Modular invariants for real quadratic fields and Kloosterman sums

A symplectic restriction problem

“…In this paper we consider a new and somewhat different instance of the interplay between mass equidistribution and L-functions in the context of certain Siegel modular forms F for the symplectic group Sp 4 (Z): we investigate the L 2 -restriction of a Saito-Kurokawa lift F(Z ) on the 6-dimensional Siegel upper half space H (2) to the 3-dimensional subspace where the argument Z = X + iY ∈ H (2) is restricted to its imaginary part. This is geometrically a very natural set-up, it is a direct higherdimensional analogue of the classical problem of bounding a cusp form f for SL 2 (Z) on the vertical geodesic, mentioned at the beginning; cf.…”

“…We start by stating the corresponding period formula. For an even positive integer k let S (2) k denote the space of Siegel modular forms of degree 2 of weight k for the group Sp 4 (Z), equipped with the standard Petersson inner product; see Sect. 2.…”

“…2. We think of k as tending to infinity and are interested in asymptotic results with respect to k. We restrict the argument of a cusp form F ∈ S (2) k to its imaginary part iY with Y ∈ P(R) where P(R), equipped with the measure dY /(det Y ) 3/2 , is the set of positive definite symmetric 2-by-2 matrices. Consider the restriction norm…”

“…Letting H denote the usual upper half plane, we observe that SL 2 (Z)\P(R) ∼ = SL 2 (Z)\H × R >0 has infinite measure; see (6.1) below. The factor (2) ) vol(SL 2 (Z)\H)…”

“…accounts for the fact that, in accordance with the literature, we choose the standard measures on Sp 4 (Z)\H (2) and SL 2 (Z)\H which are not probability measures. Let denote a set of spectral components of L 2 (SL 2 (Z)\H) consisting of the constant function √ 3/π , an orthonormal basis of Hecke-Maaß cusp forms and the Eisenstein series E(., 1/2 + it) for t ∈ R. The set is equipped with the counting measure on its discrete part and with the measure dt/4π on its continuous part.…”