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In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup of the topological group. Automorphic forms are a generalization of the idea of periodic functions in Euclidean space to general topological groups. Modular forms are automorphic forms defined over the groups SL(2, R) or PSL(2, R) with the discrete subgroup being the modular group, or one of its congruence subgroups; in this sense the theory of automorphic forms is an extension of the theory of modular forms. More generally, one can use the adelic approach is a way of dealing with the whole family of congruence subgroups at once. From this point of view, an automorphic form over the group G(A''F'') for an algebraic group G and an algebraic number field ''F'', is a complex-valued function on G(A''F'') that is left invariant under G(F) and satisfies certain smoothness and growth conditions. Poincaré first discovered automorphic forms as generalizations of trigonometric and elliptic functions. Through the Langlands conjectures automorphic forms play an important role in modern number theory. ==Formulation== An automorphic form is a function ''F'' on ''G'' (with values in some fixed finite-dimensional vector space ''V'', in the vector-valued case), subject to three kinds of conditions: # to transform under translation by elements according to the given factor of automorphy ''j''; # to be an eigenfunction of certain Casimir operators on ''G''; and # to satisfy some conditions on growth at infinity. It is the first of these that makes ''F'' ''automorphic'', that is, satisfy an interesting functional equation relating ''F''(''g'') with ''F''(γ''g'') for . In the vector-valued case the specification can involve a finite-dimensional group representation ρ acting on the components to 'twist' them. The Casimir operator condition says that some Laplacians have ''F'' as eigenfunction; this ensures that ''F'' has excellent analytic properties, but whether it is actually a complex-analytic function depends on the particular case. The third condition is to handle the case where ''G''/Γ is not compact but has cusps. The formulation requires the general notion of ''factor of automorphy'' ''j'' for Γ, which is a type of 1-cocycle in the language of group cohomology. The values of ''j'' may be complex numbers, or in fact complex square matrices, corresponding to the possibility of vector-valued automorphic forms. The cocycle condition imposed on the factor of automorphy is something that can be routinely checked, when ''j'' is derived from a Jacobian matrix, by means of the chain rule. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Automorphic form」の詳細全文を読む スポンサード リンク
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