Bounds for the Bergman Kernel and the Sup-Norm of Holomorphic Siegel Cusp Forms

Das, S and Krishna, H (2024) Bounds for the Bergman Kernel and the Sup-Norm of Holomorphic Siegel Cusp Forms. In: International Mathematics Research Notices, 2024 (7). pp. 6140-6175.

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Abstract

We prove �polynomial in k� bounds on the size of the Bergman kernel for the space of holomorphic Siegel cusp forms of degree n and weight k. When n = 1, 2 our bounds agree with the conjectural bounds, while the lower bounds match for all n � 1. For an L2-normalized Siegel cusp form F of degree 2, our bound for its sup-norm is Oϵ(k9/4+ϵ). Further, we show that in any compact set Ω (which does not depend on k) contained in the Siegel fundamental domain of Sp(2, Z) on the Siegel upper half space, the sup-norm of F is OΩ(k3/2�η) for some η > 0, going beyond the �generic� bound in this setting. © The Author(s) 2024. Published by Oxford University Press. All rights reserved.

Item Type:	Journal Article
Publication:	International Mathematics Research Notices
Publisher:	Oxford University Press
Additional Information:	The copyright for this article belongs to Oxford University Press.
Department/Centre:	Division of Physical & Mathematical Sciences > Mathematics
Date Deposited:	11 Jul 2024 05:43
Last Modified:	11 Jul 2024 05:43
URI:	http://eprints.iisc.ac.in/id/eprint/84803

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