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on $\mathrm{GL}_2$ Ramanujan conjecture exponent ($C_{56}$)

The statement

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 $\delta_2=0$ .

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.