Research Projects

Number theory is a crucial aspect of mathematics, encompassing algebra, representation theory, harmonic analysis, cryptography, and physics. It has applications in error correcting codes and secret codes. Analytic number theory focuses on L-functions, which reveal statistics of zero distributions and imply distribution results for sequences of integers. The non-vanishing of L-functions associated to Dirichlet characters at 1 implies infinitely many prime numbers in arithmetic progressions. The Riemann zeta function is a special case of an L-function, which is connected to the Riemann hypothesis. Modular forms, complex-valued holomorphic functions defined on the complex upper half-plane with infinite symmetries, are important objects in modern mathematics. They appear in various contexts, such as Fermat's Last theorem, q-hypergeometric series, partitions, and elliptic curves. By using the Fourier coefficients of modular forms, L-functions can be associated with them in various ways, such as understanding rational solutions to equations defining an elliptic curve. Special types of modular forms include Poincare series, which span certain subspaces of modular forms called cusp forms. Siegel modular forms, introduced in 1935, study the analytic theory of quadratic forms. This project aims to investigate two key problems: studying the analytic properties of certain L-functions associated to Siegel, Hermitian, and Jacobi modular forms, and the non-vanishing of Hermitian Poincare series.