On geometric density of Hecke eigenvalues for certain cusp forms

Abstract

Abstract. In this paper we prove certain density results for Hecke eigenvalues as well as we give estimates on the length of modules for Hecke algebra acting on the cusp forms constructed out of Poincaré series for a semisimple group G over a number field k. The cusp forms discusses here are taken from [15] and they generalize usual cuspidal modular forms S k (Γ) of weight k ≥ 3 for a Fuchsian group Γ [17].

On the structure of the space of cusp forms for a semisimple group over a number field

“…Now, as in the proof ( [13], Lemma 2-27), we show the claim in (3)(4)(5)(6)(7)(8)(9)(10)(11)(12). Now, (3)(4)(5)(6)(7)(8)(9)(10)(11) and (3)(4)(5)(6)(7)(8)(9)(10)(11)(12) show that n≥1 I(ϕ S , L …”

“…The first claim follows directly from Lemma 3-8 (iii). The other claim has the same proof as ( [13], Theorem 3-1). Now, we begin the proof of Theorem 1-…”

“…Finally, we prove (iii). The proof of (iii) can be done on the (simplified) lines of the proof of ( [13], Theorem 2-8 (ii)). We give some hints and leave the details to the reader.…”

“…It is well-known that the Satake isomorphism can be used to identify this algebra with the algebra of regular functions on a certain complex affine algebraic variety (see [4]; see also [13], Section 1). We denote this variety by Spec max H T .…”

“…Then there exists a well-known notion of Bernstein classes for G(k v ) (see for example [11] The proof of Theorem 1-1 is given in Section 3. It is based on a spectral decomposition of Poincaré series for compactly supported cuspidal functions in L 2 (G(k) \ G(A)), a method that was already successfully applied to study cusp forms in [11] and [12], combined with a method of reducing the problem to Sarithmetic cuspidal forms [13], Theorem 0-3 and Theorem 0-4. The idea for the studying of H T -modules of Poincaré series to obtain results such as Theorem 1-1 is taken from [13], but the main problem here is to find the correct form of a cuspidal compactly supported Poincaré series in order to prove Theorem 1-1 (see .…”