Project description
L-functions are complex analytic functions that have become central in modern number theory, bridging across arithmetic, geometry, algebra and analysis. They are the starring players in two of the 1-million-dollar Millennium problems: The Riemann Hypothesis and The Birch and Swinnerton-Dyer (BSD) Conjecture.
The Riemann zeta function is the meromorphic function of a complex variable s, defined e.g. here. It turns out that the values of this highly transcendental function are rational numbers that satisfy beautiful, deep congruence properties modulo powers of any prime number p. This is the starting point for, and the 1-dimensional case of, the study of p-adic L-functions and Iwasawa theory.
At its heart, Iwasawa theory seeks to find mysterious connections between algebraic objects (e.g. the solutions to algebraic equations) and analytic ones (L-functions). It has led to some of the most remarkable results in modern number theory, for example Kolyvagin's proof of BSD in analytic ranks 0 and 1.
Our understanding of p-adic L-functions is fairly limited beyond dimensions 1 and 2. This project would seek to study p-adic L-functions in higher-dimensional settings.
Interested students are strongly encouraged to get in touch with me via email, where I can share more details about Iwasawa theory and p-adic L-functions.