Ramanujan–Petersson conjecture
In mathematics, the Ramanujan conjecture, due to , states that Ramanujan's tau function given by the Fourier coefficients of the cusp form of weight where formula_2, satisfies when is a prime number. The generalized Ramanujan conjecture or Ramanujan–Petersson conjecture, introduced by , is a generalization to other modular forms or automorphic forms. Ramanujan L-function. The Riemann zeta function and the Dirichlet L-function satisfy the Euler product, and due to their completely multiplicative property Are there L-functions other than the Riemann zeta function and the Dirichlet L-functions satisfying the above relations? Indeed, the L -functions of automorphic forms satisfy the Euler product (1) but they do not satisfy (2) because they do not have the completely multiplicative property. However, Ramanujan discovered that the L-function of the modular discriminant satisfies the modified relation where is Ramanujan's tau function. The term is thought of as the difference from the completely multiplicative property. The above L-function is called Ramanujan's L-function. Ramanujan conjecture. Ramanujan conjectured the following: Ramanujan observed that the quadratic equation of in the denominator of RHS of , would have always imaginary roots from many examples. The relationship between roots and  coefficients of quadratic equations leads the third relation, called Ramanujan's conjecture. Moreover, for the Ramanujan tau function, let the roots of the above quadratic equation be and , then which looks like the Riemann Hypothesis. It implies an estimate that is only slightly weaker for all the , namely for any : In 1917, L. Mordell proved the first two relations using techniques from complex analysis, specifically what are now known as Hecke operators. The third statement followed from the proof of the Weil conjectures by . 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 . 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. In 1937, Erich Hecke used Hecke operators to generalize the method of Mordell's proof of the first two conjectures to the automorphic L-function of the discrete subgroups of . For any modular form one can form the Dirichlet series For a modular form of weight  for , absolutely converges in , because . Since is a modular form of weight , turns out to be an entire and satisfies the functional equation: this was proved by Wilton in 1929. This correspondence between and is one to one (). Let for , then is related with via the Mellin transformation This correspondence relates the Dirichlet series that satisfy the above functional equation with the automorphic form of a discrete subgroup of . In the case Hans Petersson introduced a metric on the space of modular forms, called the Petersson metric (also see Weil-Petersson metric). This conjecture was named after him. Under the  Petersson metric it is shown that we can define the orthogonality on the space of modular forms as the space of cusp forms and its orthogonal space and they have finite dimensions. Furthermore, we can concretely calculate the dimension of the space of holomorphic modular forms, using the Riemann–Roch theorem (see the dimensions of modular forms). The Ramanujan–Petersson conjecture for Maass forms is still open (as of 2016) because Deligne's method, which works well in the holomorphic case, does not work in the real analytic case. Ramanujan–Petersson conjecture for automorphic forms. reformulated the Ramanujan–Petersson conjecture  in terms of automorphic representations for 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. and showed that the conjecture was also false even for some quasi-split and split groups, by constructing automorphic forms for the unitary group and the symplectic group that  are non-tempered almost everywhere, related to the representation . After the counterexamples were found, 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 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 , consider a unitary cuspidal automorphic representation: The Bernstein–Zelevinsky classification tells us that each p-adic can be obtained via unitary parabolic induction from a representation Here each formula_14 is a representation of , over the place , of the form with formula_16 tempered. Given , a Ramanujan bound is a number such that Langlands classification can  be used for the archimedean places. The generalized Ramanujan conjecture is equivalent to the bound . For reductive groups other than , the generalized Ramanujan conjecture would follow from principle of Langlands functoriality. An important example are the classical groups, where the best possible bounds were obtained by 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 over a global function field leads towards a proof of the Ramanujan–Petersson conjecture. Lafforgue (2002) successfully extended Drinfeld's shtuka technique to the case of in positive characteristic . Via a different technique that extends the Langlands-Shahidi method to include global function fields, proves the Ramanujan conjecture for the classical groups. Applications. An 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 implies Selberg's conjecture about eigenvalues of the Laplacian for some discrete groups.