Numerical Computation of a Certain Dirichlet Series Attached to Siegel Modular Forms of Degree Two

The Rankin convolution type Dirichlet series D-F,D-G(s) of Siegel modular forms F and G of degree two, which was introduced by Kohnen and the second author, is computed numerically for various F and G. In particular, we prove that the series D-F,D-G(s), which shares the same functional equation and analytic behavior with the spinor L-functions of eigenforms of the same weight are not linear combinations of those. In order to conduct these experiments a numerical method to compute the Petersson scalar products of Jacobi Forms is developed and discussed in detail.

Computations of Vector-Valued Siegel Modular Forms

We carry out some computations of vector-valued Siegel modular forms of degree two, weight (k, 2) and level one, and highlight three experimental results: (1) we identify a rational eigenform in a three-dimensional space of cusp forms; (2) we observe that non-cuspidal eigenforms of level one are not always rational; (3) we verify a number of cases of conjectures about congruences between classical modular forms and Siegel modular forms. Our approach is based on Satoh's description of the module of vector-valued Siegel modular forms of weight (k, 2) and an explicit description of the Hecke action on Fourier expansions. (C) 2013 Elsevier Inc. All rights reserved.

The Design of XML-Based Model and Experiment Description Languages for Network Simulation

The Simulation Automation Framework for Experiments (SAFE) is a project created to raise the level of abstraction in network simulation tools and thereby address issues that undermine credibility. SAFE incorporates best practices in network simulationto automate the experimental process and to guide users in the development of sound scientific studies using the popular ns-3 network simulator. My contributions to the SAFE project: the design of two XML-based languages called NEDL (ns-3 Experiment Description Language) and NSTL (ns-3 Script Templating Language), which facilitate the description of experiments and network simulationmodels, respectively. The languages provide a foundation for the construction of better interfaces between the user and the ns-3 simulator. They also provide input to a mechanism which automates the execution of network simulation experiments. Additionally,this thesis demonstrates that one can develop tools to generate ns-3 scripts in Python or C++ automatically from NSTL model descriptions.

Verifying Harder's Conjecture for Classical and Siegel Modular Forms

A conjecture by Harder shows a surprising congruence between the coefficients of “classical” modular forms and the Hecke eigenvalues of corresponding Siegel modular forms, contigent upon “large primes” dividing the critical values of the given classical modular form. Harder’s Conjecture has already been verified for one-dimensional spaces of classical and Siegel modular forms (along with some two-dimensional cases), and for primes p 37. We verify the conjecture for higher-dimensional spaces, and up to a comparable prime p.