The Bergman Kernel of Siegel Modular Forms: Bounds on the Sup-norm

Abstract

The primary goal of this dissertation is to establish bounds for the sup-norm of the Bergman kernel of Siegel modular forms. Upper and lower bounds for them are studied in the weight as well as the level aspect. We get the optimal bound in the weight aspect for degree 2 Siegel modular forms of weight $k$ and show that the maximum size of the sup-norm to be $k^{9/2}$. For higher degrees, a somewhat weaker result is provided. Under the Resnikoff-Saldana conjecture (refined with dependence on the weight), which provides the best possible bounds on Fourier coefficients of Siegel cusp forms, our bounds become optimal. Further, the amplification technique is employed to improve the generic sup-norm bound for individual Hecke eigenforms however, with sup-norm being taken over a compact set instead.

In the level aspect, the variation in sup-norm of the Bergman kernel for congruent subgroups $\Gamma_{2,0}(p)$ are studied and bounds for them are provided.