The Euler system of cyclotomic units (and a more general Kolyvagin system counterpart) is a collection of "derivatives" of cyclotomic units in a cyclotomic number field whose factorizations gives bounds on the size of the ideal class group. The result is actually stronger than this: for any character of the galois group, we can get a bound on the size of the corresponding eigenspace. A major result from this line of attack is that the size of the p-part of the chi-component of the ideal class group equal the size of the p-part of the chi-component of the group of units mod cyclotomic units. Eventually, I will also be interested in the Euler system of Heegner points used in the study of elliptic curves as well.
-adic Dynamics and Formal Groups
This is the topic and title of my undergraduate thesis, which can be downloaded here in PDF format. The basic premise is that we are looking at power series (with very few restrictions) over a finite algebraic extension of the
-adic rationals, and we want to know
when they commute. Loosely speaking, Lubin's conjecture states that
if we have the case where an invertible power series commutes with a
non-invertible one, there must exist a formal group over that
extension such that both the aforementioned invertible power series
and non-invertible power series are both endomorphisms of the same
formal group.
Kontsevich and Zagier have written a great introduction to periods, which you can download here. The set of periods is essentially the set of all complex numbers that can be written as the integral of an algebraic function over an algebraic domain in
for some
. The periods fill an essential gap in the typical hierarchy of
numbers, starting with the whole numbers and extending into the
complexes. The algebraic numbers typically form the last step on this
route, so the set of periods forms a nice addition, being an
uncountable subset of the complex numbers that contains all the
algebraic numbers and has nice analytic properties.
Of particular interest to me is the dimension of a period,
being (loosely) defined as the smallest for which we can find an
algebraic function and an -dimensional algebraic domain such that
integrating that function over that domain gives us the particular
value. Algebraic numbers have dimension 0, and known periods have
dimension 1. A basic unanswered question is whether or not there
exist any periods of dimension greater than 1, and to find one.
-functions
Following progress on the Riemann zeta function and the Riemann hypothesis has been a long-time hobby. Though I don't currently do research in this field, I am impressed with and fascinated by the techniques of using analytic number theory to prove highly non-obvious theorems about primes and their distributions. The Riemann hypothesis is one such possibility, and Dirichlet's theorem on primes in arithmetic progressions is another.
An asymmetric key exchange is a protocol by which two people can communicate over an open wire, letting anyone listen in, and yet still manage to generate a secret that only the two of them know, obtaining an arbitrarily high level of security. Asymmetric key exchanges are unbelievably cool to begin with, and throw in elliptic curves, and it's bound to be a good time. The theory hinges on the definition of the group law on an elliptic curve, whereby points on the curve can be added together to get other points. The fundamental result is that it's trivial to figure out how to add points together, but extremely difficult (i.e. time-consuming) to ``divide'' them. Once the power of this difficulty is harnessed, its not difficult to devise a protocol (say, the Diffie-Hellman key exchange for elliptic curves) which will generate your secret key.