Kuznietsov trace formula and weighted distribution of Hecke eigenvalues
نویسندگان
چکیده
منابع مشابه
2 The Trace Formula and Hecke Operators
This lecture is intended as a general introduction to the trace formula. We shall describe a formula that is a natural generalization of the Selberg trace formula for compact quotient. Selberg also established trace formulas for noncompact quotients of rank 1, and our formula can be regarded as an analogue for general rank of these. As an application, we shall look at the "finite case" of the t...
متن کاملThe Eichler-Selberg Trace Formula for Level-One Hecke Operators
This paper explains the steps involved in the proof of the Eichler-Selberg Trace Formula for Hecke operators of level one. It is based on an appendix in Serge Lang’s Introduction to Modular Forms written by Don Zagier, though I also draw heavily from sections of Toshitsune Miyake’s Modular Forms and Xueli Wang’s and Dingyi Pei’sModular Forms with Integral and Half-Integral Weights. Section 2 su...
متن کاملThe Distribution of the Eigenvalues of Hecke Operators
τ(n)e(nz). The two equations were proven for τ(n) by Mordell, using what are now known as the Hecke operators. The inequality was proven by Deligne as a consequence of his proof of the Weil conjectures. Those results determine everything about af (n) except for the distribution of the af (p) ∈ [−2, 2]. Define θf (p) ∈ [0, π] by af (p) = 2 cos θf (p). It is conjectured that for each f the θf (p)...
متن کاملSign Changes of Hecke Eigenvalues
in the Maass case. In [14] Kowalski, Lau, Soundararajan and Wu investigate the problem of the first sign change of λf (n) for holomorphic f . They remark on the similarities with the problem of the least quadratic residue. This motivates the point of view that the signs of λf (n) are GL(2) analogues of real characters. The frequency of signs and sign changes and other related questions have bee...
متن کاملThe Trace Formula and the Distribution of Eigenvalues of Schrr Odinger Operators on Manifolds All of Whose Geodesics Are Closed
We investigate the behaviour of the remainder term R(E) in the Weyl formula #fnjE n Eg = Vol(M) (4) d=2 ?(d=2 + 1) E d=2 + R(E) for the eigenvalues E n of a Schrr odinger operator on a d-dimensional compact Rieman-nian manifold all of whose geodesics are closed. We show that R(E) is of the form E (d?1)=2 (p E), where (x) is an almost periodic function of Besicovitch class B 2 which has a limit ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 2004
ISSN: 0022-314X
DOI: 10.1016/s0022-314x(03)00149-5