Behavior of primes in division fields of elliptic curves

Abstract

Given elliptic curve E, we consider a family of field extension generated by n-torsion points of E, called division fields. The main objective of this thesis is determine the behavior of primes of good reduction in these extensions. Depending on a divisibility criterion, we get a case distinction into unramified and (possibly) ramified primes. In the unramified case we find a matrix representative of the Frobenius conjugacy class in the Galois group, using information about the endomorphism ring of the reduced curve. We will generalize this to abelian varieties, which introduces some difficulties. Finally, we give a lower bound for the ramification index in the ramified case, using Newton polygons.