Ramanujan's $\tau$ conjecture states that $$\tau(n)=O_\epsilon(n^{\frac{11}2+\epsilon}),$$ which is a consequence of Deligne's proof of Weil conjectures. Answers in http://math.stackexchange.com/questions/1205419/status-of-taun-before-deligne/1205516 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?

Posted originally in here http://math.stackexchange.com/questions/1206558/etale-cohomology-approach-on-taun where comments suggested to post in MO.

Ramanujan's $\tau$ conjecture states that $$\tau(n)=O_\epsilon(n^{\frac{11}2+\epsilon}),$$ which is a consequence of Deligne's proof of Weil conjectures. Answers in https://math.stackexchange.com/questions/1205419/status-of-taun-before-deligne/1205516 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?

Posted originally in here https://math.stackexchange.com/questions/1206558/etale-cohomology-approach-on-taun where comments suggested to post in MO.

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 http://math.stackexchange.com/questions/1205419/status-of-taun-before-deligne/1205516 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?

Posted originally in here http://math.stackexchange.com/questions/1206558/etale-cohomology-approach-on-taun where comments suggested to post in MO.

Ramanujan's $\tau$ conjecture states that $$\tau(n)=O_\epsilon(n^{\frac{11}2+\epsilon}),$$ which is a consequence of Deligne's proof of Weil conjectures. Answers in http://math.stackexchange.com/questions/1205419/status-of-taun-before-deligne/1205516 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?

Posted originally in here http://math.stackexchange.com/questions/1206558/etale-cohomology-approach-on-taun where comments suggested to post in MO.