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

Anamby, P and Das, S (2023) Jacobi forms, Saito-Kurokawa lifts, their Pullbacks and sup-norms on average. In: Research in Mathematical Sciences, 10 (1).

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Official URL: https://doi.org/10.1007/s40687-023-00377-z


We formulate a precise conjecture about the size of the L∞-mass of the space of Jacobi forms on Hn× Cg×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⊗ χD) 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 χD is a real quadratic Dirichlet character. We also prove that the size of the space of Saito-Kurokawa lifts (of even weight k) is k5 / 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 k2. In an appendix, the same question is answered for the pullbacks of the whole space Sk2, the size here being k3.

Item Type: Journal Article
Publication: Research in Mathematical Sciences
Publisher: Springer Science and Business Media Deutschland GmbH
Additional Information: The copyright for this article belongs to the Authors.
Keywords: Bergman kernel; Central values of twisted L-functions; Eichler-Zagier maps; Jacobi forms; Saito-Kurokawa lifts; Sup-norm
Department/Centre: Division of Physical & Mathematical Sciences > Mathematics
Date Deposited: 15 Mar 2023 09:14
Last Modified: 15 Mar 2023 09:14
URI: https://eprints.iisc.ac.in/id/eprint/80974

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