Goldfeld's conjecture

Goldfeld's conjecture pertains to the distribution of ranks of elliptic curves over rational numbers, suggesting that half of these curves have rank zero (no rational points other than the trivial point), and the other half have rank one. In essence, it predicts a balance in the complexity of solutions these special types of equations can have, with most having minimal or no rational solutions. This conjecture relates to understanding how often these curves have points with rational coordinates and has deep implications for number theory and the study of solutions to polynomial equations.