View source for Ramanujan–Petersson conjecture

In [[Mathematics|mathematics]], the '''Ramanujan conjecture''', due to {{harvs|txt|first=Srinivasa |last=Ramanujan|year=1916|loc=p. 176}}, states that Ramanujan's tau function given by the Fourier coefficients {{math|''τ''(''n'')}} of the [[Cusp form|cusp form]] {{math|Δ(''z'')}} of weight {{math|12}}
:<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

:<math>|\tau(p)| \leq 2p^{11/2},</math>

when {{mvar|p}} is a [[Prime number|prime number]]. The '''generalized Ramanujan conjecture''' or '''Ramanujan–Petersson conjecture''', introduced by {{harvs|txt|last=Petersson|year=1930}}, 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]],

{{NumBlk|::|<math>L(s,a)=\prod_p\left(1+\frac{a(p)}{p^s}+\frac{a(p^2)}{p^{2s}}+\dots\right)</math>|{{EquationRef|1}}}}

and due to their completely multiplicative property

{{NumBlk|::|<math>L(s,a)=\prod_p\left(1-\frac{a(p)}{p^s}\right)^{-1}.</math>|{{EquationRef|2}}}}

Are there L-functions other than the Riemann zeta function and the Dirichlet L-functions satisfying the above relations? Indeed, the [[Automorphic L-function|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 form#The modular discriminant|modular discriminant]] satisfies the modified relation

{{NumBlk|::|<math>L(s,\tau)=\prod_p\left(1-\frac{\tau(p)}{p^s}+\frac{1}{p^{2s-11}}\right)^{-1},</math>|{{EquationRef|3}}}}

where {{math|''τ''(''p'')}} is [[Ramanujan tau function|Ramanujan's tau function]]. The term

:<math>\frac{1}{p^{2s-11}}</math>

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:
#{{math|''τ''}} is multiplicative,
#{{math|''τ''}} is not completely multiplicative but for prime {{mvar|p}} and {{mvar|j}} in {{math|'''N'''}} we have: {{math|''τ''(''p''<sup>&thinsp;''j''+1</sup>) {{=}} ''τ''(''p'')''τ''(''p<sup>&thinsp;j</sup>''&thinsp;) − ''p''<sup>11</sup>''τ''(''p''<sup>&thinsp;''j''−1</sup>&thinsp;)}}, and
#{{math|{{!}}''τ''(''p''){{!}} ≤ 2''p''<sup>11/2</sup>}}.

Ramanujan observed that the quadratic equation of {{math|''u'' {{=}} ''p''<sup>−''s''</sup>}} in the denominator of RHS of {{EquationNote|3|(3)}},

: <math>1-\tau(p)u+p^{11}u^2</math>

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 {{mvar|α}} and {{mvar|β}}, 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 {{math|''τ''(''n'')}}, namely for any {{math|''ε'' > 0}}:

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

In 1917, [[Biography:Louis J. Mordell|L. Mordell]] proved the first two relations using techniques from complex analysis, specifically what are now known as [[Hecke operator]]s. The third statement followed from the proof of the [[Weil conjectures]] by {{harvtxt|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, [[Biography:Goro Shimura|Goro Shimura]], and [[Biography:Yasutaka Ihara|Yasutaka Ihara]], followed by {{harvtxt|Deligne|1971}}. The existence of the connection inspired some of the deep work in the late 1960s when the consequences of the [[Étale cohomology|étale cohomology]] theory were being worked out.

==Ramanujan–Petersson conjecture for modular forms==
In 1937, [[Biography:Erich Hecke|Erich Hecke]] used [[Hecke operator]]s to generalize the method of Mordell's proof of the first two conjectures to the [[Automorphic L-function|automorphic L-function]] of the discrete subgroups {{math|Γ}} of {{math|SL(2, '''Z''')}}. For any [[Modular form|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 {{math|''f''&thinsp;(''z'')}} of weight {{math|''k'' ≥ 2}} for {{math|Γ}}, {{math|''φ''(''s'')}} absolutely converges in {{math|Re(''s'') > ''k''}}, because {{math|''a<sub>n</sub>'' {{=}} O(''n''<sup>''k''−1+''ε''</sup>)}}. Since {{mvar|f}} is a modular form of weight {{mvar|k}}, {{math|(''s'' − ''k'')''φ''(''s'')}} turns out to be an [[Entire function|entire]] and {{math|''R''(''s'') {{=}} (2{{pi}})<sup>−''s''</sup>Γ(''s'')''φ''(''s'')}} satisfies the '''functional equation''':

:<math>R(k-s)=(-1)^{k/2}R(s);</math>

this was proved by Wilton in 1929. This correspondence between {{mvar|f}} and {{mvar|φ}} is one to one ({{math|''a''<sub>0</sub> {{=}} (−1)<sup>''k''/2</sup>&thinsp;Res<sub>''s''{{=}}''k''</sub>&thinsp;''R''(''s'')}}). Let {{math|''g''(''x'') {{=}} ''f''&thinsp;(''ix'') −''a''<sub>0</sub>}} for {{math|''x'' > 0}}, then {{math|''g''(''x'')}} is related with {{math|''R''(''s'')}} via the [[Mellin transform]]ation

:<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 {{math|SL(2, '''Z''')}}.

In the case {{math|''k'' ≥ 3}} [[Biography:Hans Petersson|Hans Petersson]] introduced a metric on the space of modular forms, called the [[Petersson inner product|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 form|cusp form]]s 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 [[Modular form#Consequences|the dimensions of modular forms]]).

{{harvtxt|Deligne|1971}} 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 subgroup|congruence subgroup]]s has a similar formulation, with exponent {{math|(''k'' − 1)/2}} where {{mvar|k}} is the weight of the form. These results also follow from the [[Weil conjectures]], except for the case {{math|''k'' {{=}} 1}}, where it is a result of {{harvtxt|Deligne|Serre|1974}}.

The Ramanujan–Petersson conjecture for Maass forms is still open (as of 2022) 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==
{{harvtxt|Satake|1966}} reformulated the Ramanujan–Petersson conjecture in terms of automorphic representations for {{math|GL(2)}} 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. {{harvtxt|Kurokawa|1978}} and {{harvtxt|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|unitary group]] {{math|U(2, 1)}} and the [[Symplectic group|symplectic group]] {{math|Sp(4)}} that are non-tempered almost everywhere, related to the representation {{math|[[Θ10|''θ''<sub>10</sub>]]}}.

After the counterexamples were found, {{harvtxt|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|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 [[Biography:Robert Langlands|Langlands]] that establishing functoriality of symmetric powers of automorphic representations of {{math|GL(''n'')}} 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|number theory]]. In order to understand the '''Ramanujan bounds''' for {{math|GL(''n'')}}, consider a unitary cuspidal automorphic representation:

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

The [[Bernstein–Zelevinsky classification]] tells us that each p-adic {{math|{{pi}}<sub>''v''</sub>}} 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 {{math|GL(''n<sub>i</sub>'')}}, over the place {{mvar|v}}, 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 {{math|''n'' ≥ 2}}, a '''Ramanujan bound''' is a number {{math|''δ'' ≥ 0}} 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 {{math|''δ'' {{=}} 0}}.

{{harvtxt|Jacquet|Piatetski-Shapiro|Shalika|1983}} obtain a first bound of {{math|''δ'' ≤ 1/2}} for the [[General linear group|general linear group]] {{math|GL(''n'')}}, known as the trivial bound. An important breakthrough was made by {{harvtxt|Luo|Rudnick|Sarnak|1999}}, who currently hold the best general bound of {{math|''δ'' ≡ 1/2 − (''n''<sup>2</sup>+1)<sup>−1</sup>}} for arbitrary {{mvar|n}} and any number field. In the case of {{math|GL(2)}}, Kim and Sarnak established the breakthrough bound of {{math|''δ'' {{=}} 7/64}} when the number field is the field of rational numbers, which is obtained as a consequence of the functoriality result of {{harvtxt|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 {{harvtxt|Blomer|Brumley|2011}}.

For [[Reductive group|reductive group]]s other than {{math|GL(''n'')}}, 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 {{harvtxt|Cogdell|Kim|Piatetski-Shapiro|Shahidi|2004}} as a consequence of their Langlands [[Langlands–Shahidi method|functorial lift]].

==The Ramanujan–Petersson conjecture over global function fields==
[[Biography:Vladimir Drinfeld|Drinfeld's]] proof of the global Langlands correspondence for {{math|GL(2)}} over a global function field leads towards a proof of the Ramanujan–Petersson conjecture. [[Lafforgue's theorem|Lafforgue (2002)]] successfully extended [[Drinfeld module|Drinfeld's shtuka]] technique to the case of {{math|GL(''n'')}} in positive characteristic. Via a different technique that extends the [[Langlands–Shahidi method]] to include global function fields, {{harvtxt|Lomelí|2009}} proves the Ramanujan conjecture for the classical groups.

==Applications==
An application of the Ramanujan conjecture is the explicit construction of [[Ramanujan graph]]s by [[Biography:Alexander Lubotzky|Lubotzky]], Phillips and [[Biography:Peter Sarnak|Sarnak]]. Indeed, the name "Ramanujan graph" was derived from this connection. Another application is that the Ramanujan–Petersson conjecture for the [[General linear group|general linear group]] {{math|GL(''n'')}} implies Selberg's conjecture about eigenvalues of the Laplacian for some discrete groups.

==References==
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