Methods of sheaf theory for solving Diophantine equations

Question

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

Answer 1

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.

Answer 2

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.