My name is Jim and I’m a grad student in arithmetic geometry and algebraic number theory at the University of Georgia. I’m studying for my oral exams and I’ve been invited to use this space, as Charles has, to explore topics which appear on the syllabus for my oral exam, notably topics in elliptic curves and algebraic number theory.
Elliptic curves have already been discussed in http://rigtriv.wordpress.com/2007/10/15/elliptic-curves/ , but one might ask what business algebraic number theory has in a blog on algebraic geometry. This is answered in Neukirch, who views algebraic number theory as the study of 0- and 1-dimensional schemes. In algebraic number theory, one studies field extensions and subrings with fraction field , with fraction field hoping to gain information about how primes of split into primes in . Essentially, we are asking about properties of the map of affine schemes . Very often we restrict to the case where and is a finite index subring of the ring of integers , so that the map induced by inclusion is a normalization map.
We consider problems in arithmetic through this lens to be able to tackle them with all the considerable machinery of algebraic geometry, and this approach has yielded many successes spanning many decades, most famously in the proof of Fermat’s Last Theorem and very recently in the case of the finite field Kakeya problem. I will make no attempt to approach these problems, at least in the near future. My plans for posting over the next few months will be to first review some constructions and theorems from the theory of elliptic curves, then switching to some algebriac number theory before tackling the subject of complex multiplication. After I have gone through the syllabus for my orals, I will continue writing as Charles does tackling different topics as they come along(although I’ll certainly want to do posts on quaternion and central simple algebras, modular forms and the like).
I hope you will all like my posts. Next time I will define an elliptic curve as a smooth genus 1 curve over a field with a rational point and write about the chord-and-tangent group law as well as the jacobian.