(where q=e2πiz) satisfies
Ramanujan's conjecture implies an estimate that is only slightly weaker for all the , namely for any .
This conjecture of Ramanujan followed from the proof of the Weil conjectures by Deligne (1974). The formulations required to show that 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 Deligne (1968). 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.
Ramanujan–Petersson conjecture for modular forms
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 (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 & Serre (1974).
The Ramanujan–Petersson conjecture for Maass wave forms is still open (as of 2013).
Ramanujan–Petersson conjecture for automorphic forms
Satake (1966) reformulated the Ramanujan–Petersson conjecture in terms of automorphic representations for GL2 as saying that the local components of automorphic representations lie in the principal series, and suggested this condition as a generalization of the Ramanujan–Petersson conjecture to automorphic forms on other groups. Another way of saying this is that the local components of cusp forms should be tempered. However, several authors found counter-examples for anisotropic groups where the component at infinity was not tempered. Kurokawa (1978) and Howe & Piatetski-Shapiro (1979) showed that the conjecture was also false even for some quasi-split and split groups, by constructing automorphic forms for the unitary group U2,1 and the symplectic group Sp4 that are non-tempered almost everywhere, related to the representation θ10
After the counterexamples were found, Piatetski-Shapiro (1979) suggested that a reformulation of the conjecture should still hold. The current formulation of the generalized Ramanujan conjecture is for a globally generic cuspidal automorphic representation of a connected reductive group, where the generic assumption means that the representation admits a Whittaker model. It states that each local component of such a representation should be tempered. It is an observation due to Langlands that establishing functoriality of symmetric powers of automorphic representations of GLn will give a proof of the Ramanujan–Petersson conjecture.
Bounds towards Ramanujan over number fields
Obtaining the best possible bounds towards the generalized Ramanujan conjecture in the case of number fields has caught the attention of many mathematicians. Each improvement is considered a milestone in the world of modern Number Theory. In order to understand the Ramanujan bounds for GLn, consider a unitary cuspidal automorphic representation π = ⊗' πv. The Bernstein–Zelevinsky classification tells us that each p-adic can be obtained via unitary parabolic induction from a representation . Here each is a representation of GLni, over the place v, of the form with tempered. Given n ≥ 2, a Ramanujan bound is a number δ ≥ 0 such that . Langlands classification can be used for the archimedean places. The generalized Ramanujan conjecture is equivalent to the bound δ = 0.
Jacquet, Piatetski-Shapiro & Shalika (1981) obtain a first bound of δ ≤ 1/2 for the general linear group GLn, known as the trivial bound. An important breakthrough was made by Luo, Rudnick & Sarnak (1999), who currently hold the best general bound of δ ≡ 1/2 - 1/(n2+1) for arbitrary n and any number field. In the case of GL2, Kim and Sarnak established the breakthrough bound of δ = 7/64 when the number field is the field of rational numbers, which is obtained as a consequence of the functoriality result of Kim (2002) on the symmetric fourth obtained via the Langlands-Shahidi method. Generalizing the Kim-Sarnak bounds to an arbitrary number field is possible by the results of Blomer & Brumley (2011).
For reductive groups other than GLn, the generalized Ramanujan conjecture will follow from principle of Langlands functoriality. An important example are the classical groups, where the best possible bounds were obtained by Cogdell et al. (2004) as a consequence of their Langlands functorial lift.
The Ramanujan-Petersson conjecture over global function fields
Drinfeld's proof of the global Langlands correspondence for GL(2) over a global function field leads towards a proof of the Ramanujan–Petersson conjecture. In a magnificent tour de force, Lafforgue (2002) successfully extended Drinfeld's shtuka technique to the case of GL(n) in positive characteristic. Via a different technique that extends the Langlands-Shahidi method to include global function fields, Lomelí (2009) proves the Ramanujan conjecture for the classical groups.
The most celebrated application of the Ramanujan conjecture is the explicit construction of Ramanujan graphs by Lubotzky, Phillips and Sarnak. Indeed, the name "Ramanujan graph" was derived from this connection. Another application is that the Ramanujan–Petersson conjecture for the general linear group GLn implies Selberg's conjecture about eigenvalues of the Laplacian for some discrete groups.
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