Number Theory Seminar

Title: Elliptic curves and isogenies.

Abstract: Two elliptic curves $E$ and $E'$ defined over a number field $K$  
are isomorphic over the algebraic closure of $K$ if and only if they  
have the same j-invariant. A natural question is: how is this  
invariant transformed by general isogenies? We prove a new height  
bound on the difference of heights of the j-invariants of isogenous  
elliptic curves, and derive several consequences, for instance bounds  
for the height of modular polynomials and for Vélu's formulas. If time  
permits, we will add a remark on Mordell-Weil ranks of elliptic curves.