Arithmetic Algebraic Geometry

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We will study several problems related to algebraic cycles and Abelian varieties, L-functions and mathematics arising from Information Technology.******1) In the arithmetic and geometry of Abelian varieties, conjectures about algebraic cycles are fundamental. The classical conjectures of Hodge and Tate give us methods of recognizing subvarieties (and their linear combinations) in algebraic varieties. In the case of Abelian varieties, the additional group structure and the simple structure of the cohomology makes people more explicit formulations. Our recent work about understanding the relationship between cohomological spaces of cycles on an Abelian variety and on its reduction modulo a prime is aimed at increasing our understanding of algebraic cycles, though this goal may be far off. However, there are many problems which seem quite accessible (such as proving that a cohomology class can be declared to be a Tate class by checking the action of a finite number of Frobenius elements) and this project will focus on a few of those, based on recent work by the author.******2) Another fundamental tool in number theory is that of L-function, an analytic object built from local data, which mysteriously learns about global data. In our project, we will actually look at spaces of L-functions that have additional algebraic structure (they form a ring). The purpose of studying collections of L-functions is to then ascertain what properties can actually be 'deformed' along a collection. Again, this kind of question is still too vague and too distant, and to get to the point where we might start thinking about such questions, we need to analyze the analysis and hopefully the geometry and topology of spaces of L-functions. Some first steps have been taken with the definitions of the Selberg and Lindelof classes. In this project, we will continue to work to clarify the algebraic structure of the Lindelof class. ******3) There are many mathematical questions that are inspired by problems in information technology. One is to make explicit the arithmetic on Abelian varieties over finite fields. These are higher dimensional versions of elliptic curves and one might hope to form a public-key encryption scheme using such objects. There are of course some impediments to this such as new attacks on higher dimensional varieties and the possible construction of a quantum computer. Nevertheless, understanding explicitly, and hence computationally, the arithmetic on these higher dimensional varieties may for the foundation of a new kind of encryption scheme, not exactly modeled on the discrete logarithm, but perhaps on the more quantum-resistant model using isogenies. In addition to this kind of fundamental mathematical work, there are also problems involving algorithms to ensure security and privacy in transactions such as e-payments and e-health.