Adelic Modular Forms
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چکیده
Hecke’s theory is concerned with a family of finite-dimensional vector spaces Sk(N,χ), indexed by weights, levels, and characters. The Hecke operators on such spaces already provide a very rich theory. It will be very advantageous to pass to the adelic setting, however, for the same reasons that Hecke characters on number fields should be studied in the adelic setting (rather than as homomorphisms out of the group of fractional ideals). In Tate’s thesis, we learned that once we view Hecke characters as idele class characters, we can apply the tools of harmonic analysis on locally compact groups. We’re going to do something very similar with modular forms: we’ll view them as special functions (called automorphic forms) on an adelic group GL(2). Some advantages of this point of view are (at least):
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