Ramanujan–Petersson conjecture

In mathematics, the Ramanujan-Petersson conjecture is conjecture concerning growth rate of coefficients of modular forms and more generally, automorphic forms. Name of conjecture comes from Srinivasa Ramanujan who proposed it for Ramanujan tau function and Hans Petersson, who generalized it for coefficients of modular forms.

In version for modular forms, it says that for any cusp form of weight ${\displaystyle k}$ and every ${\displaystyle \epsilon >0}$ if ${\displaystyle a_{n}}$ are Fourier coefficients of this form, we have:

${\displaystyle a_{n}=\left(n^{{\frac {k-1}{2}}+\epsilon }\right)}$

Generalization for automorphic forms is more sophisticated due to found counterexamples for the simplest propositions. Current version was proposed by |Howe]] and Piatetski-Shapiro, it says that for a globally generic cuspidal automorphic representation of a connected reductive group that admits a Whittaker model, each local component of representation is tempered.

For modular forms conjecture was proven due to extensive work of Erich Hecke, Michio Kuga and Pierre Deligne. Despite many similarities between modular forms and Maass forms, counterpert of conjecture for Maass forms is still open problem, because Deligne method that solves holomorphic case, don't work in real-analytic case of Maass forms. Generalization of conjecture for automorphic forms is also open problem.

Ramanujan conjecture

Let ${\textstyle q=e^{2\pi iz}}$. The discriminant modular form is usually defined by:

${\displaystyle \Delta (z)=q\prod _{n>0}\left(1-q^{n}\right)^{24}=\eta (z)^{24}}$

Where ${\textstyle \eta (z)}$ is Dedekind eta function. ${\textstyle \Delta (z)}$ is holomorphic cusp modular form of weight 12 and level 1. Ramanujan tau function for natural number is defined as coefficient of Fourier series of this modular form:

${\displaystyle \Delta (z)=\sum _{n=1}^{\infty }\tau (n)q^{n}}$

The first few terms in this series are:

${\displaystyle \Delta (z)=q-24q^{2}+252q^{3}-1472q^{4}+4830q^{5}-\cdots }$

Ramanujan (1916) conjectured the following:

1. ${\textstyle \tau }$ is multiplicative,
2. ${\textstyle \tau }$ is not completely multiplicative but for prime ${\textstyle p}$ and ${\textstyle j\geq 2}$ we have recursive dependence: ${\textstyle \tau (p^{j+1})=\tau (p)\tau (p^{j})-p^{11}\tau (p^{j-1})}$
3. For any prime number: ${\textstyle \vert \tau (p)\vert \leq 2p^{\frac {11}{2}}}$.

Due to lack of complete multiplicativity of tau function, estimation in full generality of conjecture is sligtly weaker:

${\displaystyle \tau (n)=O\left(n^{{\frac {11}{2}}+\varepsilon }\right).}$

and is not simple consequence of the following three conjectures. The formulations required to show that full generality estimation is actually a consequence of them were delicate, and not at all obvious.

In 1917, L. Mordell proved the first two relations using techniques from complex analysis, specifically using what are now known as Hecke operators. Proof that full generality estimation follows from three previous conjectures was given by Michio Kuga with contributions also by Mikio Sato, Goro Shimura, and Yasutaka Ihara, followed by Deligne (1971). The third statement followed from the proof of the Weil conjectures (more precise, counterpart of Riemann Hypothesis for local zeta functions) by Deligne (1974).

Ramanujan L-function

Original Ramanujan hypothesis was inspired by his research on L-function, nowadays called Ramanujan L-function. It can be defined as a Dirichlet series for Ramanujan tau function:

${\displaystyle L(s,\tau )=\sum _{n=1}^{\infty }{\frac {\tau (n)}{n^{s}}}}$

This series arises naturally as a Mellin transform of ${\displaystyle \Delta (z)}$. From general estimation for tau function:

${\textstyle \sum _{k=1}^{n}\tau (k)=O\left(n^{{\frac {13}{2}}+\epsilon }\right)}$

then this series converges absolutely for ${\textstyle \operatorname {Re} (s)>{\frac {13}{2}}}$ On the rest of complex plane Ramanujan L-function can be defined by analytic continuation of this series. As another L-functions, Ramanujan L-function satisfies functional equation:

${\displaystyle \left({\frac {1}{2\pi }}\right)^{s}\Gamma (s)L(s)=\left({\frac {1}{2\pi }}\right)^{12-s}\Gamma (12-s)L(12-s)}$

From mulitplicave property of tau function this series in the domain of convergence can be written as Euler product:

${\displaystyle L(s,\tau )=\prod _{p}\sum _{n=0}^{\infty }{\frac {\tau (p^{n})}{p^{ns}}}}$

Since tau function is not completely multiplicative, then sums cannot be changed to geometric series like in case of Riemann zeta function or Dirichlet L-functions. However, using recursive dependence for powers of primes, this sum can be simplified and we can write:

${\displaystyle L(s,\tau )=\prod _{p}{\frac {1}{1-\tau (p)p^{-s}+p^{11-2s}}}}$

Ramanujan used all properties above only conditionally. He checked some of quadratic equations obtained from denominators in formula for L-function obtained by substitution ${\displaystyle u=p^{-s}}$:

${\displaystyle 1+\tau (p)u-p^{11}u^{2}=0}$

He observed that if such equation have non-real roots or double real root, then from its discriminant follows:

${\displaystyle \Delta =\tau (p)^{2}-4p^{11}\leq 0}$

${\displaystyle \vert \tau (p)\vert \leq 2p^{\frac {11}{2}}}$

Equivalently, since all coefficients of equation are real and roots are conjugated, we can say that this holds if roots of this equation lie on the same straight line in complex plane and:

${\displaystyle \operatorname {Re} (u_{1})=\operatorname {Re} (u_{2})}$

Exacly this property follows from Riemann hypothesis counterpart for local zeta functions that was proven by Deligne.

Ramanujan also conjectured that analogously to Riemann hypothesis all nontrivial zeros should lie on critical line ${\textstyle \operatorname {Re} =6}$ and observed that it implies:

${\displaystyle \tau (n)=O\left(n^{{\frac {11}{2}}+\varepsilon }\right)}$

Riemann hypothesis for ${\textstyle L(s,\tau )}$ is still unproven, but this estimation was proven unconditionally.

Ramanujan–Petersson conjecture for modular forms

Having any modular form ${\displaystyle f(z)}$ of weight ${\displaystyle k\geq 2}$ for arithmetic subgroup ${\displaystyle \Gamma \subseteq SL(2,\mathbb {Z} )}$, we can write it as Fourier series:

${\displaystyle f(z)=\sum _{n=0}^{\infty }a_{n}q^{n}\quad {\text{where: }}q=e^{2\pi iz}}$

If we have ${\textstyle a_{0}=0}$, we say that ${\displaystyle f(z)}$ is cusp form. In 1937, Erich Hecke used Hecke operators to generalize the method of Mordell's proof of the first two conjectures for holomorphic cusp forms. The more general Ramanujan–Petersson conjecture for holomorphic cusp forms for congruence subgroups was proposed in Petersson (1930) and has a similar formulation:

${\displaystyle a_{n}=O\left(n^{{\frac {k-1}{2}}+\epsilon }\right)}$

Deligne (1971) used the Eichler–Shimura isomorphism to reduce this conjecture to the Weil conjectures that he later proved. This is also true for case ${\displaystyle k=1}$, where it is a result of Deligne & Serre (1974).

In the case ${\displaystyle k\geq 3}$ Hans Petersson introduced a metric on the space of modular forms, called the Petersson metric (also see Weil–Petersson metric). Under the Petersson metric it is shown that we can define the orthogonality on the space of modular forms as on the subspace 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 version of 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 techniques from automorphic distribution theory.

L-functions for modular forms

Having modular form:

${\displaystyle f(z)=\sum _{n=0}^{\infty }a_{n}q^{n}\quad {\text{where: }}q=e^{2\pi iz}}$

one can form the Dirichlet series:

${\displaystyle \varphi (s)=\sum _{n=1}^{\infty }{\frac {a_{n}}{n^{s}}}.}$

This correspondence between ${\displaystyle f}$ and ${\displaystyle \varphi }$ is one to one. Denote:

${\displaystyle R(s)=(2\pi )^{-s}\Gamma (s)\varphi (s)}$

${\displaystyle g(x)=f(ix)-a_{0}\quad {\text{for: }}x>0}$

And ${\displaystyle R(s)}$ is related with ${\displaystyle g(x)}$ via the Mellin transformation:

${\displaystyle R(s)=\int _{0}^{\infty }g(x)x^{s-1}\,dx}$

Applying inverse Mellin transform we get:

${\displaystyle g(x)={\frac {1}{2\pi i}}\int _{\operatorname {Re} (s)=\sigma _{0}}R(s)x^{-s}\,ds.}$

Because ${\displaystyle a_{n}=O(n^{k-1+\epsilon })}$, ${\displaystyle \varphi (s)}$ converges absolutely for ${\displaystyle \operatorname {Re} >k}$. In 1929 Wilton proven that this Dirichlet series have analytic continuation, has at most one simple pole at ${\displaystyle s=k}$ (for non-cusp forms) with residue:

${\displaystyle a_{0}=(-1)^{\frac {k}{2}}{\text{Res}}_{s=k}R(s)=(-1)^{\frac {k}{2}}{\frac {(2\pi )^{k}}{(k-1)!}}{\text{Res}}_{s=k}\varphi (s)}$

and using notation above satisfies functional equation:

${\displaystyle R(s-k)=(-1)^{\frac {k}{2}}R(s)}$

Mellin transform relates the Dirichlet series that satisfy the above functional equation with the automorphic form of a discrete subgroup of ${\displaystyle SL(2,\mathbb {Z} )}$. From residue formula we see that exactly for cusp forms this series have continuation to entire function. This shows that Ramanujan-Petersson conjecture fails for non-cusp forms, since every Dirichlet series with coefficients satisfying it is absolutely convergent for ${\displaystyle \operatorname {Re} >{\frac {k+1}{2}}}$ and if ${\displaystyle k>1}$ it means it cannot have pole at ${\displaystyle s=k}$.

From multiplicative property of coefficients follows existence of Euler product for this L-function and from recursive property follows rewriting sums of powers of primes as reciprocals of polynomials of ${\displaystyle p^{-s}}$.

Ramanujan–Petersson conjecture for automorphic forms

Satake (1966) reformulated the Ramanujan–Petersson conjecture in terms of automorphic representations for 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.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 U(2, 1) and the symplectic group Sp(4) that are non-tempered almost everywhere, related to the representation θ₁₀.

After the counterexamples were found, Howe & Piatetski-Shapiro (1979) suggested that a reformulation of the conjecture should still hold. The current formulation 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 GL(n) will give a proof of this version of the Ramanujan–Petersson conjecture. For reductive groups other than GL(n), the generalized Ramanujan conjecture would follow from principle of Langlands functoriality.

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. 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.

Bounds towards Ramanujan over number fields

In order to understand the Ramanujan bounds for GL(n), consider a unitary cuspidal automorphic representation:

${\displaystyle \pi =\bigotimes \pi _{v}.}$

Langlands classification can be used for the archimedean places. The Bernstein–Zelevinsky classification tells us that each p-adic place π_v can be obtained via unitary parabolic induction from a representation:

${\displaystyle \tau _{1,v}\otimes \cdots \otimes \tau _{d,v}.}$

Here each ${\displaystyle \tau _{i,v}}$ is a representation of GL(n_i), over the place v, of the form

${\displaystyle \tau _{i_{0},v}\otimes \left|\det \right|_{v}^{\sigma _{i,v}}}$

with ${\displaystyle \tau _{i_{0},v}}$ tempered. Given n ≥ 2, a Ramanujan bound is a number δ ≥ 0 such that

${\displaystyle \max _{i}\left|\sigma _{i,v}\right|\leq \delta .}$

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. The generalized Ramanujan-Petersson conjecture is equivalent to the bound:

${\displaystyle \delta =0}$

Jacquet, Piatetskii-Shapiro & Shalika (1983) obtained a first bound of:

${\displaystyle \delta ={\frac {1}{2}}}$

for the general linear group GL(n), known as the trivial bound. An important breakthrough was made by Luo, Rudnick & Sarnak (1999), who currently hold the best general bound of:

${\displaystyle \delta ={\frac {1}{2}}-{\frac {1}{n^{2}+1}}}$

for arbitrary n and any number field.

In special case of GL(2), best established bound is δ = 7/64. Original result of Kim and Sarnak in Kim (2002) was for number field being rational numbers, which was obtained as a consequence of the functoriality result of on the symmetric fourth obtained via the Langlands–Shahidi method. Generalization of the Kim-Sarnak result to an arbitrary number field is possible by the results of Blomer & Brumley (2011).

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.

Consequences

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 consequence of Ramanujan–Petersson conjecture for the general linear group GL(n) is Selberg's 1/4 conjecture about eigenvalues of the Laplacian for some discrete groups.