K)//GL_n(\mathcal{O})$ (constant sheaves on this stack form unramified automorphic representations)	“	$Bun_{GL_n(\mathbb{C})}(\Sigma)$ (moduli stack of bundles on the curve $\Sigma$, by Weil uniformization theorem)
	Tamagawa-Weil for number fields	Tamagawa-Weil for function fields
theta functions
	Hecke theta function		functional determinant line bundle of Dirac operator/chiral Laplace operator on $\Sigma$
zeta functions
	Dedekind zeta function	Weil zeta function	zeta function of a Riemann surface/of the Laplace operator on $\Sigma$
higher dimensional spaces
zeta functions	Hasse-Weil zeta function

Application in string theory

In string theory partition functions tend to be automorphic forms for U-duality groups. See the references below

Related entries

• modular form, topological modular form, topological automorphic form

• automorphic L-function

• Langlands duality

References

General

Introductions and surveys include

• Pierre Deligne, Fromed Modulaires et representations de $GL(2)$ ()

• Stephen Gelbart, starting on p. 20 (196) of An elementary introduction to the Langlands program, Bull. Amer. Math. Soc. (N.S.) 10 (1984), no. 2, 177–219 (web)

• Nolan Wallach, Introductory lectures on automorphic forms (pdf)

• E. Kowalski, section 3 of Automorphic forms, L-functions and number theory (March 12–16) Three Introductory lectures (pdf)

• Dorian Goldfeld, Joseph Hundley, chapter 2 of Automorphic representations and L-functions for the general linear group, Cambridge Studies in Advanced Mathematics 129, 2011 (pdf)

• Daniel Bump, Automorphic forms and representations

• David Loeffler, Computing with algebraic automorphic forms, 2011 (pdf)

• Kimball Martin, A brief overview of modular and automorphic forms,2013 pdf

• Daniel Litt, Automorphic forms notes, part I (pdf)

• Denis Trotabas, Modular Forms and Automorphic Representations (2010) (pdf)

• Werner Muller, Spectral theory of automorphic forms (2010) (pdf)

• Toshitsune Miyake’s Modular Forms 1976 (English version 1989) (review pdf)

Review in the context of the geometric Langlands correspondence is in

• Edward Frenkel, Lectures on the Langlands Program and Conformal Field Theory, in Frontiers in number theory, physics, and geometry II, Springer Berlin Heidelberg, 2007. 387-533. (arXiv:hep-th/0512172)

The generalization of theta functions to automorphic forms is due to

• André Weil, Sur certaines groups d’operateur unitaires, Acta. Math. 111 (1964), 143-211

see Gelbhart 84, page 35 (211) for review.

Further developments here include

• Stephen Kudla, Relations between automorphic forms produced by theta-functions, in Modular Functions of One Variable VI, Lecture Notes in Math. 627, Springer, 1977, 277–285.

• Stephen Kudla, Theta functions and Hilbert modular forms,Nagoya Math. J. 69 (1978) 97-106

• Jeffrey Stopple, Theta and $L$-function splittings, Acta Arithmetica LXXII.2 (1995) (pdf)

In string theory

The relation between string theory on Riemann surfaces and automorphic forms was first highlighted in

• Edward Witten, Quantum field theory, Grassmannians, and algebraic curves, Comm. Math. Phys. Volume 113, Number 4 (1988), 529-700 (Euclid)

See also

• Michael Green, Jorge G. Russo, Pierre Vanhove, Automorphic properties of low energy string amplitudes in various dimensions (arXiv:1001.2535)