Poincaré series for SO(n, 1)

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

“…There is a different type of (partial) Poincaré series developed by Selberg [19] in a classical settings as a spectral tool for analyzing arithmetic questions, and further studied by many people (see for example [8], [9], and [10], Chapter 11).…”

“…We show that f is a cuspidal function on G ∞ . We apply the Fourier expansion (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16).…”

“…Since (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17) implies that the series δ∈K ∞ E l δ (f ) converges uniformly on compacta to f on G ∞ and (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19) …”

Spectral large sieve inequalities for Hecke congruence subgroups ofSL(2,Z[i])

“…In Theorem 2.16 of [7] Cogdell, Li, Piatetski-Shapiro and Sarnak have shown that if a and b are Γ-equivalent cusps then the Linnik-Selberg series Z a,b ω,ω ′ (s) can be meromorphically continued into all of C. In the case of the analogous series involving generalised Kloosterman sums associated with any group Γ ′ < SL(2, R) that is a congruence subgroup with respect to SL(2, Z), the corresponding meromorphic continuation was obtained by Selberg, in Section 3 of [41].…”

“…By (1.4.2)-(1.4.3), the operator F c ω maps each f in the space C ∞ (Γ\G) to an even function F c ω f in the space C ∞ (N \G, ω) = {h ∈ C ∞ (G) : h(ng) = ψ ω (n)h(g) for n ∈ N, g ∈ G} (1.4. 7) and commutes with the actions (as differential operators upon those spaces) of the elements of U(g). Consequently F c ω maps functions f ∈ A Γ (Υ; ℓ, q) to functions F c ω f lying in the complex vector space W ω (Υ; ℓ, q) = {h ∈ C ∞ (N \G, ω) : h is of K-type (ℓ, q) with character Υ} .…”