Inequalities relating invariants of elliptic curves

Abstract: In this talk we discuss explicit inequalities which relate heights, discriminants and conductors of elliptic curves E over a number field K. In the first part we present an explicit height-conductor inequality for E 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 \(\mathbb{P}^1-\lbrace 0,1,\infty\rbrace\).