The sup-norm problem on the Siegel modular space of rank two

Abstract

Abstract. Let F be a square integrable Maaß form on the Siegel upper half space H of rank 2 for the Siegel modular group Sp 4 (Z) with Laplace eigenvalue λ. If, in addition, F is a joint eigenfunction of the Hecke algebra and Ω is a compact set in Sp 4 (Z)\H, we show the bound F | Ω ∞ ≪ Ω (1 + λ) 1−δ for some global constant δ > 0. As an auxiliary result of independent interest we prove new uniform bounds for spherical functions on semisimple Lie groups.

Integers represented by positive-definite quadratic forms and Petersson inner products

“…For each such choice, R(x, y, z, w) is a binary quadratic polynomial (with coefficients that are polynomial in n). The proof of Lemma 8 in [5] implies that if P = ax 2 + bxy + cy 2 + dx+ ey + f , with b 2 −4ac < 0, then number of solutions to P (x, y) = 0 is at most 6d(|4a…”

Subconvexity for sup-norms of cusp forms on $$\mathrm{PGL}(n)$$ PGL ( n )

“…By the Harish-Chandra inversion formula together with the uniform bounds for elementary spherical functions in [BP,Theorem 2], we see that the inverse spherical transform f µ : K\G/K → C off µ has compact support and satisfies the decay property…”

“…We are then left with an inhomogeneous binary problem in y 1 , y 2 whose (positive definite) quadratic homogeneous part has discriminant |D| h 2 1 . By [BP,Corollary 9] with δ = 0 there are at most h 1 ,hn m ε choices for y 1 , y 2 .…”

Analytic Properties of Spherical Cusp Forms on GL(n)

“…Following [BlMa,Section 2] and [BlPo,Sections 2 & 6], we can construct a smooth, bi-O n (R)-invariant function f µ : PGL n (R) → C supported in a fixed compact subset Ω (which is independent of µ ∈ a * C ) with the following properties. On the one hand, the function obeys (cf.…”

“…On the other hand, its spherical transform f µ : a * C /S n → C, defined as in [He,(17) of Section II.3] or [BlPo,(2.3)], satisfies f µ (µ) 1,f µ (κ) 0 for all Langlands parameters κ ∈ a * C occurring in L 2 (X n ), including possibly non-tempered parameters. Then, using positivity in Selberg's pre-trace formula (see [BlPo,(6.1)]), or more directly by a Mercer-type pre-trace inequality (cf. [BHMM,(3.15)]), we obtain (68) |φ(z)| 2 γ∈PGLn(Z) f µ (z −1 γz).…”