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 $n$.
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.
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 $n$.