Automorphic forms of higher order. (English) Zbl 1239.11052
Introduction: Higher-order modular forms have in recent years arisen in various contexts
and they have been studied as analytic functions. In particular their spaces have been investigated. Parallel to that, \(L\)-functions were attached to higher-order forms and some of their basic aspects studied. In the first author’s paper [J. Reine Angew. Math. 629, 221–235 (2009; Zbl 1254.11051)], a cohomology theory for higher invariants is developed, and an Eichler-Shimura isomorphism for higher order forms is established.
The current paper serves the following purposes,
– to establish a theory of Hecke operators on higher order forms,
– to extend the definition of higher forms beyond parabolic invariants,
– to clarify the role of representation theoretic methods in the theory, and
– to introduce new convolution products of L-functions of higher forms.
The first item fills a long-standing gap in the theory of higher order forms by constructing a natural Hecke action. This is surprising, as there is no adelic counterpart of higher order forms. The Hecke operators form bounded self-adjoint operators on direct limits of spaces of higher order forms. It is an on-going long-term project of the authors to better understand the spectral decompositions of Hecke operators. For the second item, one gets that in the case of higher order forms, the crucial Fourier expansion is replaced by a “Fourier-Taylor-expansion” which is introduced in this paper. For the third item, representation theoretic methods, it turns out that a intervention of Lie groups, as Dieudonné terms it, is possible in the theory and, in fact, higher-order forms can be incorporated into the same representational context as the classical automorphic forms. One would thus expect a distinction from classical forms in the intervention of adeles. Indeed, it turns out that there is no intervention of adeles, as there are no higher forms on the adelic level. The last item in the list, the convolution product, is inspired by the second insofar as the \(L\)-functions of higher-order forms are special cases of the convolution products. We show analyticity and functional equation in greater generality.
The current paper serves the following purposes,
– to establish a theory of Hecke operators on higher order forms,
– to extend the definition of higher forms beyond parabolic invariants,
– to clarify the role of representation theoretic methods in the theory, and
– to introduce new convolution products of L-functions of higher forms.
The first item fills a long-standing gap in the theory of higher order forms by constructing a natural Hecke action. This is surprising, as there is no adelic counterpart of higher order forms. The Hecke operators form bounded self-adjoint operators on direct limits of spaces of higher order forms. It is an on-going long-term project of the authors to better understand the spectral decompositions of Hecke operators. For the second item, one gets that in the case of higher order forms, the crucial Fourier expansion is replaced by a “Fourier-Taylor-expansion” which is introduced in this paper. For the third item, representation theoretic methods, it turns out that a intervention of Lie groups, as Dieudonné terms it, is possible in the theory and, in fact, higher-order forms can be incorporated into the same representational context as the classical automorphic forms. One would thus expect a distinction from classical forms in the intervention of adeles. Indeed, it turns out that there is no intervention of adeles, as there are no higher forms on the adelic level. The last item in the list, the convolution product, is inspired by the second insofar as the \(L\)-functions of higher-order forms are special cases of the convolution products. We show analyticity and functional equation in greater generality.
MSC:
| 11F12 | Automorphic forms, one variable |
| 11F25 | Hecke-Petersson operators, differential operators (one variable) |
| 11F66 | Langlands \(L\)-functions; one variable Dirichlet series and functional equations |
