Automorphic Forms on SL2 (R)

On geometric density of Hecke eigenvalues for certain cusp forms

“…Then we have the following: (2)(3)(4)(5)(6)(7)(8)(9)(10). Then, since ϕ generates U ϕ and belongs to U L 1 ϕ (D), we see the following: (2-12) (2-1)), we see that (2-11) is a decomposition of H T -modules.…”

“…Then, since ϕ generates U ϕ and belongs to U L 1 ϕ (D), we see the following: (2-12) (2-1)), we see that (2-11) is a decomposition of H T -modules. Moreover, (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12) implies that a submodule of U L 1 ϕ (D) generated by ϕ has a non-trivial projections to all U L 1 i (D). Since U i is irreducible, we can decompose into the restricted tensor product of irreducible representations U i U i,∞ ⊗ w∈V f U i,w (see [11]).…”

“…It was an extension and refinement of a classical method of Poincaré series ( [2], [6], [7]). In [17], we apply those considerations to write down examples of cuspidal modular in S k (Γ) of weight k ≥ 3 for an arbitrary Fuchsian group Γ of the first kind.…”

On a construction of certain classes of cuspidal automorphic forms via Poincaré series

“…As in the settings of semisimple Lie groups (see [6], Theorem 5.4, and [5], Theorem 6.1), our method of proof also follows the ideas of Harish-Chandra and Borel (see Section 2, Theorem 2-1 and Corollary 2-2) but it gives a slightly stronger result even in the classical settings since we obtain boundness of the constructed automorphic forms (compare to [6], Theorem 5.4 and Remarks 5.5, or its variant for SL(2, R) given in [5], Theorem 6.1 (ii)). The cuspidality in the classical settings is discussed in [23], Lemma 3.2, under the stronger assumption that function is both left and right K ∞ -finite.…”

“…In this paper we discuss the construction of cusp forms using an extension and refinement of a classical method of Poincaré series ( [6], [4], [5], [18], [23]). We adopt the adelic point of view.…”