Abstract: In the groundbreaking and straightforward-titled 1985 paper “Il n’y a pas de variété abélienne sur Z” (“There are no abelian schemes over Z”), Fontaine studied the ramification of the p^n-torsion group A
p n
of an abelian variety A defined over certain “smaller” number fields, having everywhere good reduction. The upshot is, such a group scheme Ap n
is mildly ramified, so mild that NO such group schemes exist. As a consequence, he showed that there is no abelian varieties over Q having everywhere good reduction. In this talk, I will give essential underlying ideas in the proof of such results. Recent generalization of Schoof will be also introduced.
