Weil conjectures
used the Eichler–Shimura isomorphism to reduce the Ramanujan conjecture to the Weil conjectures that he later proved.　The more general Ramanujan–Petersson conjecture for holomorphic cusp forms in the theory of elliptic modular forms for congruence subgroups has a similar formulation, with exponent where is the weight of the form. These results also follow from the Weil conjectures, except for the case , where it is a result of .