One of the central themes in modern algebraic number theory is that local information about a mathematical structure can often be pieced together in surprising ways to yield global information about the structure. The passage from local to global in number theory is deep, mysterious, and not well-understood. This general philosophy can be made concrete through precise conjectures on values of L-series, which are certain functions defined in terms of the local behavior of mathematical structures. These conjectures assert that the values of L-series at integers reveal global arithmetic invariants of the structures. While there are plentiful and very general conjectures regarding special values of L-series, only special cases of the conjectures have been proven. This research project aims to generalize these results and deepen understanding in this important area of number theory.
Some of the most important and well-known conjectures in number theory concern the special values of L-series, including the conjectures of Birch and Swinnerton-Dyer, Stark, Beilinson, and Bloch and Kato. For example, the Birch and Swinnerton-Dyer Conjecture predicts that the value at s=1 of the L-series defined in terms of the number of points on an elliptic curve over the integers modulo p for all primes p carries information on the size of the set of points of the elliptic curve over the rational numbers. In many cases, the L-series in question have alternate manifestations known as p-adic L-series, which are defined by applying a different notion of analysis from that encountered in standard Euclidean geometry. Special-value formulae for classical and p-adic L-series hold an important place at the center of modern number theory research. In this project, the investigator will continue to study conjectures on the special values of p-adic L-functions. The methods involve studying certain explicit families of Eisenstein series and build on work on the Iwasawa Main Conjecture. The project aims to develop new techniques to prove p-adic special-value conjectures, as well as various generalizations and refinements.
