Deligne's solution of the Ramanujan conjecture 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 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.)