Elliptic curves are fundamental objects in arithmetic geometry, together with their higher dimensional generalizations, abelian varieties, and their function field analogues, Drinfeld modules. The extraordinary interest in these objects emerges from their remarkable analytic, algebraic, arithmetic and geometric properties. Jean-Pierre Serre was the first to investigate elliptic curves from an analytic number theoretic point of view, in a systematic and innovative way, mostly motivated by conjectures proposed by Serge Lang and Hale Trotter in the 1970s in analogy with classical deep open problems in number theory. Serre's research on analytic properties for elliptic curves was subsequently continued, among others, by Rajiv Gupta, M. Ram Murty and V. Kumar Murty, and, more recently, by the PI herself. The PI proposes to expand and deepen her prior research investigations to higher dimensional abelian varieties and Drinfeld modules, through analytic tools, such as sieves, used in an arithmetic geometrical context. Her research efforts will corroborate with her educational endeavours, such as doctoral and postdoctoral mentoring, and will be made available to a broad audience through conference presentations and workshop organization.
The proposed research lies in the general area of number theory, a distinguished branch of mathematics with a marvelous long history and exciting current developments and applications. The central objects of study in number theory are prime numbers and Diophantine equations. The proposed research projects concern thorough studies of prime numbers with additional interesting properties coming from geometric contexts. Apart from the intrinsic interest in the main theoretical questions underlying these projects, there is yet an additional appealing interest emerging from the practical implications of progress on these questions to fields such as cryptography and coding theory.
