A "flowchart" for handling Diophantine equations

Question

There's no algorithm that correctly decides if a Diophantine equation does or doesn't have a solution. Still, many equations can be successfully analyzed, and I'm wondering if anyone wrote down a "cookbook" for dealing with Diophantine equations of various shapes and forms, including the higher-degree, higher-dimensionality ones.

Given a system of polynomial equations with integer coefficients, we may wish to determine if there are any solutions in integers, and if so, whether there are finitely or infinitely many, and whether they can be explicitly described; we may also wish to determine if there are any solutions in rational numbers, and if so, whether there are finitely many, etc.

• If the system is linear, do this (easy).
• If there is just one variable, do that (easy).
• If there's one quadratic equation in two variables (or a homogenous one in three variables), there's again an explicit procedure: check if there's a singularity, determine if there are integer solutions at all (Hasse Minkowski), parametrize the curve, etc. I think all questions can be effectively answered in the case of genus 0 curves.
• If it's an elliptic curve, follow these steps... (I don't think all questions can be algorithmically answered, at present).
• Higher genus curve? What do you do? Find the Jacobian? What else?
• Higher dimensional surfaces and varieties? What do you do? Which heuristics do you try, what are some useful families of equations that can be attacked?

All of those pieces are well covered in the literature - I'm just wondering if there's a good resource that succinctly describes the various alternatives that we may be able to handle.

Note: this older question has similar goals, but it stops short of giving details on how to handle genus 0 and genus 1 and says nothing much about higher genera and higher dimensional varieties.

Answer 1

There are a lot of references which describe "the various possibilities", and it is difficult where to start. One of the first references one finds is the article Open Diophantine Problems by Michel Waldschmidt. It gives a survey about problems, methods, heuristics, etc. The following "flowchart" is taken from the literature (it is by no means complete):

1.) Definition of a "Diophantine Equation" by an algebraic equation of the from $F(x_1,\ldots ,x_n)=0$, where $F$ is a given polynomial in the ring $\mathbb{Z}[x_1,\ldots ,x_n]$, and the equation is to be solved either in integers or rational numbers (which of both is more interesting depends on the particular problem).

2.) Questions about solvability:
a.) Is there any solution (integral or rational) at all ?
b.) Are there finitely many or infinitely many solutions ?
c.) Which structure does the set of all solutions have ?
d.) Is there an algorithm giving in principle a complete list of all solutions ?

3.) Methods: There are almost as many methods as Diophantine equations; algebriac, analytic, geometric, arithmetic methods etc.
a.) The modular method: Applications of ideas of Shimura, Frey, Ribet, Wiles, etc. leading to the proof of FLT, i.e., modular forms, elliptic curves, Galois representations of the absolute Galois group $$ \rho\colon Gal(\overline{\mathbb{Q}}/\mathbb{Q})\rightarrow GL_d(\mathbb{F}) $$ and so on.
b.) Cohomological obstructions: An old (and simple) method to show that a particular Diophantine equation has no solution, is to show that there exists a local obstruction, i.e., a Brauer-Manin obstruction; the study of Brauer groups of certain varieties over global fields.
c.) Mordell-Weil sieve methods: In particular for curves $C:f(x,y)=0$ one can use the knowledge on the Mordell-Weil group of the Jacobi variety of a curve with local informations, e.g., by reduction modulo a prime $p$, for many primes $p$. This gives often strong results concerning rational points on the curve, which can be used algorithmically.
4.) An example Perhaps it is nice to see an "easy" example. Let $$ f(x,y)=y^2-x^3-7823 $$ Then the Diophantine equation $f(x,y)=0$ has no integer solutions; all rational solutions are generated by a single, fundamental solution, namely by $(x,y)$ with $$ x=\frac{2263582143321421502100209233517777}{11981673410095561^2},\; $$

$$ y=\frac{186398152584623305624837551485596770028144776655756}{11981673410095561^3}. $$ The Mordell-Weil group over $\mathbb{Q}$ is cyclic of rank $1$.