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.