Numerical Computations with the Trace Formula and the Selberg Eigenvalue Conjecture
نویسنده
چکیده
We verify the Selberg eigenvalue conjecture for congruence groups of small squarefree conductor, improving on a result of Huxley [20]. The main tool is the Selberg trace formula which, unlike previous geometric methods, allows for treatment of cases where the eigenvalue 1 4 is present. We present a few other sample applications, including the classification of even 2-dimensional Galois representations of small squarefree conductor.
منابع مشابه
On a Generalization of the Selberg Trace Formula
appear (the uj run over an orthonormal basis of automorphic Laplaceeigenforms), so our formula (Theorem 1) is a duality between such integrals and certain geodesic integrals of u. New integral transformations are involved depending on the Laplace-eigenvalue of u. We invert these integral transformations in Section 5, Theorem 2. We develop the formula for finite volume Fuchsian groups, so (as in...
متن کاملSome Explicit Cases of the Selberg Trace Formula for Vector Valued Functions
The trace formula for SL{2,Z) can be developed for vector-valued functions which satisfy an automorphic condition involving a group representation n . This paper makes this version explicit for the class of representations which can be realized as representations of the finite group PSL(2,Z/q) for some prime q . The body of the paper is devoted to computing, for the singular representations n ,...
متن کاملA Simple Trace Formula for Algebraic Modular Forms
We derive an elementary formula for the trace of a Hecke operator acting on a space of algebraic modular forms, as a sum of character values. We describe explicit computations in the case of the unitary group U(4), allowing the determination of the eigenvalues of a certain Hecke operator. This produces numerical evidence for a U(2, 2) analogue of Harder’s conjecture, on congruences between Heck...
متن کاملThe Trace Formula for Noncompact Quotient
1. In [12] and [13] Selberg introduced a trace formula for a compact, locally symmetric space of negative curvature. There is a natural algebra of operators on any such space which commute with the Laplacian. The Selberg trace formula gives the trace of these operators. Selberg also pointed out the importance of deriving such a formula when the symmetric space is assumed only to have finite vol...
متن کاملThe Selberg Trace Formula and Selberg Zeta-Function for Cofinite Kleinian Groups with Finite Dimensional Unitary Representations
For cofinite Kleinian groups, with finite-dimensional unitary representations, we derive the Selberg trace formula. As an application we define the corresponding Selberg zeta-function and compute its divisor, thus generalizing results of Elstrodt, Grunewald and Mennicke to non-trivial unitary representations. We show that the presence of cuspidal elliptic elements sometimes adds ramification po...
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 2007