Ramanujan–Petersson conjecture

In mathematics, the Ramanujan conjecture states that the Fourier coefficients $au(n)$ of the cusp form $Delta(z)$ of weight 12, defined in modular form theory, satisfy

:$|\; au(p)|\; leq\; 2p^\{11/2\},$

when $p$ is a prime number. This implies an estimate that is only slightly weaker for all the $au(n)$, namely $O(n^\{frac\{11\}\{2\}+varepsilon\})$ for any $varepsilon\; >\; 0$. This conjecture of Ramanujan was confirmed by the proof of the Weil conjectures in 1973. The formulations required to show it was a consequence were delicate and not at all obvious. It was the work of Michio Kuga with 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 cohomology theory 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 $(k-1)/2$ where $k$ 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 $GSp\_4$, 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 conjectures is in terms which explain the mechanism leading to the known kind of counterexample.

Applications

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.