GL_2 Ramanujan conjecture exponent
Description of constant
We define $C_{56} = \delta_2$ to be the smallest real number $\delta \ge 0$ such that the following uniform bound toward the Generalized Ramanujan Conjecture holds.
(Hypothesis $H_2(\delta)$, specialization of [BB2011]) For every number field $F$, every cuspidal automorphic representation $\pi$ of $\mathrm{GL}_2(\mathbb A_F)$ with unitary central character, and every place $v$ of $F$, the local component $\pi_v$ is “$\delta$-tempered”.
One convenient way to quantify “$\delta$-tempered” is via the Langlands classification. Write the (generic, unitary) representation $\pi_v$ as a parabolic induction \(\pi_v \simeq \operatorname{Ind}\bigl(\tau_1|\det|^{\sigma_1}\,\otimes\,\tau_2|\det|^{\sigma_2}\bigr)\) with $\tau_1,\tau_2$ tempered and real exponents $\sigma_1 \ge \sigma_2$. Set \(m(\pi,v) := \max(|\sigma_1|,|\sigma_2|) = \sigma_1.\) Then $H_2(\delta)$ is the assertion that $m(\pi,v)\le \delta$ for all $\pi,v$, and $\delta_2$ is the infimum of admissible $\delta$.
At an unramified finite place $v$ with residue field size $N(v)$, the representation $\pi_v$ has Satake parameters ${\alpha_v,\beta_v}$ with $|\alpha_v\beta_v|=1$ and \(|\alpha_v| = N(v)^{t_v},\qquad |\beta_v| = N(v)^{-t_v}\) for some $t_v\in[0,\tfrac12]$. In this case $m(\pi,v)=t_v$, so $t_v\le \delta_2$ is equivalent to \(|\alpha_v|,|\beta_v| \le N(v)^{\delta_2}.\) Equivalently, the Hecke eigenvalue $\lambda_v := \alpha_v+\beta_v$ satisfies \(|\lambda_v| \le N(v)^{\delta_2}+N(v)^{-\delta_2}\le 2N(v)^{\delta_2}.\) In the classical language of Hecke–Maass newforms (over $\mathbb Q$), this corresponds to bounds of the shape \(|\lambda_f(n)| \ll_{\varepsilon,f} d(n)\,n^{\delta_2+\varepsilon},\) where $d(n)$ is the divisor function.
Conjecturally (and implied by the Langlands functoriality conjectures), one has $\delta_2=0$.
Known upper bounds
| Bound | Reference | Comments |
|---|---|---|
| $\tfrac12$ | [JS1981] | “Trivial” bound coming from Rankin–Selberg theory / unitarity; holds in general rank as $H_n(\tfrac12)$ [BB2011]. |
| $\tfrac12-\tfrac{1}{2^2+1}=\tfrac{3}{10}=0.3$ | [LRS1999] (ramified extension: [MS2004]) | Specialization of the Luo–Rudnick–Sarnak bound $\delta_n\le \tfrac12-\tfrac{1}{n^2+1}$ to $n=2$; extended to ramified places by Müller–Speh. |
| $\tfrac19\approx 0.11111$ | [KiSh2002] | Bound obtained from functoriality results for low symmetric powers (often stated for unramified places; see discussion in [Sar2005]). |
| $\tfrac{7}{64}=0.109375$ | [KS2003] (over $\mathbb Q$), 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 $\mathrm{GL}2(\mathbb A{\mathbb Q})$ (and Blomer–Brumley extend it to arbitrary number fields). |
Known lower bounds
| Bound | Reference | Comments |
|---|---|---|
| $0$ | Trivial | Conjectured to be sharp (Generalized Ramanujan / Selberg). |
Additional comments and links
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Relation to Selberg’s $\tfrac14$ eigenvalue conjecture. For weight $0$ Maaß cusp forms for congruence subgroups, the archimedean parameter bound $m(\pi,\infty)\le \theta$ implies a spectral gap \(\lambda_1 \ge \tfrac14-\theta^2.\) In particular, $\theta=\tfrac{7}{64}$ yields \(\lambda_1 \ge \tfrac14-\bigl(\tfrac{7}{64}\bigr)^2 = \tfrac{975}{4096}\approx 0.238037\ldots\) see [Sar2005] and also [Li2006] for this numerical value.
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Holomorphic forms. For classical holomorphic cusp forms on $\mathrm{GL}_2/\mathbb Q$ (i.e. $\pi_\infty$ holomorphic discrete series), the full Ramanujan–Petersson conjecture is known ($\delta=0$), by Deligne (and Deligne–Serre in weight 1); see [Sar2005] for a discussion.
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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 $H_n(\delta)$ in [BB2011] and the remarks in [Sar2005].
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Terminology. In analytic number theory, $\delta_2$ is often denoted by $\theta$ and referred to as “the (best known) Ramanujan exponent for $\mathrm{GL}_2$”.
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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).
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[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.
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[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 $\mathrm{GL}_4$ and the symmetric fourth of $\mathrm{GL}_2$. J. Amer. Math. Soc. 16 (2003), no. 1, 139–183. DOI: 10.1090/S0894-0347-02-00410-1.
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[Li2006] Li, Xian-Jin. On exceptional eigenvalues of the Laplacian for $\Gamma_0(N)$. arXiv:math/0610120.
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[LRS1999] Luo, Wenzhi; Rudnick, Zeev; Sarnak, Peter. On the generalized Ramanujan conjecture for $\mathrm{GL}(n)$. 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.
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[MS2004] Müller, Werner; Speh, Birgit. Absolute convergence of the spectral side of the Arthur trace formula for $\mathrm{GL}_n$. Geom. Funct. Anal. 14 (2004), 58–93. DOI: 10.1007/s00039-004-0452-0.
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[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
Contribution notes
Prepared with assistance from ChatGPT 5.2 Pro.