geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
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.
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 $K = Stab_G(\{i\}) \simeq SO(2)/\{\pm I\}$
In view of this, one finds that every modular function $f \colon \mathfrak{H} \to \mathbb{C}$ lifts to a function
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.
The formulation of modular forms as automorphic forms for $G = PSL(2,\mathbb{R})$ above has in turn an equivalent formulation in terms of certain automorphic forms on $GL(2,\mathbb{A})$, where $\mathbb{A}$ is the ring of adeles (e.g. Martin 13, p. 8, also Goldfeld-Hundley 11, lemma 5.5.10, Bump, section 3.6, Gelbhart 84, p. 22): we have
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.
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
finiteness – the functions invariant under the action of the maximal compact subgroup span a finite dimensional vector space;
central character – the action by the center is is controled by (…something…);
growth – the functions are bounded functions;
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
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.
For the special case of $n = 1$ in the discussion of adelic automorphic forms above, the group
is the group of ideles and the quotient
is the idele class group. Automorphic forms in this case are effectively Dirichlet characters in disguise… (Goldfeld-Hundley 11, theorem 2.1.9).
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, function 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 functions) | meromorphic functions on complex plane | |
$p$ (prime number/non-archimedean place) | $x \in \mathbb{F}_p$ | $x \in \mathbb{C}$ | |
$\infty$ (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 | |||
$\mathbb{Z}_p$ (p-adic integers) | $\mathbb{F}_q[ [ t -x ] ]$ (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 |
In string theory partition functions tend to be automorphic forms for U-duality groups. See the references below
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)
Toshitsune Miyake’s Modular Forms 1976 (English version 1989) (review pdf)
Review in the context of the geometric Langlands correspondence is in
The generalization of theta functions to automorphic forms is due to
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)
The relation between string theory on Riemann surfaces and automorphic forms was first highlighted in
See also