This is probably far-fetched, but the Weil-Shimura-Taniyama conjecture was a global conjecture (now solved), whose proof implied the non-existence of solutions for a famous Diophantine equation $x^n+y^n=z^n$ for sufficiently large $n$.
Showing that someting is automorphic requires some general tools from the Langlands program, such as the Arthur trace formula or converse theorems. The latter require analytic continuation, functional equations and boundedness in vertical stripes, which are purely global features.
The Arthur trace formula or the more specialized Eichler-Selberg trace formula are also a global construction, which requires an understanding of all local places, and glue this information together in terms of the conjugacy classes, of say $GL_2(\mathbb{Q})$. It allows you to match automorphic coeffecients (Hecke eigenvalues) with arithmetic coeffecients (I guess: these are the eigenvalues of an action on the cohomology theory) of the Hasse Weil Zeta function, as soon as you find a good geometric realization of the cohomology theory for the later.
