Heights and conductors of elliptic curves

Abstract: In the first part we present for any elliptic curve over a number field an explicit height-conductor inequality based on the theory of logarithmic forms. In the second part we consider the case K=Q: We explain how one can use modularity to get a sharper height-conductor inequality and we deduce explicit height bounds for S-integral points on Mordell curves and on \(\mathbb{P}^1-\lbrace 0,1,\infty\rbrace\).