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.

Answer 1

CW just a list

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Answer 2

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.