Arithmetic Study Group: An informal introduction to the Shafarevich Conjecture
14 October 2008 14:15 in CM 101
The Shafarevich Conjecture pertains to the non-existence of abelian varieties over $\Bbb Q$ with everywhere good reduction. We discuss its generalization to proper smooth schemes over $\Bbb Z$ (i.e. projective varieties over $\Bbb Q$ with everywhere good reduction) and recent progress towards varieties which have bad semi-stable reduction in p=3 and good reduction in all $p\ne 3$. Main attention will be paid to the roles of finite flat group schemes, crystalline and semi-stable representations on the one side and of Odlyzko estimates for minimal discriminants of algebraic number fields on the other side.