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

Abstract

We formulate a precise conjecture about the size of the $$L^\infty $$ -mass of the space of Jacobi forms on $$\mathbb {H}_n \times \mathbb C^{g \times n}$$ of matrix index S of size g. This $$L^\infty $$ -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 $$n=1$$ , $$g \ge 1$$ . 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 $$L(1/2, f \otimes \chi _D)$$ with f varying over newforms of level a prime p and even weight k as $$k,p \rightarrow \infty $$ and D being (explicitly) polynomially bounded by k, p. Here $$\chi _D$$ 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 $$\mathbb {H}\times \mathbb {H}$$ 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 $$k^3$$ .