Test vectors for trilinear forms when at least one representation is not supercuspidal

Abstract

Given three irreducible, admissible, infinite dimensional complex representations of GL 2 (F ), with F a local field, the space of trilinear functionals invariant by the group has dimension at most one. When it is one we provide an explicit vector on which the functional does not vanish assuming that not all three representations are supercuspidal.

Quantum variance on quaternion algebras, I

“…By estimating the integral of | Eis * (f )(z)Φ(z)| over a Siegel domain using §10. 13, it follows that…”

“…If k = l, so that V kl ∞ := 0, the strong multiplicity one theorem on Y furnishes an odd prime p for which λ Ψ k (p) = λ Ψ l (p). 12 The local data defining the lift Ψ k → h k is unramified at p, which is known by Eichler's commutation relations 13 to imply that h k is an eigenfunction of Shimura's T p 2 Hecke operator with eigenvalue proportional to λ Ψ k (p). Since T p 2 is self-adjoint for the Petersson inner product, the vanishing h k , h l = 0 follows.…”

“…Note that ϕ → θ Ẽ′ s , ϕ is linear, unlike ϕ → µ ϕ (E s ), and that the non-linear identity (13), like the triple product formula, applies only when ϕ is an eigenfunction. 9 Consider now the pair of unitary Eisenstein series Ψ 1 , Ψ 2 := E 1/2+it1 , E 1/2+it2 attached to t 1 , t 2 ∈ R. Let F be an orthonormal basis of eigenfunctions for the discrete spectrum of L 2 (SL 2 (Z)\H); write • • • for the corresponding contribution of the continuous spectrum, and ignore (for the purposes of this formal discussion) that most of the sums/integrals written below are divergent.…”

“…9 Consider now the pair of unitary Eisenstein series Ψ 1 , Ψ 2 := E 1/2+it1 , E 1/2+it2 attached to t 1 , t 2 ∈ R. Let F be an orthonormal basis of eigenfunctions for the discrete spectrum of L 2 (SL 2 (Z)\H); write • • • for the corresponding contribution of the continuous spectrum, and ignore (for the purposes of this formal discussion) that most of the sums/integrals written below are divergent. From (13) and Parseval, one obtains with h 1 , h…”

The orbit method and analysis of automorphic forms

“…Solutions to this problem have appeared in period-based approaches to the subconvexity period (see, e.g., [54, §3.6.1], [64, §2.17.1], [11], [73], [31], [18], [63,Rmk 50], [66, Thm 1.2]); the point is that the period formula (1.7) and a "trivial" estimate for the global period [H] vu often suffice to recover the convexity bound for L-function, suggesting the possibility for improvement in arithmetic settings via Hecke amplification.…”