Ramanujan-Petersson conjecture
Encyclopedia
In mathematics
, the Ramanujan conjecture, due to , states that Ramanujan's tau function given by the Fourier coefficients
of the cusp form
of weight 12
(where q=e2πiz) 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.
, namely 
for any 
.
This conjecture of Ramanujan followed from the proof of the Weil conjectures
by . The formulations required to show 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.
s 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 .
The Ramanujan–Petersson conjecture for Maass wave form
s is still open (as of 2011).
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
U2,1 and the symplectic group
that are non-tempered almost everywhere, related to the representation θ10
suggested that the generalized Ramanujan conjecture should still hold for generic cuspidal automorphic representations of a quasi-split reductive group
, where a generic cusp form is roughly one with a Whittaker model
. It states that each local component of such a representation should be tempered. Lafforgue's theorem
implies that the generalized Ramanujan conjecture is true for the general linear group
over a global function field, by an argument due to . There are known bounds over a number field. Ramanujan bounds for groups other than 
can be obtained as an application of known cases of Langlands functoriality
.
The Ramanujan–Petersson conjecture for general linear groups implies Selberg's conjecture
about eigenvalues of the Laplacian for some discrete groups. In turn, the Ramanujan–Petersson conjecture for general linear groups follows from the Arthur conjectures
.
s by Lubotzky
, Phillips and Sarnak
. Indeed, the name "Ramanujan graph" was derived from this connection.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Ramanujan conjecture, due to , states that Ramanujan's tau function given by the Fourier coefficients
of the cusp formCusp form
In number theory, a branch of mathematics, a cusp form is a particular kind of modular form, distinguished in the case of modular forms for the modular group by the vanishing in the Fourier series expansion \Sigma a_n q^n...
of weight 12(where q=e2πiz) satisfies
when
is a prime numberPrime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
. The generalized Ramanujan conjecture or Ramanujan–Petersson conjecture, introduced by , is a generalization to other modular forms or automorphic forms.
Ramanujan conjecture
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
Weil conjectures
In mathematics, the Weil conjectures were some highly-influential proposals by on the generating functions derived from counting the number of points on algebraic varieties over finite fields....
by . The formulations required to show it was a consequence were delicate and not at all obvious. It was the work of Michio Kuga
Michio Kuga
Michio Kuga is a mathematician who received his Ph.D. from University of Tokyo in 1960. His work is linked to the Ramanujan conjecture....
with contributions also by Mikio Sato
Mikio Sato
is a Japanese mathematician, who started the field of algebraic analysis. He studied at the University of Tokyo, and then did graduate study in physics as a student of Shin'ichiro Tomonaga...
, Goro Shimura
Goro Shimura
is a Japanese mathematician, and currently a professor emeritus of mathematics at Princeton University.Shimura was a colleague and a friend of Yutaka Taniyama...
, 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
Étale cohomology
In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures...
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 subgroupCongruence subgroup
In mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. A very simple example would be invertible 2x2 integer matrices of determinant 1, such that the off-diagonal entries are even.An importance class of congruence...
s 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 .
The Ramanujan–Petersson conjecture for Maass wave form
Maass wave form
In mathematics, a Maass wave form is a function on the upper half plane that transforms like a modular form but need not be holomorphic. They were first studied by .-Definition:...
s is still open (as of 2011).
Ramanujan–Petersson conjecture for automorphic forms
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. and 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
In mathematics, the unitary group of degree n, denoted U, is the group of n×n unitary matrices, with the group operation that of matrix multiplication. The unitary group is a subgroup of the general linear group GL...
U2,1 and the symplectic group
Symplectic group
In mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups, denoted Sp and Sp. The latter is sometimes called the compact symplectic group to distinguish it from the former. Many authors prefer slightly different notations, usually...
that are non-tempered almost everywhere, related to the representation θ10Θ10
In representation theory, a branch of mathematics, θ10 is a cuspidal unipotent complex irreducible representation of the symplectic group Sp4 over a finite, local, or global field....
suggested that the generalized Ramanujan conjecture should still hold for generic cuspidal automorphic representations of a quasi-split reductive group
Reductive group
In mathematics, a reductive group is an algebraic group G over an algebraically closed field such that the unipotent radical of G is trivial . Any semisimple algebraic group is reductive, as is any algebraic torus and any general linear group...
, where a generic cusp form is roughly one with a Whittaker model
Whittaker model
In representation theory, a branch of mathematics, the Whittaker model is a realization of a representation of a reductive algebraic group such as GL2 over a finite or local or global field on a space of functions on the group. It is named after E. T...
. It states that each local component of such a representation should be tempered. Lafforgue's theorem
Lafforgue's theorem
In mathematics, Lafforgue's theorem, due to Laurent Lafforgue, completes the Langlands program for general linear groups over function fields, by giving a correspondence between automorphic forms on these groups and representations of Galois groups....
implies that the generalized Ramanujan conjecture is true for the general linear group
General linear group
In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible...
over a global function field, by an argument due to . There are known bounds over a number field. Ramanujan bounds for groups other than can be obtained as an application of known cases of Langlands functorialityLanglands–Shahidi method
In mathematics, the Langlands–Shahidi method provides the means to define automorphic L-functions in many cases that arise with connected reductive groups over a number field. This includes Rankin–Selberg products for cuspidal automorphic representations of general linear groups...
.
The Ramanujan–Petersson conjecture for general linear groups implies Selberg's conjecture
Selberg's conjecture
In mathematics, the Selberg conjecture, named after Atle Selberg, is about the density of zeros of the Riemann zeta function \zeta\bigl. It is known that the function has infinitely many zeroes on this line in the complex plane: the point at issue is how densely they are clustered...
about eigenvalues of the Laplacian for some discrete groups. In turn, the Ramanujan–Petersson conjecture for general linear groups follows from the Arthur conjectures
Arthur conjectures
In mathematics, the Arthur conjectures are some conjectures about automorphic representations of reductive groups over the adeles and unitary representations of reductive groups over local fields made by , motivated by the Arthur-Selberg trace formula....
.
Applications
The most celebrated application of the Ramanujan conjecture is the explicit construction of Ramanujan graphRamanujan graph
A Ramanujan graph, named after Srinivasa Ramanujan, is a regular graph whose spectral gap is almost as large as possible . Such graphs are excellent spectral expanders....
s by Lubotzky
Alexander Lubotzky
Professor Alexander Lubotzky is an Israeli academic and former politician. A former head of the Mathematics Institute at the Hebrew University of Jerusalem, he served as a member of the Knesset for The Third Way party between 1996 and 1999.-Education:...
, Phillips and Sarnak
Peter Sarnak
Peter Clive Sarnak is a South African-born mathematician. He has been Eugene Higgins Professor of Mathematics at Princeton University since 2002, succeeding Andrew Wiles, and is an editor of the Annals of Mathematics...
. Indeed, the name "Ramanujan graph" was derived from this connection.