Math Conjectures

Generalized Ramanujan Conjecture for GLn\mathrm{GL}_n

OPENLandmarkConjectureProposed c. 1967 · Standard version

Canonical statement

For every n≥2n\ge2, every irreducible unitary cuspidal automorphic representation π=⨂v′πv\pi=\bigotimes'_v\pi_v of GLn(AQ)\mathrm{GL}_n(\mathbb A_{\mathbb Q}), where AQ\mathbb A_{\mathbb Q} is the adèle ring of Q\mathbb Q, and every place vv of Q\mathbb Q, the local representation πv\pi_v is tempered; equivalently, all its KvK_v-finite matrix coefficients belong to L2+ε(GLn(Qv)/Zv)L^{2+\varepsilon}(\mathrm{GL}_n(\mathbb Q_v)/Z_v) for every ε>0\varepsilon>0, where KvK_v is a maximal compact subgroup and ZvZ_v is the center.
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For every \(n\ge2\), every irreducible unitary cuspidal automorphic representation \(\pi=\bigotimes'_v\pi_v\) of \(\mathrm{GL}_n(\mathbb A_{\mathbb Q})\), where \(\mathbb A_{\mathbb Q}\) is the adèle ring of \(\mathbb Q\), and every place \(v\) of \(\mathbb Q\), the local representation \(\pi_v\) is tempered; equivalently, all its \(K_v\)-finite matrix coefficients belong to \(L^{2+\varepsilon}(\mathrm{GL}_n(\mathbb Q_v)/Z_v)\) for every \(\varepsilon>0\), where \(K_v\) is a maximal compact subgroup and \(Z_v\) is the center.
NT-AUTO-030:[Ramanujan1916Tau][Deligne1974WeilI][Sarnak2005Ramanujan][LuoRudnickSarnak1999]
Number theoryAlgebra and representation theory

Notes

Ramanujan conjectured in 1916 that his tau function satisfies ∣τ(p)∣≤2p11/2|\tau(p)|\le 2p^{11/2} for every prime pp [Ramanujan1916Tau]. The generalized Ramanujan conjecture, which took its modern shape around 1967 as automorphic forms were recast in adelic representation-theoretic language, asserts that for every irreducible unitary cuspidal automorphic representation π\pi of GLn(AQ)\mathrm{GL}_n(\mathbb A_{\mathbb Q}), every local component πv\pi_v is tempered, meaning its matrix coefficients lie in L2+εL^{2+\varepsilon} modulo the center for all ε>0\varepsilon>0.

Deligne's proof of the Weil conjectures settled the case of holomorphic modular forms on GL2\mathrm{GL}_2, including Ramanujan's original claim [Deligne1974WeilI]. For general nn, Luo, Rudnick and Sarnak obtained bounds toward temperedness valid at every place [Sarnak2005Ramanujan], and the surrounding landscape of approximations and functoriality implications is surveyed by Sarnak [LuoRudnickSarnak1999].

Only partial progress toward temperedness is known beyond these cases, and a full resolution is generally expected to require functoriality or genuinely new arithmetic input; the conjecture remains open for general Maass forms and for cuspidal representations in higher rank.

References (4)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.