Reduction of abelian varieties and a conjecture of Agashe

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.