Contents

Idea

Quite generally, automorphic forms are suitably well-behaved functions on coset spaces $G/K$, hence functions on groups $G$ which are invariant with respect to the action of some subgroup $K \hookrightarrow G$.

Modular forms

For modular group/congruence subgroups $K$ of the real general linear group in dimension 2, $G = SL(2,\mathbb{R})$, modular forms may be identified with such functions on $SL(2,\mathbb{R})/K$ (see at modular form – as automorphic forms) and this is where the concept of automorphic forms originates.

In harmonic analysis

In harmonic analysis one typically considers topological groups $G$ with discrete group subgroups $K$ and considers continuous functions, typically bounded..

Automorphic representation in number theory

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, 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)

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

See at Langlands correspondence for more on this.

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)$\mathbb{F}_q[t]$ (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)$\mathbb{F}_q(t)$ (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
genus of the rational numbers = 0genus of the Riemann sphere = 0
formal neighbourhoods
$\mathbb{Z}_p$ (p-adic integers)$\mathbb{F}_q[ [ t -x ] ]$ (power series around $x$)$\mathbb{C}[ [t-x] ]$ (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)$\mathbb{F}_q((t-x))$ (Laurent series around $x$)$\mathbb{C}((t-x))$ (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 )$\underset{x \in \mathbb{C}}{\prod^\prime} \mathbb{C}((t-x))$ (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 )$\underset{x \in \mathbb{C}}{\prod^\prime} GL_1(\mathbb{C}((t-x)))$
zeta functions
Riemann zeta functionGoss 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$$K_\Sigma$ (sheaf of rational functions on complex curve $\Sigma$)
$\mathcal{O}_K$ (ring of integers)$\mathcal{O}_{\Sigma}$ (structure sheaf)
$Spec_{an}(\mathcal{O}_K) \to Spec(\mathbb{Z})$ (spectrum with archimedean places)$\Sigma$ (arithmetic curve)$\Sigma \to \mathbb{C}P^1$ (complex curve being branched cover of Riemann sphere)
genus of a number fieldgenus of an algebraic curvegenus of a surface
formal neighbourhoods
$v$ prime ideal in ring of integers $\mathcal{O}_K$$x \in \Sigma$$x \in \Sigma$
$K_v$ (formal completion at $v$)$\mathbb{C}((t_x))$ (function algebra on punctured formal disk around $x$)
$\mathcal{O}_{K_v}$ (ring of integers of formal completion)$\mathbb{C}[ [ t_x ] ]$ (function algebra on formal disk around $x$)
$\mathbb{A}_K$ (ring of adeles)$\prod^\prime_{x\in \Sigma} \mathbb{C}((t_x))$ (restricted product of function rings on all punctured formal disks around all points in $\Sigma$)
$\mathcal{O}$$\prod_{x\in \Sigma} \mathbb{C}[ [t_x] ]$ (function ring on all formal disks around all points in $\Sigma$)
$\mathbb{I}_K = GL_1(\mathbb{A}_K)$ (group of ideles)$\prod^\prime_{x\in \Sigma} GL_1(\mathbb{C}((t_x)))$
Galois theory
Galois group$\pi_1(\Sigma)$ fundamental group
Galois representationflat connection (“local system”) on $\Sigma$
class field theory
class field theorygeometric class field theory
Hilbert reciprocity lawArtin reciprocity lawWeil 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 correspondencefunction field Langlands correspondencegeometric 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 fieldsTamagawa-Weil for function fields
zeta functions
Dedekind zeta functionWeil zeta functionzeta function of a Riemann surface

Application in string theory

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

References

General

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

• 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)

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)