Automorphic forms and representations for GL(n, A_ℚ)

Simultaneous non-vanishing and sign changes of Fourier coefficients of modular forms

“…x λ 2 f (p)λ 2 g (p). For our purpose, the following trivial upper bound is sufficient which we get using (2).…”

On sums involving Fourier coefficients of Maass forms for $\mathrm{SL}(3,\mathbb Z)$

“…where d * z is the GL(n, R)-invariant measure on the generalised upper-half plane H n ≃ SL(n, R)/SO(n, R), see Section 1.5 of [6]. Here the group GL(n, R) acts on H n by left matrix multiplication.…”

“…For the proofs of these facts, see [6,Theorem 9.3.11. ] For a Hecke eigenfunction, one can use Möbius inversion to show that the relation…”

“…is the group of (n − 1) × (n − 1) upper triangular matrices with ones on the diagonal and integral entries above the diagonal, ψ (1,...,1,mn−1/|mn−1|) is a certain character, and W Jacquet is the Jacquet-Whittaker function of type ν for the character ψ (1,...,1,mn−1/|mn−1|) . For more details we refer to Goldfeld's book [6]. We further assume that the Maass cusp form f is an eigenfunction for the full Hecke ring and normalised so that A(1, ..., 1) = 1.…”

The shape of cubic fields

“…This representation shows that F φ is left invariant under SL 2 (Z). By applying the Hecke operators and Casimir operator to φ under the integral, it follows that F φ has the same Hecke and Casimir eigenvalues as φ, and hence, by strong multiplicity 1, that F φ = Λ f,φ φ for some constant Λ f,φ , see [12]. To demonstrate that the constant can be taken non-zero, write h = k θ 1 a t k θ 2 = k θ 1 k θ 2 (a t ) k θ 2 , where (a t ) k θ 2 = k −θ 2 a t k θ 2 .…”