Аннотация:
This talk will consist of two
parts. First, reduction properties of abelian varieties defined over
a field K that have a K-rational point of order p will be
studied. Here K is a field of characteristic 0 equipped with a
discrete valuation which has algebraically closed residue field of
characteristic p. After presenting a general result, we will focus
on the dimension 1 case and classify the possible Kodaira types of
reduction that can occur. In the second part of the talk, we will
discuss a conjecture of Agashe, which is a consequence of the Birch
and Swinnerton-Dyer (BSD) conjecture for elliptic curves over the
rationals. We will present a theorem that proves Agashe's
conjecture. The connection between the two parts is that we can put
restrictions on torsion subgroups of certain twists of elliptic
curves using reduction.