In mathematics, Selberg's conjecture, also known as Selberg's eigenvalue conjecture, conjectured by , states that the

eigenvalues In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...

of the

Laplace operator In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is t ...

Maass wave form In mathematics, Maass forms or Maass wave forms are studied in the theory of automorphic forms. Maass forms are complex-valued smooth functions of the upper half plane, which transform in a similar way under the operation of a discrete subgroup \ ...

s of

congruence 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 2 × 2 integer matrices of determinant 1, in which the ...

s are at least 1/4. Selberg showed that the eigenvalues are at least 3/16. Subsequent works improved the bound, and the best bound currently known is 975/4096≈0.238..., due to Kim and Sarnak (2003). The generalized Ramanujan conjecture for the

general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible ...

implies Selberg's conjecture. More precisely, Selberg's conjecture is essentially the generalized Ramanujan conjecture for the group GL₂ over the rationals at the infinite place, and says that the component at infinity of the corresponding representation is a principal series representation of GL₂(R) (rather than a complementary series representation). The generalized Ramanujan conjecture in turn follows from the

Langlands functoriality conjecture In representation theory and algebraic number theory, the Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry. Proposed by , it seeks to relate Galois groups in algebraic num ...

, and this has led to some progress on Selberg's conjecture.

References

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External links

* {{Cite web , title=Selberg conjecture - Encyclopedia of Mathematics , url=https://encyclopediaofmath.org/wiki/Selberg_conjecture , access-date=2022-06-08 , website=encyclopediaofmath.org Automorphic forms Conjectures