Automorphic forms on GL(2)
نویسندگان
چکیده
For us it is imperative not to consider functions on the upper half plane but rather to consider functions on GL(2,Q)\GL(2,A(Q)) where A(Q) is the adéle ring of Q. We also replace Q by an arbitrary number field or function field (in one variable over a finite field) F . One can introduce [3] a space of functions, called automorphic forms, and the notion that an irreducible representation π ofGL(2,A(F )) is a constituent of the space of automorphic forms onGL(2, F )\GL(2,A(F )).
منابع مشابه
Lecture 16: Review of representation theory
• The theory of admissible representations of GL(2,Qp) (or more generally, GL(2, F ) with F/Qp a finite extension). • The theory of automorphic representations of GL(2); in particular, the correspondence between Hecke eigenforms in the classical sense and automorphic representations. • The Jacquet-Langlands correspondence, relating automorphic forms on GL(2) with those on a division algebra. • ...
متن کاملClassical automorphic forms and representations of SL ( 2 )
This essay will explain the relationship between classical automorphic forms and representations of GL 2 (R). The classical theory of automorphic forms, in spite of initial appearances, is about the group GL 2 , not SL 2. The classical theory is concerned with functions on the upper half plane, which is acted on by fractional linear transformations in GL pos 2 (R), and it happens that the inter...
متن کاملTruncation and Maaß-Selberg Relations
This bears upon the construction of non-trivial residual square-integrable automorphic forms coming from cuspforms on smaller groups, anticipating that such automorphic forms occur as residues of Eisenstein series. For example, we can see why there is no interesting (i.e., non-constant) non-cuspidal discrete spectrum for GL(2) nor for GL(3), but only for GL(4) and larger groups. Namely, the Eis...
متن کاملAnalysis on arithmetic quotients: SL(2) Classical and adelic automorphic forms
This essay will exhibit the realization of discrete series representations of SL 2 (R) on spaces of holomor-phic functions and relate them to automorphic forms for certain arithmetic groups. In a second essay I shall relate these in turn to representations of adèle groups, and more generally make remarks about the connection between arithmetic quotients and adelic quotients. By now the realizat...
متن کاملModular Forms and - adic Representations ∗
the circumstance that L-functions can be introduced not only in the context of automorphic forms, with which he has had some experience, but also in the context of diophantine geometry. That this circumstance can be the source of deep problems was, I believe, first perceived by E. Artin. He was, to be sure, concerned with forms on GL(1) and with varieties of dimension 0. This remains the only c...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 1970