On the Fourier Coefficients of Automorphic Forms on Gl2
نویسندگان
چکیده
Given E/F a quadratic extension of number fields and a cuspidal representation π of GL2(AE), we give a full description of the fibers of the Asai transfer of π. We then determine the extent to which the Hecke eigenvalues of all the Hecke operators indexed by integral ideals in F determine the representation π.
منابع مشابه
Toroidal Automorphic Forms, Waldspurger Periods and Double Dirichlet Series
The space of toroidal automorphic forms was introduced by Zagier in the 1970s: a GL2-automorphic form is toroidal if it has vanishing constant Fourier coefficients along all embedded non-split tori. The interest in this space stems (amongst others) from the fact that an Eisenstein series of weight s is toroidal for a given torus precisely if s is a non-trivial zero of the zeta function of the q...
متن کاملWeek 2: Betti Cohomology of Shimura Varieties — the Matsushima Formula
Rough exposition of the goal of this course. Hecke actions on Shimura varieties and their cohomology groups. The Matsushima formula in the classical setting; relative Lie algebra cohomology, spectral decomposition on L(Γ\G). The Matsushima formula in the adelic setting — automorphic representations appear in the Betti cohomology of Shimura varieties. Admissible (g, U)-modules and cohomological ...
متن کاملBounds for the multiplicities of cohomological automorphic forms on GL2
We prove a power saving for the dimension of the space of cohomological automorphic forms of fixed level and growing weight on GL2 over any number field that is not totally real. Our proof involves the theory of p-adically completed cohomology developed by Calegari and Emerton and a bound for the growth of coinvariants in certain finitely generated noncommutative Iwasawa modules.
متن کاملApplications of a Pre–trace Formula to Estimates on Maass Cusp Forms
By using spectral expansions in global automorphic Levi–Sobolev spaces, we estimate an average of the first Fourier coefficients of Maass cusp forms for SL2(Z), producing a soft estimate on the first numerical Fourier coefficients of Maass cusp forms, which is an example of a general technique for estimates on compact periods via application of a pre–trace formula. Incidentally, this shows that...
متن کاملNotes on the Generalized Ramanujan Conjectures
Ramanujan’s original conjecture is concerned with the estimation of Fourier coefficients of the weight 12 holomorphic cusp form ∆ for SL(2,Z) on the upper half plane H. The conjecture may be reformulated in terms of the size of the eigenvalues of the corresponding Hecke operators or equivalently in terms of the local representations which are components of the automorphic representation associa...
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 2011