Special Values of L-functions

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The subject of L-functions has its origin in the study of the Riemann zeta function, and is largely driven by the Riemann hypothesis, a conjecture that continues to elude mathematicians for more than 150 years. Over the last 50 years, a significant part of research in number theory has been centred around the special values of L-functions. This trend will surely continue in view of the bounty of conjectures that this topic encompasses and the major implications they would have on our understanding of number fields, representations, Abelian varieties and automorphic forms. For example, the vanishing or nonvanishing of L-functions and their twists at the center of the critical strip have rich arithmetic implications, as it is vividly portrayed by the Birch and Swinnerton-Dyer Conjecture and its generalizations.This research is primarily motivated by questions regarding the nonvanishing of L-functions in the critical strip. A crucial step in answering such questions is the study of the moments associated with these values; a topic that is interesting in its own right and plays a pivotal role in my research. A conjecture of Chowla (1965) states that the central values of Dirichlet L-functions are nonzero. While this conjecture remains open, there has been remarkable progress towards it (e.g. Soundararajan positive proportion result). Likewise, it is also believed that the central values of modular L-functions are nonvanishing unless there is a trivial (e.g. sign in the functional equation) or an arithmetical reason for these values to vanish. In this research, I explore families of L-functions associated with Hilbert modular forms. I am especially focused on the convolution of two such forms which are varying in the weight or the level aspects. A major component of this research program is to study various moments of certain families of central L-values associated with Hilbert modular forms of varying weight. This will allow us to achieve strong results regarding the size of these values and their non-vanishing. In this research, I also study the value-distribution of certain families of L-functions. On this front, I made some progress pertaining to the particular family associated with cubic Hecke characters. I plan to extend my results to include higher order characters. I also plan to investigate this problem for quadratic twists of automorphic L-functions using the modern probabilistic framework. Another objective of this research program is to establish a mean value theorem for certain long Dirichlet polynomials. This topic is intimately related to moments of L-functions and convolution sums. Finally, I plan to continue my research on half-integral weight modular forms by computing their spaces in more general settings.This research program will enrich our understanding of various aspects of the special values of L-functions while moving this beautiful area of number theory forward.