Arithmetic Invariants and Their Non-Triviality

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This project concerns research in number theory, a subject that has many interesting connections to more applied areas such as cryptography and physics. Number theorists study arithmetic objects by attaching to them invariants in order to make them clearly visible. Each important object of interest in number theory has associated to it certain mathematical objects called its L-functions. As they are functions of complex or p-adic variables, one can evaluate them at integers, getting concrete numbers associated with the object under study. There are two fundamental problems concerning invariants in number theory:A. Find a relation (or an identity) among two or more arithmetic invariants of different nature;B. Distinguish between the non-triviality and triviality of important arithmetic invariants.This project will develop an algebraic theory dealing with Problem B. In particular, the research will study non-triviality of values of zeta functions and their derivatives. By a well-known principle (for example, the Birch-Swinnerton Dyer conjecture), these L-values encode solutions of deep Diophantine problems, and if we can show non-vanishing or vanishing of zeta values, we should be able to predict how many rational solutions modular and elliptic equations have. This research project aims to develop a systematic theory for distinguishing between the non-triviality and triviality of important arithmetic invariants. Earlier work of the investigator and a collaborator developed new understanding of p-adic Galois representations and Hecke algebras, p-adic analytic families of modular forms and their L-functions, and analysis of arithmetic invariants. This research project will follow on these developments. For example, the project will investigate how to measure the size of the image of modular Galois representation with coefficients in a big Hecke algebra (a nontriviality question concerning the image). As another example, the work will study non-vanishing (and non-vanishing modulo a prime) of p-adic L-functions and non-triviality of modular attempts of creating rational points in rational elliptic curves and abelian varieties.