A naive number theory question

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!

A generalization of Principal Ideal Theorem

Today I am amused by a ingenious generalization of Principal Ideal Theorem.

Recall that the Principal Ideal Theorem says over a noetherian ring, any
prime ideal minimal over a ideal generated by r elements has height less
than or equal to r. Eagon generalized this theorem to determinantal ideals
in his thesis. The statement is as follows.

Let M be a r*s matrix over a northerian ring, let I(t) denote the ideals
generated by the t*t minors of M, then any prime ideal minimal over I(t)
has height less than or equal to (r-t+1)(s-t+1).

The Principal Ideal Theorem is the special case when t=1.

The original proof was given by Eagon in “Ideals defined by matrices and a
certain complex associated with them” using genelized Koszul complex.
There is also a short proof in the appendix of Matsumura’ s Commutative
Ring Theory”.

                                                                           Shuchao

                                                                          04/23/08