Executive Summary : | Modular forms and their generalizations are one of the most important concepts in modern mathematics. In the words of a great mathematician Martin Eichler: "There are five basic operations in arithmetic: addition, subtraction, multiplication, division, and modular forms". A key approach to understand modular forms is to understand their Fourier coefficients, Hecke eigenvalues and inter-relations between these two objects. Basic questions about Fourier coefficients and Hecke eigenvalues can lead to deep mathematics and can be hard to answer. For example, a conjecture of Ramanujan (from 1916) about the upper bound of Fourier coefficients of modular forms was solved by Deligne in 1974 as a consequence of his Fields medal winning result. Another example is the Lehmer’s conjecture for non-vanishing of the Ramanujan tau-function (Fourier coefficients of a special modular form which is known as Ramanujan Delta function) which is still open. In a recent preprint, we gave a new interpretation of Lehmer’s conjecture in term of sign change of the tau-function. This is derived from a more general result which connects the phenomenon of sign change of Fourier coefficients and multiplicity one theorem. Some important generalizations of (elliptic) modular forms are Siegel modular forms and Hilbert modular forms.
The first main aim of the proposed research is to make progress towards various conjectures in the theory of Siegel modular forms which can be viewed as generalizations of the Ramanujan's conjecture mentioned in the earlier paragraph. Further we want to study inter-relations among these conjectures and suggest new ways to approach these conjectures. Our next proposal is to generalize the connection of sign change and multiplicity one theorem in the context of Hilbert modular forms. Lastly, we propose to study location and nature of zeros of Siegel Eisenstein series which is relatively unexplored till-date. Since modular forms are ubiquitous in Mathematics and appear also in certain areas in theoretical physics, the academic communities in India and abroad will be benefited from the outcome of the project. The successful completion of the project will lead to better understanding of Siegel and Hilbert modular forms which have great importance in mathematics, particularly, in Number Theory. |