Ramanujan–Petersson conjecture

Template:Short description In mathematics, the Ramanujan conjecture, due to Template:Harvs, states that Ramanujan's tau function given by the Fourier coefficients Template:Math of the cusp form Template:Math of weight Template:Math

<math>\Delta(z)= \sum_{n>0}\tau(n)q^n=q\prod_{n>0}\left (1-q^n \right)^{24} = q-24q^2+252q^3- 1472q^4 + 4830q^5-\cdots,</math>

where <math>q=e^{2\pi iz}</math>, satisfies

when Template:Mvar is a prime number. The generalized Ramanujan conjecture or Ramanujan–Petersson conjecture, introduced by Template:Harvs, is a generalization to other modular forms or automorphic forms.

Ramanujan L-functionEdit

The Riemann zeta function and the Dirichlet L-function satisfy the Euler product,

Template:NumBlk+\dots\right)</math>|Template:EquationRef}}

and due to their completely multiplicative property

Template:NumBlk

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

Template:NumBlk\right)^{-1},</math>|Template:EquationRef}}

where Template:Math 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 conjectureEdit

Ramanujan conjectured the following:

Template:Math is multiplicative,
Template:Math is not completely multiplicative but for prime Template:Mvar and Template:Mvar in Template:Math we have: Template:Math, and
Template:Math.

Ramanujan observed that the quadratic equation of Template:Math in the denominator of RHS of Template:EquationNote,

would have always imaginary roots from many examples. The relationship between roots and coefficients of quadratic equations leads to the third relation, called Ramanujan's conjecture. Moreover, for the Ramanujan tau function, let the roots of the above quadratic equation be Template:Mvar and Template:Mvar, then

<math>\operatorname{Re}(\alpha)=\operatorname{Re}(\beta)=p^{11/2},</math>

which looks like the Riemann Hypothesis. It implies an estimate that is only slightly weaker for all the Template:Math, namely for any Template:Math:

<math>O \left (n^{11/2+\varepsilon} \right ).</math>

In 1917, L. Mordell proved the first two relations using techniques from complex analysis, specifically using what are now known as Hecke operators. The third statement followed from the proof of the Weil conjectures by Template:Harvtxt. 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 Template:Harvtxt. 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 formsEdit

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 Template:Math of Template:Math. For any modular form

<math>f(z)=\sum^\infty_{n=0}a_nq^n \qquad q=e^{2\pi iz},</math>

one can form the Dirichlet series

<math>\varphi(s)=\sum^\infty_{n=1} \frac{a_n}{n^s}.</math>

For a modular form Template:Math of weight Template:Math for Template:Math, Template:Math absolutely converges in Template:Math, because Template:Math. Since Template:Mvar is a modular form of weight Template:Mvar, Template:Math turns out to be an entire and Template:Math satisfies the functional equation:

this was proved by Wilton in 1929. This correspondence between Template:Mvar and Template:Mvar is one to one (Template:Math). Let Template:Math for Template:Math, then Template:Math is related with Template:Math via the Mellin transformation

<math>R(s)=\int^\infty_0g(x)x^{s-1} \, dx \Leftrightarrow g(x) = \frac{1}{2\pi i} \int_{\operatorname{Re}(s)=\sigma_0}R(s)x^{-s} \, ds.</math>

This correspondence relates the Dirichlet series that satisfy the above functional equation with the automorphic form of a discrete subgroup of Template:Math.

In the case Template:Math 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).

Template:Harvtxt 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 Template:Math where Template:Mvar is the weight of the form. These results also follow from the Weil conjectures, except for the case Template:Math, where it is a result of Template:Harvtxt.

The Ramanujan–Petersson conjecture for Maass forms is still open (as of 2025) because Deligne's method, which works well in the holomorphic case, does not work in the real analytic case. A proof has recently been claimed by André Unterberger using tecniques from automorphic distribution theory.

Ramanujan–Petersson conjecture for automorphic formsEdit

Template:Harvtxt reformulated the Ramanujan–Petersson conjecture in terms of automorphic representations for Template:Math 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. Template:Harvtxt and Template:Harvtxt showed that the conjecture was also false even for some quasi-split and split groups, by constructing automorphic forms for the unitary group Template:Math and the symplectic group Template:Math that are non-tempered almost everywhere, related to the representation Template:Math.

After the counterexamples were found, Template:Harvtxt 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 Template:Math will give a proof of the Ramanujan–Petersson conjecture.

Bounds towards Ramanujan over number fieldsEdit

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 Template:Math, consider a unitary cuspidal automorphic representation:

<math>\pi = \bigotimes \pi_v.</math>

The Bernstein–Zelevinsky classification tells us that each p-adic Template:Math can be obtained via unitary parabolic induction from a representation

<math>\tau_{1,v} \otimes \cdots \otimes \tau_{d,v}.</math>

Here each <math>\tau_{i,v}</math> is a representation of Template:Math, over the place Template:Mvar, of the form

<math>\tau_{i_0,v} \otimes \left|\det\right|_v^{\sigma_{i,v}}</math>

with <math>\tau_{i_0,v}</math> tempered. Given Template:Math, a Ramanujan bound is a number Template:Math such that

<math>\max_i \left |\sigma_{i,v} \right | \leq \delta.</math>

Langlands classification can be used for the archimedean places. The generalized Ramanujan conjecture is equivalent to the bound Template:Math.

Template:Harvtxt obtain a first bound of Template:Math for the general linear group Template:Math, known as the trivial bound. An important breakthrough was made by Template:Harvtxt, who currently hold the best general bound of Template:Math for arbitrary Template:Mvar and any number field. In the case of Template:Math, Kim and Sarnak established the breakthrough bound of Template:Math when the number field is the field of rational numbers, which is obtained as a consequence of the functoriality result of Template:Harvtxt 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 Template:Harvtxt.

For reductive groups other than Template:Math, 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 Template:Harvtxt as a consequence of their Langlands functorial lift.

The Ramanujan–Petersson conjecture over global function fieldsEdit

Drinfeld's proof of the global Langlands correspondence for Template:Math 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 Template:Math in positive characteristic. Via a different technique that extends the Langlands–Shahidi method to include global function fields, Template:Harvtxt proves the Ramanujan conjecture for the classical groups.

ApplicationsEdit

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 Template:Math implies Selberg's conjecture about eigenvalues of the Laplacian for some discrete groups.

ReferencesEdit

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