### IV.5 Arithmetic Geometry

### *Jordan S. Ellenberg*

### 1 Diophantine Problems, Alone and in Teams

Our goal is to sketch some of the essential ideas of arithmetic geometry; we begin with a problem which, on the face of it, involves no geometry and only a bit of arithmetic.

**Problem.** Show that the equation

has no solution in nonzero rational numbers *x*, *y*, *z*.

(Note that it is only in the coefficient 7 that (1) differs from the Pythagorean equation *x*^{2} + *y*^{2} = *z*^{2}, which we know has *infinitely* many solutions. It is a feature of arithmetic geometry that modest changes of this kind can have drastic effects!)

**Solution.** Suppose *x*, *y*, *z* are rational numbers ...