#### PhD Thesis colloquium

##### Venue: LH-1, Mathematics Department (Hybrid mode)

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 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 $k^{9/2}$. 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 an individual Hecke eigen-forms however, with the sup-norm being taken over a compact set of the Siegel’s fundamental domain instead. In the level aspect, the variation in sup-norm of the Bergman kernel for congruent subgroups $\Gamma_0^2(p)$ are studied and bounds for them are provided. We further consider this problem for the case of Saito-Kurokawa lifts and obtain suitable results.

Contact: +91 (80) 2293 2711, +91 (80) 2293 2265 ;     E-mail: chair.math[at]iisc[dot]ac[dot]in
Last updated: 03 Feb 2023