geometric representation theory
representation, 2-representation, β-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
Be?linson-Bernstein localization?
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.
Henri PoincarΓ© 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 , 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 βmodular formβ and βautomorphic formβ 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).
But the upper half plane is itself the coset of the projective linear group by the subgroup
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 (βautomorphyβ), 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.
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.
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 βwell-behavedβ typically means
finiteness β the functions invariant under the action of the maximal compact subgroup span a finite dimensional vector space;
central character β the action by the center is is controled by (β¦somethingβ¦);
growth β the functions are bounded functions;
cuspidality β (β¦)
(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 βunramifiedβ 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
is the idele class group. Automorphic forms in this case are effectively Dirichlet characters in disguise⦠(Goldfeld-Hundley 11, theorem 2.1.9).
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 ()
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, 177β219 (web)
Nolan Wallach, Introductory lectures on automorphic forms (pdf)
E. Kowalski, section 3 of Automorphic forms, L-functions and number theory (March 12β16) 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 Miyakeβs 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, 277β285.
Stephen Kudla, Theta functions and Hilbert modular forms,Nagoya Math. J. 69 (1978) 97-106
Jeffrey Stopple, Theta and -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
Last revised on February 8, 2023 at 14:30:06. See the history of this page for a list of all contributions to it.