- Ramanujan–Petersson conjecture
mathematics, the Ramanujan conjecture states that the Fourier coefficients of the cusp formof weight 12, defined in modular formtheory, satisfy
when is a
prime number. This implies an estimate that is only slightly weaker for all the , namely for any . This conjecture of Ramanujanwas confirmed by the proof of the Weil conjecturesin 1973. The formulations required to show it was a consequence were delicate and not at all obvious. It was the work of Michio Kugawith contributions also by Mikio Sato, Goro Shimura, and Yasutaka Ihara, followed by Pierre Deligne. The existence of the connection inspired some of the deep work in the late 1960s when the consequences of the étale cohomologytheory were being worked out.
The more general Ramanujan–Petersson conjecture for 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 k = 1, where it is a result of Deligne and Jean-Pierre Serre. It is named for Hans Petersson(1902 – 1984).
In the language of
automorphic representations, a very broad generalisation is possible; but it was shown to be too optimistic, by the particular case of , i.e. the similitude group of the four-dimensional symplectic group, for which counter-examples were found. The appropriate generalised form for the Ramanujan conjecture is still though hoped for; the formulation of the Arthur conjecturesis in terms which explain the mechanism leading to the known kind of counterexample.
The most celebrated application of the Ramanujan conjecture is the explicit construction of
Ramanujan graphs by Lubotzky, Phillips and Sarnak. In fact, this conjecture gave a name to the graphs.
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