Quite generally, automorphic forms are suitably well-behaved functions on a quotient space where is typically a discrete group, hence suitable functions on which are invariant under the action of a discrete group. The precise definition has evolved a good bit through time.
More generally, automorphic forms in the modern sense are suitable functions on a coset space , hence functions on groups which are invariant with respect to the action of the subgroup . The archetypical example here are modular forms regarded as functions on where 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 and such representations are then called automorphic representations (e.g. Martin 13, p. 9) , specifically so if is the general linear group with coefficients in a ring of adeles of some global field and . 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 with coefficients in the integral adeles, see below.
By a standard definition, a modular form is a holomorphic function on the upper half plane satisfying a specified transformation property under the action of a given congruence subgroup of the modular group (e.g. Martin 13, definition 1, Litt, def. 1).
In view of this, one finds that every modular function lifts to a function
hence to a function on which is actually invariant with respect to the -action (“automorphy”), but which instead now satisfies some transformation property with respect to the action of , as well as some well-behavedness property
This is the incarnation as an automorphic function of the modular function (e.g. Martin 13, around def. 3, Litt, section 2). For emphasis these automorphic forms on equivalent to modular forms are called classical modular forms.
Where by the above an ordinary modular form is equivalently a suitably periodic function on , one may observe that the real numbers 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 of all these completions at once. This means to consider functions on , for 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
where 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.
More generally, for the general linear group , for any and with coefficients in a ring of adeles of some number field , and for the subgroup , then sufficiently well-behaved functions on form representations of which are called automorphic representations. Here “well-behaved” typically means
central character – the action by the center is is controled by (…something…);
growth – the functions are bounded functions;
cuspidality – (…)
(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 of the integral adeles (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 in the discussion of adelic automorphic forms above, the group
is the group of ideles and the quotient
|number fields (“function fields of curves over F1”)||function fields of curves over finite fields (arithmetic curves)||Riemann surfaces/complex curves|
|affine and projective line|
|(integers)||(polynomials, function algebra on affine line )||(holomorphic functions on complex plane)|
|(rational numbers)||(rational functions)||meromorphic functions on complex plane|
|(prime number/non-archimedean place)|
|(place at infinity)|
|(Spec(Z))||(affine line)||complex plane|
|(projective line)||Riemann sphere|
|(Fermat quotient)||(coordinate derivation)||“|
|genus of the rational numbers = 0||genus of the Riemann sphere = 0|
|(p-adic integers)||(power series around )||(holomorphic functions on formal disk around )|
|(“-arithmetic jet space” of at )||formal disks in|
|(p-adic numbers)||(Laurent series around )||(holomorphic functions on punctured formal disk around )|
|(ring of adeles)||( adeles of function field )||(restricted product of holomorphic functions on all punctured formal disks, finitely of which do not extend to the unpunctured disks)|
|(group of ideles)||( ideles of function field )|
|Jacobi theta function|
|Riemann zeta function||Goss zeta function|
|branched covering curves|
|a number field ( a possibly ramified finite dimensional field extension)||a function field of an algebraic curve over||(sheaf of rational functions on complex curve )|
|(ring of integers)||(structure sheaf)|
|(spectrum with archimedean places)||(arithmetic curve)||(complex curve being branched cover of Riemann sphere)|
|(lift of Frobenius morphism/Lambda-ring structure)||“|
|genus of a number field||genus of an algebraic curve||genus of a surface|
|prime ideal in ring of integers|
|(formal completion at )||(function algebra on punctured formal disk around )|
|(ring of integers of formal completion)||(function algebra on formal disk around )|
|(ring of adeles)||(restricted product of function rings on all punctured formal disks around all points in )|
|(function ring on all formal disks around all points in )|
|(group of ideles)|
|Galois group||“||fundamental group|
|Galois representation||“||flat connection (“local system”) on|
|class field theory|
|class field theory||“||geometric class field theory|
|Hilbert reciprocity law||Artin reciprocity law||Weil reciprocity law|
|(idele class group)||“|
|“||(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|
|(constant sheaves on this stack form unramified automorphic representations)||“||(moduli stack of bundles on the curve , by Weil uniformization theorem)|
|Tamagawa-Weil for number fields||Tamagawa-Weil for function fields|
|Hecke theta function||functional determinant line bundle of Dirac operator/chiral Laplace operator on|
|Dedekind zeta function||Weil zeta function||zeta function of a Riemann surface/of the Laplace operator on|
|higher dimensional spaces|
|zeta functions||Hasse-Weil zeta function|
Introductions and surveys include
Pierre Deligne, Fromed Modulaires et representations de ()
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)
Daniel Bump, Automorphic forms and representations
David Loeffler, Computing with algebraic automorphic forms, 2011 (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