2022/06/05 by Soumya Das, Das, Soumya, Hariram Krishna +1
Mathematics · #Advanced Algebra and Geometry #Analytic and geometric function theory #Algebraic Geometry and Number Theory
paper · pdf · doi:10.48550/arxiv.2206.02190
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 on the aforementioned size, while the lower bounds match for all n ≥ 1. For an L2-normalised 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, \mathbb 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.