Topics in Arithmetical Algebraic Geometry

We will look at some foundational material in 
arithmetical algebraic geometry, leading to recent 
developments around the ranks of elliptic curves.  

Possible topics:

	basics on elliptic curves
	complex multiplication and the dream of Kronecker's youth
	Jacobians and abelian varieties
	Shimura-Taniyama theory of complex multiplication
	Heegner points and the BSD conjecture in rank 1
	Fancier Heegner points, Iwasawa theory, and ranks in dihedral towers
	Growth of ranks in non-abelian towers

This will be a seminar-style course with active student 
participation.  It should be of interest to students 
planning to work in number theory or algebraic geometry.  

Prerequisites: some knowledge of basic algebraic geometry 
and number theory (together with a willingness to work) 
should suffice.