Math 2.0

Number theory Conjecture

Selberg's 1/4 conjecture

Builds on Congruence subgroupModular groupUpper half-planeLaplace–Beltrami operatorMaass wave formEigenvalue

Statement

Let Γ\Gamma be a congruence subgroup of the modular group SL2(ℤ)\mathrm{SL}_2(\Z), acting on the upper half-plane ℍ\mathbb H. Let Δ=−y2(∂x2+∂y2)\Delta = -y^2 (\partial_x^2 + \partial_y^2) be the hyperbolic Laplace–Beltrami operator on L2(Γ\ℍ)L^2(\Gamma \backslash \mathbb H). The conjecture states that every nonzero eigenvalue λ\lambda of Δ\Delta on L2(Γ\ℍ)L^2(\Gamma \backslash \mathbb H) satisfies λ≥14\lambda \ge \tfrac14.

Remarks

The eigenfunctions in question are the Maass cusp forms for Γ\Gamma, since the continuous spectrum is [14,∞)[\tfrac14, \infty). Selberg proved the bound λ≥316\lambda \ge \tfrac{3}{16}. The best bound known is λ≥9754096\lambda \ge \tfrac{975}{4096}, due to Kim and Sarnak. The conjecture is known for SL2(ℤ)\mathrm{SL}_2(\Z) itself and for congruence subgroups of small level. It is the archimedean case of the generalised Ramanujan conjecture for GL2\mathrm{GL}_2: an eigenvalue λ=14−s2\lambda = \tfrac14 - s^2 with 0<s<120 < s < \tfrac12 would give a non-tempered local component at the real place. The arithmetic hypothesis matters: for general cofinite Fuchsian groups, eigenvalues below 14\tfrac14 can occur.

Reading path

To understand this page from the ground up, read these 20 pages in order. The foundations come first.

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Complex numbers, object, no evidence levelComplex numbersobjectCongruence subgroup, concept, no evidence levelCongruence subgroupconceptEigenvalue, concept, no evidence levelEigenvalueconceptGeneral linear group, object, no evidence levelGeneral linear groupobjectIdentity matrix, object, no evidence levelIdentity matrixobjectLaplace–Beltrami operator, concept, no evidence levelLaplace–Beltrami oper…conceptMaass wave form, concept, no evidence levelMaass wave formconceptMatrix, definition, no evidence levelMatrixdefinitionMöbius transformation, concept, no evidence levelMöbius transformationconceptQuotient group, concept, no evidence levelQuotient groupconceptVector space, definition, no evidence levelVector spacedefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionLinear map, definition, no evidence levelLinear mapdefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionDeterminant, definition, no evidence levelDeterminantdefinitionSpecial linear group, object, no evidence levelSpecial linear groupobjectUpper half-plane, object, no evidence levelUpper half-planeobjectModular group, object, no evidence levelModular groupobjectSelberg's 1/4 conjecture, conjecture, no evidence levelSelberg's 1/4 conject…conjecture
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Complex numbers object
  2. Congruence subgroup concept
  3. Eigenvalue concept
  4. General linear group object
  5. Identity matrix object
  6. Laplace–Beltrami operator concept
  7. Maass wave form concept
  8. Matrix definition
  9. Möbius transformation concept
  10. Quotient group concept
  11. Vector space definition
  12. Square matrix definition
  13. Linear map definition
  14. Multilinear map definition
  15. Alternating multilinear map definition
  16. Determinant definition
  17. Special linear group object
  18. Upper half-plane object
  19. Modular group object
  20. Selberg's 1/4 conjecture conjecture

Further reading

  1. Web https://en.wikipedia.org/wiki/Selberg%27s_1/4_conjecture
About this page
TagB7K7NY8NTJ
URIm2://B7K7NY8NTJ
Kindconjecture
WritingW0 stub
MSC11F72, 11F12, 58J50
StatusThis copy of the site has no build data. Run m2 status B7K7NY8NTJ.
AboutLaplace–Beltrami operator, Congruence subgroup
UsesCongruence subgroup, Modular group, Upper half-plane, Laplace–Beltrami operator, Maass wave form, Eigenvalue
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