Ramanujan's $\tau$ conjecture states that $$\tau(n)\sim O(n^{\frac{11}2+\epsilon})$$ which is a consequence of Deligne's proof of Weil conjectures. Answers in Status of $\tau(n)$ before Deligne tell that best exponent before Deligne reached $\frac{29}5$.
What was it that cohomological approach gave that broke the traditional approach barrier to reach $5.5$ in exponent?