Deligne's solution of the [Ramanujan conjecture][1] on estimate of coefficients of an  automorphic form.

To state it, define for $|q|<1$  
$$D(q):=q\prod_{n=1}^\infty(1-q^n)^{24}=\sum_{n=1}^\infty \tau(n)q^n.$$
Then the Ramanujan's conjecture (or Deligne's theorem) says that
for any prime number $p$ one has
$$|\tau(p)|<2p^{11/2}.$$
Deligne used a surprising interpretation of $\tau(p)$ as a trace of an element (called Frobenius element) of some Galois group in cohomology of an arithmetic variety with coefficients in a sheaf. To get the estimate, Deligne used the [Weil conjectures][2] predicting the behavior of eigenvalues of the Frobenius element. (The corresponding statement of the Weil conjectures Deligne proved few year after his work on the Ramanujan conjecture.)


  [1]: https://en.wikipedia.org/wiki/Ramanujan%E2%80%93Petersson_conjecture
  [2]: https://en.wikipedia.org/wiki/Weil_conjectures