I began to read some basic number theory few days ago. So many different number theory problems finally reduce to the rational points on elliptic curve, e.g., Fermat’s Last Theorem n=4 case by simple transformations. I have huge interest in the following comments. An elliptic curve over Q has only a finite number of integral point (Mordell, Siegel), e.g., Y^2 = X^3- X, which correspond to the genus 1 curve in complex case. However, e.g. Y^2 = X^3 +X or Y^2 = X^3, they are rational curves with genus 0, and at the same time, they have infinitely many integral points on their real locus. The book said that the geometrical difference is related to the arithmetical difference. Is there something deep behind this? I hope some one could give me a hint. Thanks!
