 Bombieri–Lang conjecture

The Bombieri–Lang conjecture is a concept in number theory that relates to the distribution of rational points on certain types of algebraic varieties. Specifically, it deals with the behavior of rational points on algebraic varieties defined over number fields and has implications for understanding the ranks of abelian varieties and the distribution of solutions to Diophantine equations. The conjecture can be stated in a few steps for certain types of varieties, particularly for curves and higher-dimensional varieties.