What are some examples of sheaf theory used to either provide solutions to Diophantine equations, or to state that no such solutions exist?

5$\begingroup$ Well, the Weil conjectures are proved using sheaf theory, but tht's a long story... $\endgroup$ – abx May 13 at 16:26

6$\begingroup$ OK, let me try to be more explicit. Arithmetic geometry is an important part of algebraic geometry, with stunning applications to number theory. It is based in an essential way on the techniques of sheaf cohomology. Listing examples would take forever. $\endgroup$ – abx May 13 at 17:12

3$\begingroup$ May be my comment was not clear enough.. You have mentioned that you have spent 3 years on this idea... I was asking you how did you spend those 3 years... For "whether I've learned anything in grad school." I do not know what to say... I do not know how that is relevant to my comment.. $\endgroup$ – Praphulla Koushik May 13 at 17:24

2$\begingroup$ How about BrauerManin obstructions to rational points? Those come from Azumaya algebras, which are sheaves of modules (satisfying a bunch of criteria, etc.). $\endgroup$ – RP_ May 13 at 21:42

2$\begingroup$ This paper uses sheaves (at least in the form of vector bundles with flat connections) to find all the rational points on an explicit curve. arxiv.org/pdf/1711.05846.pdf . My paper with Brian Lawrence arxiv.org/abs/2004.09046 uses sheaves and perverse sheaves to prove finiteness of integral points for a family of Diophantine equation. The BrauerManin obstruction that RP_ mentioned is a good example  specifically, it can often be used to show a Diophantine equation has no solutions. $\endgroup$ – Will Sawin May 14 at 1:55
Your question may be get closed, but before that happens, I will leave an answer that might help you. Taken very literally the answer to your question is likely no. It is known that there can be no algorithm to decide whether a Diophantine equation has an integer solution (Hilbert's 10 problem), which makes it unlikely that there would be a sheaf theoretic test for this. However, as already pointed out in the comments, sheaf theoretical methods have been extraordinarily useful in studying Diophantine equations. For example, to repeat what abx said in the comments, the solution to the Weil conjectures by Grothendieck and Deligne is one particularly striking example. Instead of looking at solutions over $\mathbb{Z}$, one studies the number of solutions in $\mathbb{Z}/(p)$ and its extensions. In some cases, such as for hypersurfaces, the conjectures yield explicit bounds on the number of solutions. To attack the conjectures, one needs a generalization of sheaf theory along with the corresponding cohomology theory, called étale cohomology. This is a very big topic, but perhaps this gives you some idea.
Here is a possible explanation from Chapter 1 section 4 of "The Geometry of Schemes" by Eisenbud and Harris.
The apparently abstract idea of the functor of points has its root in the study of solutions of equations. Let $X = \text{Spec}( R)$ be an affine scheme, where $R = \mathbb{Z}[x_1, x_2,...]/(f_1, f_2,...)$. If T is any other ring (one should think of $T = \mathbb{Z}, \mathbb{Z}/(p), \mathbb{Z}_{(p)}, \hat {\mathbb{Z}}_{(p)}, \mathbb{Q}_p, \mathbb{R}, \mathbb{C},$ and so on), then a morphism from $\text{Spec}( T)$ to $\text{Spec}( R )$ is the same as a ring homomorphism from $R$ to $T$, and this is determined by the images $a_i$ of the $x_i$. Of course, a set of elements $a_i \in T$ determines a morphism in this way if and only if they are solutions to the equations $f_i = 0$. We have shown that
\begin{equation}h_X(T ) = \left\{\!\begin{aligned}\text{sequences of elements $a_1,... \in T $ that } \\ \text{are solutions of the equations $f_i = 0$.} \end{aligned}\right\} \end{equation} Similarly, if $X$ is an arbitrary scheme, so that $X$ is the union of affine schemes $X_a$ meeting along open subsets, then a map from an affine scheme $Y$ to $X$ may be described by giving a covering of $Y$ by distinguished affine open subsets $Y_{f_a}$ and maps from $Y_{f_a}$ to $X_a$ for each $a$, agreeing on open sets (some of the $Y_{f_a}$ may, of course, be empty). Thus an element of $h_X(Y )$ may be described even in this general context as a set of solutions to systems of equations, corresponding to some of the $X_a$, with compatibility conditions satisfied by the solutions on the sets where certain polynomials are nonzero.
Even with this interpretation, the notion of the functor of points may seem an arid one: while we can phrase problems in this new language, it’s far from clear that we can solve them in it. The key to being able to work in this setting is the fact that many apparently geometric notions have natural extensions from the category of schemes to larger categories of functors. Thus, for example, we can talk about an open subfunctor of a functor, a closed subfunctor, a smooth functor, the tangent space to a functor, and so on. These notions will be developed in Chapter VI, where we will also give a better idea of how they are used.

1$\begingroup$ @ReginaldAnderson Well, even if all you wanted was to study affine schemes, you will often have to study non affine ones first  for instance the proof of Falting's theorem (or Mordell's conjecture) reduces to statements about abelian varieties and to study projective schemes, you definitely need sheaves (in the simplest case  maps to projective space are given by line bundle+sections but it goes much deeper of course). There seems to be no way around it  sheaves and schemes are crucial to algebraic number theory. $\endgroup$ – Asvin May 13 at 19:50

1$\begingroup$ Yes, they will let you solve things. Does proving that any high degree equation has only finitely many rational solutions count as "solving things"? Even if you object that it isn't very explicit, there are other ways of bounding solutions like the Chabauty method which provides very explicit bounds on how many solutions such an equation can have. Surely that counts as solving an equation! $\endgroup$ – Asvin May 13 at 19:55

1$\begingroup$ I just saw your edit: Yes, you can often construct sheaves (on the etale site!) that tell you about solutions to equations. This is the idea behind the proof of the weak MordellWeil theorem which effectively says that the torsion points of an elliptic curve inject into some Galois/etale cohomology group. There are really too many examples to list but here's one more  a recent paper proves that the average rank of elliptic curves over finite fields is some number by computing the cohomology of an explicit sheaf. $\endgroup$ – Asvin May 13 at 20:00

1$\begingroup$ A scheme is a sheaf; you need sheaves to define schemes. I think another text that will help you a lot is Hartshorne springer.com/gp/book/9780387902449; his advisor was Zariski who invented the Zariski topology which eventually lead to sheaves, which eventually lead to schemes. The book starts with sheaves and then defines schemes in terms of sheaves, "The Geometry of Schemes" does this as well. The language of sheaves was first used by Jean Pierre Serre to attack the problems of algebraic geometry and commutative algebra in a famous paper "Faisceaux algébriques cohérents." $\endgroup$ – Pedro Juan Soto May 13 at 20:01

1$\begingroup$ The idea is the following: sheaves were originally invented for other more general problems. Serre had the good idea to apply this theory to algebraic geometry (which is roughly "polynomial geometry") and Grothendieck further refined sheaves to locally ringed spaces which are isomorphic to the spectrum of a ring; i.e. affine schemes. Why sheaves? The stalks of the sheaves carry local information about the "manifold" defined by the polynomial. This is exactly the point in most of geometry (for example in Differential Geometry); i.e. to study the interplay between the local and global geometry. $\endgroup$ – Pedro Juan Soto May 13 at 20:10