# 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.