Completeness proofs for the solutions of Diophantine Equations

Question

In general, what are the strategies for showing the completeness of a solution set for Diophantine Equations?

For example, take the $\textit{Pell-type}$ equation $x^2 - dy^2 = a$. Say, you have a set of solutions $\lbrace (u,v) \rbrace$ and say that $\epsilon$ is the $\textit{fundamental unit}$ of $\mathcal{O}_{\mathbb{Q}(\sqrt{d})}$. Then what is the easiest way to show that all the solutions are generated by $\lbrace\epsilon^n(u+\sqrt{d}v) \rbrace$ and that there are $\textit{no others}$?

As a slightly more intricate example, take the case of Mordell's Equation: $y^2 = x^3 - 13$. Assuming that $y+\sqrt{-13}$ was a cube in $\mathbb{Z}(\sqrt{-13})$ allowed me to find the solutions $(x,y) = (17,\pm 70)$, but in order to show that there were no others, I had to show that the sum and product of ideals:

$(y+\sqrt{-13})+(y-\sqrt{-13}) = \mathbb{Z}(\sqrt{-13})$ and $(y+\sqrt{-13})\cap(y-\sqrt{-13}) = (y+\sqrt{-13})(y-\sqrt{-13}) = (x^3)$

Which I reckon allowed me to justify the assumption using the $CRT$.

Another interesting thing I found useful to keep track of from Keith Conrad's fantastic blurbs is parity. I also found a Theorem (whose proof I do not know) in one of Pete Clark's expositions that could be useful. (Theroem $8$ in http://alpha.math.uga.edu/~pete/4400MordellEquation.pdf)

But besides this, are there any other strategies one can use to learn more about a solution set? For instance, when can we ascertain whether or not it's finite? Are there any applications of the class number of the field here?

Thank you very much.

your claim on the Pell equation is incorrect. On the other hand, there is a finite set of solutions such that your process generates all solutions. — Will Jagy, Apr 30 '16 at 02:27
@Will My apologies. What I think I really meant was: once you have a finite set of solutions, how do you check that you can now generate all of them? — AlpArslan, Apr 30 '16 at 02:32
I made a CW answer which is just a list of links here that I have answered. Maybe start on the more recent and go backwards. — Will Jagy, Apr 30 '16 at 02:40

score 1 · Answer 1 · answered Apr 30 '16 at 06:29

The only strategy I know of uses the group structure of the points. On integral points on curves of genus $0$, sch as the Pell equation, you have a group structure coming essentially from the multiplication of algebraic numbers; and on elliptic curves, which are curves of genus $1$, you have a group structure on the set of rational points. There is little you can do in terms of strategies if your variety does not have a group structure.

Now if you have a finite set of points, you can use ascent to produce (in general) infinitely many: given $P$ and $Q$ you can form linear combinations $aP + bQ$ using the group law. For showing that these are all you use infinite descent: the group law allows you to show that the existence of an additional solution implies the existence of a point $R$ that is in a very precise sense "smaller" than those you already have (the technical term is called "height"). This reduces the task to a brute force search.

Both for curves of genus $0$ and $1$ there is an associated $L$-function $L(s)$ whose behavior at certain values ($s = 0$, $s = 1$ etc.) allows you to predict whether there are finitely or infinitely many integral (genus $0$) or rational (genus $1$) solutions. In both cases, you can use heavy machinery (cyclotomic units; Heegner points) to produce a subgroup of finite index if the group in question has rank $1$. This again exploits the underlying group structure.

There are, as you have said, direct methods of obtaining integral points on curves of genus $1$ using algebraic number theory, but these become technical very quickly and reduce to solving sets of Thue equations. This is a completely different idea, since here you do not gain anything from knowing a few solutions. Indeed, the integral points do not have a group structure in this case.