Jacobi forms, Saito-Kurokawa lifts, their Pullbacks and sup-norms on average

Abstract

We formulate a precise conjecture about the size of the L ∞ -mass of the space of Jacobi forms on H n × C g×n of matrix index S of size g. This L ∞ -mass is measured by the size of the Bergman kernel of the space. We prove the conjectured lower bound for all such n, g, S and prove the upper bound in the k aspect when n = 1, g ≥ 1. When n = 1 and g = 1, we make a more refined study of the sizes of the index-(old and) new spaces, the latter via the Waldspurger's formula. Towards this and with independent interest, we prove a power saving asymptotic formula for the averages of the twisted central L-values L(1/2, f ⊗ χ D ) with f varying over newforms of level a prime p and even weight k as k, p → ∞ and D being (explicitly) polynomially bounded by k, p. Here χ D is a real quadratic Dirichlet character. We also prove that the size of the space of Saito-Kurokawa lifts (of even weight k) is k 5/2 by three different methods (with or without the use of central L-values), and show that the size of their pullbacks to the diagonally embedded H × H is k 2 . In an appendix, the same question is answered for the pullbacks of the whole space S 2 k , the size here being k 3 .