### January 2009

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 $L/K$ and subrings $R \subset K$ with fraction field $K$, $S \subset L$ with fraction field $L$ hoping to gain information about how primes $\mathfrak{p}$ of $R$ split into primes $\mathfrak{P}$ in $S$. Essentially, we are asking about properties of the map of affine schemes $\mathrm{Spec}(S) \to \mathrm{Spec}(R)$. Very often we restrict to the case where $R = \mathbf{Z}$ and $S$ is a finite index subring of the ring of integers $\mathcal{O}_L$, so that the map $\mathrm{Spec}(\mathcal{O}_L) \to \mathrm{Spec}(S)$  induced by inclusion is a normalization map.