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 $${\pi }_v\simeq \mathrm{Ind}({\tau }_1|\det {|}^{{\sigma }_1}\otimes {\tau }_2|\det {|}^{{\sigma }_2})$$ 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 $$|{\alpha }_v|=N(v{)}^{t_v},|{\beta }_v|=N(v{)}^{-t_v}$$ for some $t_v\in [0,\frac{1}{2}]$. 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 $$|{\lambda }_f(n)|{\ll }_{\epsilon ,f}d(n)n^{{\delta }_2+\epsilon },$$ where $d(n)$ is the divisor function.

Conjecturally (and implied by the Langlands functoriality conjectures), one has ${\delta }_2=0$.

Known upper bounds

Known lower bounds

Additional comments and links

• Relation to Selberg's $\frac{1}{4}$ eigenvalue conjecture. For weight $0$ Maaß cusp forms for congruence subgroups, the archimedean parameter bound $m(\pi ,\infty )\le \theta$ implies a spectral gap $${\lambda }_1\ge \frac{1}{4}-{\theta }^2.$$ In particular, $\theta =\frac{7}{64}$ yields $${\lambda }_1\ge \frac{1}{4}-(\frac{7}{64}{)}^2=\frac{975}{4096}\approx 0.238037\dots$$ see [Sar2005] and also [Li2006] for this numerical value.

• 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.

• 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].

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

• 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 $\delta \le 1/2$.

• [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 ${\mathrm{GL}}_4$ 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.

• [Li2006] Li, Xian-Jin. On exceptional eigenvalues of the Laplacian for ${\Gamma }_0(N)$. arXiv:math/0610120.

• [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.

• [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.

• [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

What counts as progress

• A better upper or lower bound, with a proof or a construction whose value is re-computed by published code (reproducible), ideally with a certificate a deterministic checker can validate.
• A formal proof (Lean) of a known bound, or a precise error in a claimed one.
• New references for the tables above (literature claims).

Source and licence

Imported from the crowdsourced repository of optimization constants (Terence Tao and contributors), commit 2c1968cd520b, Apache License 2.0; reformatted for this page. New records should also be reported there.