Chapter 3 is devoted to the theory of heights, which is fundamental in Diophantine geometry. We explain archimedean and nonarchimedean absolute values on a number field, and prove the product formula. We define the absolute (logarithmic) Weil height. We explain heights associated to line bundles, and prove Northcott’s finiteness theorem. In the latter part of Chapter 3, we briefly introduce abelian varieties and some properties of line bundles on abelian varieties, such as the seesaw theorem and the theorem of cube. We define Neron–Tate height pairings on abelian varieties. We introduce Jacobian varieties and the Abel–Jacobi maps. We prove the Hermite–Minkowski theorem and the Mordell–Weil theorem.