Functoriality is the conjecture that automorphic forms on different groups should be related in terms of their-groups.
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Now suppose we are given a family of automorphic forms with a complex parameter " s ".
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Further, the automorphic forms for a given factor of automorphy correspond to sections of the corresponding line bundle.
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The theory is particularly well developed for symmetric spaces and provides a theory of automorphic forms ( discussed below ).
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This is almost possible and leads to a description of all automorphic forms in terms of these constructs and cusp forms.
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It requires the existence of an automorphic form, starting with an object that ( very abstractly ) lies in a category of motives.
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Automorphic forms are a generalization of modular forms to more general analytic functions, perhaps of several complex variables, with similar transformation properties.
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The Kuznetsov or " relative trace " formula connects Kloosterman sums at a deep level with the spectral theory of automorphic forms.
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Orbital integrals are an important technical tool in the theory of automorphic forms, where they enter into the formulation of various trace formulas.
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Modular forms are automorphic forms defined over the groups adelic approach as a way of dealing with the whole family of congruence subgroups at once.
