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GL2\mathrm{GL}_2 Ramanujan conjecture exponent (C56C_{56})

GL2\mathrm{GL}_2 Ramanujan conjecture exponent: determine C56C_{56} or improve its bounds; the repository records 0 (lower) and 764=0.109375\tfrac{7}{64}=0.109375 (upper).

MSC 11F70 · Representation-theoretic methods; automorphic representations over local and global fieldsMSC 11F72 · Spectral theory; trace formulas (e.g., that of Selberg)

Editorial MSC2020 classification. Original article codes and source evidence are recorded in the literature directory.

Licensed source statementAs of Repository table at commit 2c1968c (2026-09-26); open while the recorded lower and upper bounds differSource & evidence ↓
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Problem statement

We define C56=δ2C_{56} = \delta_2 to be the smallest real number δ≥0\delta \ge 0 such that the following uniform bound toward the Generalized Ramanujan Conjecture holds.

(Hypothesis H2(δ)H_2(\delta), specialization of [BB2011]) For every number field FF, every cuspidal automorphic representation π\pi of GL2(AF)\mathrm{GL}_2(\mathbb A_F) with unitary central character, and every place vv of FF, the local component πv\pi_v is "δ\delta-tempered".

One convenient way to quantify "δ\delta-tempered" is via the Langlands classification. Write the (generic, unitary) representation πv\pi_v as a parabolic induction

πv≃Ind⁡(τ1∣det⁡∣σ1 ⊗ τ2∣det⁡∣σ2)\pi_v \simeq \operatorname{Ind}\bigl(\tau_1|\det|^{\sigma_1}\,\otimes\,\tau_2|\det|^{\sigma_2}\bigr)

with τ1,τ2\tau_1,\tau_2 tempered and real exponents σ1≥σ2\sigma_1 \ge \sigma_2. Set

m(π,v):=max⁡(∣σ1∣,∣σ2∣)=σ1.m(\pi,v) := \max(|\sigma_1|,|\sigma_2|) = \sigma_1.

Then H2(δ)H_2(\delta) is the assertion that m(π,v)≤δm(\pi,v)\le \delta for all π,v\pi,v, and δ2\delta_2 is the infimum of admissible δ\delta.

At an unramified finite place vv with residue field size N(v)N(v), the representation πv\pi_v has Satake parameters {αv,βv}\{\alpha_v,\beta_v\} with ∣αvβv∣=1|\alpha_v\beta_v|=1 and

∣αv∣=N(v)tv,∣βv∣=N(v)−tv|\alpha_v| = N(v)^{t_v},\qquad |\beta_v| = N(v)^{-t_v}

for some tv∈[0,12]t_v\in[0,\tfrac12]. In this case m(π,v)=tvm(\pi,v)=t_v, so tv≤δ2t_v\le \delta_2 is equivalent to

∣αv∣,∣βv∣≤N(v)δ2.|\alpha_v|,|\beta_v| \le N(v)^{\delta_2}.

Equivalently, the Hecke eigenvalue λv:=αv+βv\lambda_v := \alpha_v+\beta_v satisfies

∣λv∣≤N(v)δ2+N(v)−δ2≤2N(v)δ2.|\lambda_v| \le N(v)^{\delta_2}+N(v)^{-\delta_2}\le 2N(v)^{\delta_2}.

In the classical language of Hecke–Maass newforms (over Q\mathbb Q), this corresponds to bounds of the shape

∣λf(n)∣≪ε,fd(n) nδ2+ε,|\lambda_f(n)| \ll_{\varepsilon,f} d(n)\,n^{\delta_2+\varepsilon},

where d(n)d(n) is the divisor function.

Conjecturally (and implied by the Langlands functoriality conjectures), one has δ2=0\delta_2=0.

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Background

  • Relation to Selberg's 14\tfrac14 eigenvalue conjecture. For weight 00 Maaß cusp forms for congruence subgroups, the archimedean parameter bound m(π,∞)≤θm(\pi,\infty)\le \theta implies a spectral gap
λ1≥14−θ2.\lambda_1 \ge \tfrac14-\theta^2.

In particular, θ=764\theta=\tfrac{7}{64} yields

λ1≥14−(764)2=9754096≈0.238037…\lambda_1 \ge \tfrac14-\bigl(\tfrac{7}{64}\bigr)^2 = \tfrac{975}{4096}\approx 0.238037\ldots

see [Sar2005] and also [Li2006] for this numerical value.

  • Holomorphic forms. For classical holomorphic cusp forms on GL2/Q\mathrm{GL}_2/\mathbb Q (i.e. π∞\pi_\infty holomorphic discrete series), the full Ramanujan–Petersson conjecture is known (δ=0\delta=0), by Deligne (and Deligne–Serre in weight 1); see [Sar2005] for a discussion.

  • Unramified vs. ramified places. Many "toward Ramanujan" bounds are first proved for unramified places; extending the same exponent to ramified places can require additional input. See the discussion around Hypothesis Hn(δ)H_n(\delta) in [BB2011] and the remarks in [Sar2005].

  • Terminology. In analytic number theory, δ2\delta_2 is often denoted by θ\theta and referred to as "the (best known) Ramanujan exponent for GL2\mathrm{GL}_2".

  • See also: the Wikipedia page on the Ramanujan–Petersson conjecture (for a quick orientation) and Sarnak's survey [Sar2005] (for a detailed representation-theoretic overview).

  • [BB2011] Blomer, Valentin; Brumley, Farrell. On the Ramanujan conjecture over number fields. Annals of Math. 174 (2011), 581–605. arXiv:1003.0559.

  • [JS1981] Jacquet, Hervé; Shalika, Joseph A. On Euler products and the classification of automorphic representations I. Amer. J. Math. 103 (1981), 499–558. Rankin–Selberg / unitarity trivial bound δ≤1/2\delta\le 1/2.

  • [KiSh2002] Kim, Henry H.; Shahidi, Freydoon. Cuspidality of symmetric powers with applications. Duke Math. J. 112 (2002), 177–197. DOI: 10.1215/S0012-9074-02-11215-0.

  • [KS2003] Kim, Henry H.; Sarnak, Peter. Refined estimates towards the Ramanujan and Selberg conjectures. Appendix 2 in: H. H. Kim, Functoriality for the exterior square of GL4\mathrm{GL}_4 and the symmetric fourth of GL2\mathrm{GL}_2. J. Amer. Math. Soc. 16 (2003), no. 1, 139–183. DOI: 10.1090/S0894-0347-02-00410-1.

  • [Li2006] Li, Xian-Jin. On exceptional eigenvalues of the Laplacian for Γ0(N)\Gamma_0(N). arXiv:math/0610120.

  • [LRS1999] Luo, Wenzhi; Rudnick, Zeev; Sarnak, Peter. On the generalized Ramanujan conjecture for GL(n)\mathrm{GL}(n). In: Automorphic Forms, Automorphic Representations, and Arithmetic (Fort Worth, TX, 1996), Proc. Sympos. Pure Math. 66, Amer. Math. Soc., Providence, RI, 1999, pp. 301–310.

  • [MS2004] Müller, Werner; Speh, Birgit. Absolute convergence of the spectral side of the Arthur trace formula for GLn\mathrm{GL}_n. Geom. Funct. Anal. 14 (2004), 58–93. DOI: 10.1007/s00039-004-0452-0.

  • [Sar2005] Sarnak, Peter. Notes on the Generalized Ramanujan Conjectures. Clay Mathematics Proceedings, Vol. 4 (2005). Available at https://web.math.princeton.edu/sarnak/FieldNotesCurrent.pdf

Progress & source assessment

Repository table (commit 2c1968c, 26 September 2026): best known lower bound 0; best known upper bound 764=0.109375\tfrac{7}{64}=0.109375. Values marked with an asterisk have minimal verification.

Known upper bounds

Bound Reference Comments
12\tfrac12 [JS1981] “Trivial” bound coming from Rankin–Selberg theory / unitarity; holds in general rank as Hn(12)H_n(\tfrac12) [BB2011].
12−122+1=310=0.3\tfrac12-\tfrac{1}{2^2+1}=\tfrac{3}{10}=0.3 [LRS1999] (ramified extension: [MS2004]) Specialization of the Luo–Rudnick–Sarnak bound δn≤12−1n2+1\delta_n\le \tfrac12-\tfrac{1}{n^2+1} to n=2n=2; extended to ramified places by Müller–Speh.
19≈0.11111\tfrac19\approx 0.11111 [KiSh2002] Bound obtained from functoriality results for low symmetric powers (often stated for unramified places; see discussion in [Sar2005]).
764=0.109375\tfrac{7}{64}=0.109375 [KS2003] (over Q\mathbb Q), extended uniformly to all number fields by [BB2011] Best known general bound. In Sarnak’s notation, this controls both finite-place Satake parameters and archimedean spectral parameters for GL2(AQ)\mathrm{GL}_2(\mathbb A_{\mathbb Q}) (and Blomer–Brumley extend it to arbitrary number fields).

Known lower bounds

Bound Reference Comments
00 Trivial Conjectured to be sharp (Generalized Ramanujan / Selberg).
References
  1. [1]Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — $\mathrm{GL}_2$ Ramanujan conjecture exponent ($C_{56}$) linkApache-2.0 (repository license); constant pages reproduced with attribution; repository references retained.

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