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automorphic form

Contents

Idea

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

By pullback of functions the linear space of such functions hence constitutes a representation of GG and such representations are then called automorphic representations, specifically so if G=GL n(𝔸 K)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)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 nGL_n with coefficients in the integral adeles, see below.

Modular forms

For modular group/congruence subgroups KK of the real general linear group in dimension 2, G=SL(2,)G = SL(2,\mathbb{R}), modular forms may be identified with such functions on SL(2,)/KSL(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 GG with discrete group subgroups KK and considers continuous functions, typically bounded..

Adelic automorphic forms in number theory

For the general linear group G=GL n(𝔸 F)G = GL_n(\mathbb{A}_F), for any nn and with coefficients in a ring of adeles 𝔸 F\mathbb{A}_F of some number field, and for the subgroup GL n(F)GL_n(F), then sufficiently well-behaved functions on GL n(F)\GL n(𝔸 F)GL_n(F)\backslash GL_n(\mathbb{A}_F) form representations of GL n(𝔸 F)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).

In particular one considers subspaces of “unramified” such functions, namely those which are in addition trivial on the subgroup of GL nGL_n of the integral adeles 𝒪 F\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)\GL n(𝔸 F)/GL n(𝒪 F). 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 geoemtry?, an analogy which leads to the conjecture of the geometric Langlands correspondence.

Dirichlet characters

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

GL 1(𝔸 F)=(𝔸 F) ×=𝕀 F GL_1(\mathbb{A}_F) = (\mathbb{A}_F)^\times = \mathbb{I}_F

is the group of ideles and the quotient

GL 1(F)\GL 1(𝔸 F)=F ×\(𝔸 F) × 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 𝔽 q\mathbb{F}_q (arithmetic curves)Riemann surfaces/complex curves
affine and projective line
\mathbb{Z} (integers)𝔽 q[z]\mathbb{F}_q[z] (polynomials, function algebra on affine line 𝔸 𝔽 q 1\mathbb{A}^1_{\mathbb{F}_q})𝒪 \mathcal{O}_{\mathbb{C}} (holomorphic functions on complex plane)
\mathbb{Q} (rational numbers)𝔽 q(z)\mathbb{F}_q(z) (rational functions)meromorphic functions on complex plane
pp (prime number/non-archimedean place)x𝔽 px \in \mathbb{F}_pxx \in \mathbb{C}
\infty (place at infinity)\infty
Spec()Spec(\mathbb{Z}) (Spec(Z))𝔸 𝔽 q 1\mathbb{A}^1_{\mathbb{F}_q} (affine line)complex plane
Spec()place Spec(\mathbb{Z}) \cup place_{\infty} 𝔽 q\mathbb{P}_{\mathbb{F}_q} (projective line)Riemann sphere
p() p()p\partial_p \coloneqq \frac{(-)^p - (-)}{p} (Fermat quotient)z\frac{\partial}{\partial z} (coordinate derivation)
genus of the rational numbers = 0genus of the Riemann sphere = 0
formal neighbourhoods
p\mathbb{Z}_p (p-adic integers)𝔽 q[[tx]]\mathbb{F}_q[ [ t -x ] ] (power series around xx)[[zx]]\mathbb{C}[ [z-x] ] (holomorphic functions on formal disk around xx)
Spf( p)×Spec()XSpf(\mathbb{Z}_p)\underset{Spec(\mathbb{Z})}{\times} X (“pp-arithmetic jet space” of XX at pp)formal disks in XX
p\mathbb{Q}_p (p-adic numbers)𝔽 q((zx))\mathbb{F}_q((z-x)) (Laurent series around xx)((zx))\mathbb{C}((z-x)) (holomorphic functions on punctured formal disk around xx)
𝔸 = pplace p\mathbb{A}_{\mathbb{Q}} = \underset{p\; place}{\prod^\prime}\mathbb{Q}_p (ring of adeles)𝔸 𝔽 q((t))\mathbb{A}_{\mathbb{F}_q((t))} ( adeles of function field ) x((zx))\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)
𝕀 =GL 1(𝔸 )\mathbb{I}_{\mathbb{Q}} = GL_1(\mathbb{A}_{\mathbb{Q}}) (group of ideles)𝕀 𝔽 q((t))\mathbb{I}_{\mathbb{F}_q((t))} ( ideles of function field ) xGL 1(((zx)))\underset{x \in \mathbb{C}}{\prod^\prime} GL_1(\mathbb{C}((z-x)))
theta functions
Jacobi theta function
zeta functions
Riemann zeta functionGoss zeta function
branched covering curves
KK a number field (K\mathbb{Q} \hookrightarrow K a possibly ramified finite dimensional field extension)KK a function field of an algebraic curve Σ\Sigma over 𝔽 p\mathbb{F}_pK ΣK_\Sigma (sheaf of rational functions on complex curve Σ\Sigma)
𝒪 K\mathcal{O}_K (ring of integers)𝒪 Σ\mathcal{O}_{\Sigma} (structure sheaf)
Spec an(𝒪 K)Spec()Spec_{an}(\mathcal{O}_K) \to Spec(\mathbb{Z}) (spectrum with archimedean places)Σ\Sigma (arithmetic curve)ΣP 1\Sigma \to \mathbb{C}P^1 (complex curve being branched cover of Riemann sphere)
() pΦ()p\frac{(-)^p - \Phi(-)}{p} (lift of Frobenius morphism/Lambda-ring structure)z\frac{\partial}{\partial z}
genus of a number fieldgenus of an algebraic curvegenus of a surface
formal neighbourhoods
vv prime ideal in ring of integers 𝒪 K\mathcal{O}_KxΣx \in \SigmaxΣx \in \Sigma
K vK_v (formal completion at vv)((z x))\mathbb{C}((z_x)) (function algebra on punctured formal disk around xx)
𝒪 K v\mathcal{O}_{K_v} (ring of integers of formal completion)[[z x]]\mathbb{C}[ [ z_x ] ] (function algebra on formal disk around xx)
𝔸 K\mathbb{A}_K (ring of adeles) xΣ ((z x))\prod^\prime_{x\in \Sigma} \mathbb{C}((z_x)) (restricted product of function rings on all punctured formal disks around all points in Σ\Sigma)
𝒪\mathcal{O} xΣ[[z x]]\prod_{x\in \Sigma} \mathbb{C}[ [z_x] ] (function ring on all formal disks around all points in Σ\Sigma)
𝕀 K=GL 1(𝔸 K)\mathbb{I}_K = GL_1(\mathbb{A}_K) (group of ideles) xΣ GL 1(((z x)))\prod^\prime_{x\in \Sigma} GL_1(\mathbb{C}((z_x)))
Galois theory
Galois groupπ 1(Σ)\pi_1(\Sigma) fundamental group
Galois representationflat connection (“local system”) on Σ\Sigma
class field theory
class field theorygeometric class field theory
Hilbert reciprocity lawArtin reciprocity lawWeil reciprocity law
GL 1(K)\GL 1(𝔸 K)GL_1(K)\backslash GL_1(\mathbb{A}_K) (idele class group)
GL 1(K)\GL 1(𝔸 K)/GL 1(𝒪)GL_1(K)\backslash GL_1(\mathbb{A}_K)/GL_1(\mathcal{O})Bun GL 1(Σ)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)\GL n(𝔸 K)//GL n(𝒪)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()(Σ)Bun_{GL_n(\mathbb{C})}(\Sigma) (moduli stack of bundles on the curve Σ\Sigma, by Weil uniformization theorem)
Tamagawa-Weil for number fieldsTamagawa-Weil for function fields
theta functions
Hecke theta functionfunctional determinant line bundle of Dirac operator/chiral Laplace operator on Σ\Sigma
zeta functions
Dedekind zeta functionWeil zeta functionzeta function of a Riemann surface/of the Laplace operator on Σ\Sigma
higher dimensional spaces
zeta functionsHasse-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

References

General

  • 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)

  • 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)

See also

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

Revised on August 23, 2014 05:14:10 by Urs Schreiber (82.113.121.240)