Generalized Ramanujan conjecture for automorphic representations

Generalized Ramanujan conjecture for automorphic representations

Definitions and mathematical statement

Write a cuspidal automorphic representation as $\pi=\bigotimes'_v\pi_v$. A unitary irreducible local representation is tempered if it is weakly contained in the regular representation; for the groups in question this also has the usual matrix-coefficient characterization modulo the center. The conjecture is \[ \pi\text{ cuspidal on }\operatorname{GL}_n({\mathbb{A}}_F),\quad \omega_\pi\text{ unitary}\quad\Longrightarrow\quad \pi_v\text{ tempered for every place }v. \] At an unramified finite place, in unitary normalization, this asks that all Satake eigenvalues have absolute value one. The target includes ramified and archimedean places as well.

Formulation and concept references: [S1].

Short English statement

Must every local component of a unitary cuspidal automorphic representation satisfy the strongest expected temperedness bound?

Research attempt

Status, normalization, and attack.

This is an unresolved attempt, dated 27 September 2026. It develops an exact growth criterion at an unramified finite place, examines whether positivity of Rankin--Selberg coefficients supplies it, and then checks why the criterion cannot simply be copied to every place in the catalogue statement. All Satake parameters below use the unitary normalization of [S1]. The elementary power-sum argument is proved here; no new automorphic estimate is claimed. The local representation facts used to interpret the criterion are standard and are referenced explicitly.

Fix a cuspidal $\pi$ as in the statement and an unramified finite place $v$. Write $q=\#(\mathcal O_F/v)$ and let $\alpha_1,\ldots,\alpha_n\ne0$ be the Satake parameters. Unitarity of the central character gives \[ \left|\prod_{j=1}^n\alpha_j\right|=1. \tag{30.1} \] Set $p_m=\sum_j\alpha_j^m$ and $R=\max_j|\alpha_j|$. The target here is $R=1$: together with $30.1$, the upper bound $R\leq1$ forces every $|\alpha_j|=1$.

1. Recovering the largest parameter from traces.

For an arbitrary finite nonzero complex multiset, one has \[ \limsup_{m\to\infty}|p_m|^{1/m}=R. \tag{30.2} \] The upper bound follows from $|p_m|\leq nR^m$. For the lower bound use the generating function, initially for $|z|<1/R$, \[ F(z)=\sum_{m\geq1}p_mz^m =\sum_{j=1}^n\frac{\alpha_jz}{1-\alpha_jz}. \tag{30.3} \] Group together equal parameters. A distinct value $\beta$ of multiplicity $r_\beta$ contributes $r_\beta\beta z/(1-\beta z)$, with a genuine pole at $z=1/\beta$. All terms belonging to other distinct values are analytic there, and $r_\beta$ is a positive integer. Thus no pole cancels. The nearest pole has modulus $1/R$, which is the Taylor series radius of convergence. The root test for power series proves $30.2$.

This is stronger than examining a few traces: cancellations can be extensive without changing the limiting growth. Equivalently, unramified temperedness holds precisely when \[ \text{for every }\varepsilon>0\text{ there exists }C_{\varepsilon,v} \text{ such that }|p_m|\leq C_{\varepsilon,v}q^{\varepsilon m} \quad(m\geq1). \tag{30.4} \] The constants may depend on this representation and place. An estimate with a single fixed positive exponent is insufficient. When temperedness holds, the sharper bound $|p_m|\leq n$ is immediate; the point of $30.4$ is that it would suffice to prove only subexponential growth.

2. The positive local Rankin--Selberg criterion.

Because $\pi_v$ is unitary, the Satake multiset of its contragredient agrees with the complex conjugate multiset. Consequently the unramified factor is \[ L_v(s,\pi\times\widetilde\pi) =\prod_{i,j}(1-\alpha_i\overline{\alpha_j}q^{-s})^{-1}, \] and in a right half-plane its logarithm has the absolutely convergent expansion \[ \log L_v(s,\pi\times\widetilde\pi) =\sum_{m\geq1}\frac{|p_m|^2}{m}q^{-ms}. \tag{30.5} \] In particular every displayed coefficient is nonnegative. From $30.2$, \[ \limsup_m\left(\frac{|p_m|^2}{m}\right)^{1/m}=R^2. \] It follows that temperedness is equivalent to the family of finite-energy conditions \[ \mathcal E_v(\sigma):= \sum_{m\geq1}\frac{|p_m|^2}{m}q^{-m\sigma}<\infty \qquad\text{for every real }\sigma>0. \tag{30.6} \] Indeed convergence for every $\sigma>0$ implies $R^2q^{-\sigma}\leq1$ for each $\sigma$, hence $R\leq1$; the converse follows from $|p_m|\leq n$. Under $30.1$, $R\geq1$, and the abscissa of convergence of this local positive series is exactly \[ \sigma_v=2\log_q R. \tag{30.7} \] No assertion about convergence on the boundary is needed.

Positivity is useful because there is no cancellation between the terms in $30.6$. It also reveals the limitation of a global argument. Even if a global positive majorant supplies convergence of this local series for every $\sigma>1$, the root test only yields $R\leq q^{1/2}$. The desired threshold is zero, not one. Sarnak's notes [S1] describe the stronger known general bound $|\log_q|\alpha_j||\leq1/2-1/(n^2+1)$ and its origins. That positive exponent is useful information but is not $30.4$. This reference is not an assertion of an exhaustive survey of later improvements in special families.

Nor can meromorphic continuation of a global $L$-function be substituted for convergence of its positive coefficient series. Analytic continuation changes the domain on which a function is defined; it does not change the radius of convergence of a particular series. For example $\sum_{m\geq1}R^{2m}z^m/m$ has its first singularity at $z=R^{-2}$ even though its expression $-\log(1-R^2z)$ can be continued along many paths beyond that circle. An argument using a continued value in place of the sum would lose the positivity premise exactly where it is needed.

3. Stress tests for the proposed positivity attack.

Consider the two parameters $(q^t,q^{-t})$, where $0<t<1/2$. They satisfy the determinant condition and are the parameters of a spherical unitary complementary-series representation of $\operatorname{GL}_2(F_v)$. The existence of such local representations is part of the local unitary picture discussed in [S1]. Here \[ p_m=q^{tm}+q^{-tm},\qquad |p_m|^2=q^{2tm}+2+q^{-2tm}, \] so all the required positivity holds but $\sigma_v=2t>0$. The corresponding local factor is \[ \frac{1}{(1-q^{2t-s})(1-q^{-s})^2(1-q^{-2t-s})}. \tag{30.8} \] Thus local unitarity, unitary determinant, and nonnegative logarithmic coefficients do not prove temperedness. These are local test objects; their occurrence in a cuspidal global representation is not being asserted. Excluding that occurrence requires a genuinely global restriction.

A separate algebraic example tests cancellation: for $R>1$ take $(R,-R,R^{-1},-R^{-1})$. Its product is one, all odd traces vanish, and \[ p_{2k}=2(R^{2k}+R^{-2k}). \tag{30.9} \] Half the traces therefore look as small as possible while $30.2$ detects the full growth. More generally multiplying both $R$ and $R^{-1}$ by every $M$th root of unity makes all traces of orders $1,\ldots,M-1$ vanish. The product of the resulting $2M$ parameters is one. This last family changes the dimension with $M$ and is only a multiset example, not a claimed family of automorphic representations. Its purpose is to show why finite trace inspection cannot replace a quantified growth theorem.

4. Ramification requires the representation, not a copied formula.

At a ramified finite place the standard local $L$-factor sees only the inertia-invariant part of the monodromy kernel of a Weil--Deligne parameter. It is not generally a degree-$n$ polynomial with all roots of modulus one even when the representation is tempered. In the convention $|\operatorname{Frob}_v|=q^{-1}$, consider the unitary Steinberg representation of $\operatorname{GL}_2(F_v)$. Its Weil--Deligne data can be written \[ r(\operatorname{Frob}_v)= \begin{pmatrix}q^{-1/2}&0\\0&q^{1/2}\end{pmatrix},\qquad Ne_2=e_1,\quad Ne_1=0, \] with trivial inertia. On $\ker N=\mathbb Ce_1$ the eigenvalue is $q^{-1/2}$, so \[ L_v(s,\mathrm{St}_2)=(1-q^{-s-1/2})^{-1}. \tag{30.10} \] Nevertheless this representation is square integrable modulo its center, hence tempered. The parameter normalization and the square-integrability criterion are explained in [E1]. The matrix calculation verifies explicitly why blindly imposing modulus one on the single standard-factor parameter would reject a valid tempered representation.

For full local scope one may instead use the Langlands classification: write the irreducible representation as the Langlands quotient of normalized induction from tempered blocks $\tau_j|\det|^{t_j}$, with decreasing real exponents. Temperedness requires all $t_j=0$ in this classification [E1]. That is a meaningful ramified target, but the proof of $30.2$ for spherical Satake traces has not bounded these block exponents. Additional twisted local factors or matrix-coefficient estimates would need separate arguments.

At archimedean places there is no residue cardinality $q$ and no sequence of Frobenius powers to which $30.6$ applies. Even the spherical real $\operatorname{GL}_2$ case contains complementary parameters $\pm t$; in the usual weight-zero Laplace normalization they correspond to $1/4-t^2$, whereas tempered principal-series parameters have zero real part. The general statement also includes nonspherical archimedean representations. A proof at all unramified finite places alone does not prove the conjecture as stated.

Verification and outcome.

The local companion script checks $30.9$ exactly for $R=2$ through order twelve. It also multiplies formal power series for $30.8$ at $q=16,t=1/4$: the logarithmic-derivative coefficients agree with $(2^m+2^{-m})^2$ through order twelve, and its coefficients are nonnegative. These checks verify normalization and bookkeeping, not automorphic realizability. The Steinberg matrices satisfy $r(\operatorname{Frob})Nr(\operatorname{Frob})^{-1}=q^{-1}N$, and restricting to their monodromy kernel gives $30.10$. The limiting statements are proved by the rational generating function, not inferred from those finite tests.

The resulting conditional route is precise: establish $30.6$ for every unramified place, then separately force the Langlands block exponents to vanish at all remaining places. No estimate doing the first task is proved here, and the second task has not been reduced to the first. Positivity, global continuation, and determinant normalization each survive explicit tests that fail temperedness. The all-place generalized Ramanujan conjecture therefore remains UNRESOLVED.

Sources

• S1 Peter Sarnak, Notes on the Generalized Ramanujan Conjectures, in Harmonic Analysis, the Trace Formula, and Shimura Varieties. https://web.math.princeton.edu/~sarnak/Preprints/SarnakFieldsNotes.pdf. Formulation source.
• S2 A note on Sarnak's density hypothesis for Sp4. https://ems.press/content/serial-article-files/53056. Further reference; not a proof endorsement.
• S3 Regularized spectral expansion of a Rankin-Selberg period and moments. https://arxiv.org/abs/2609.01358. Further reference; not a proof endorsement.

• E1 Torsten Wedhorn, The local Langlands correspondence for $\operatorname{GL}(n)$ over $p$-adic fields, arXiv:math/0011210v2 (25 November 2000), Sections 2.3, 3.2, and 4.3. https://arxiv.org/pdf/math/0011210v2. Author exposition of the local classification, temperedness, Weil--Deligne factors, and the unitary Steinberg normalization. These are established inputs, not new claims of this attempt. Consulted 27 September 2026.

Generalized Ramanujan conjecture for automorphic representations

Definitions and mathematical statement

Write a cuspidal automorphic representation as $\pi=\bigotimes'_v\pi_v$. A unitary irreducible local representation is tempered if it is weakly contained in the regular representation; for the groups in question this also has the usual matrix-coefficient characterization modulo the center. The conjecture is \[ \pi\text{ cuspidal on }\operatorname{GL}_n({\mathbb{A}}_F),\quad \omega_\pi\text{ unitary}\quad\Longrightarrow\quad \pi_v\text{ tempered for every place }v. \] At an unramified finite place, in unitary normalization, this asks that all Satake eigenvalues have absolute value one. The target includes ramified and archimedean places as well.

Formulation and concept references: [S1].

Short English statement

Must every local component of a unitary cuspidal automorphic representation satisfy the strongest expected temperedness bound?

Research attempt

Current frontier.

The target concerns every number field, every cuspidal representation with unitary central character, and every place, including ramified and archimedean ones. Sarnak's formulation and distinction between local unitarity bounds and temperedness are retained. [S1], Section 1. Two substantial recent results apply to special arithmetic classes: Boxer--Calegari--Gee--Newton--Thorne prove Ramanujan and Sato--Tate for regular algebraic cuspidal $\operatorname{GL}_2$ representations of parallel weight over CM fields, and Newton--Thorne establish all symmetric-power liftings for regular-weight Hilbert modular forms. The former appeared in 2025 and the latter in the 2026 Annals volume. [E1], [E2]. These do not cover arbitrary nonalgebraic Maass representations or arbitrary rank. The catalogue's $\operatorname{Sp}_4$ density reference concerns a different group; no pointwise $\operatorname{GL}_n$ theorem is inferred from it. Recent publication records and stated scopes were checked, without claiming a new independent audit of the long automorphy proofs.

Attack route.

At an unramified finite place, repeated tensor or symmetric powers magnify any Satake eigenvalue of modulus greater than one. If appropriate powers were automorphic and locally compatible, even a coarse uniform-in-rank bound would rule out that eigenvalue. This is a classical motivation for functoriality, not a newly discovered principle. [S1], Section 1. The attempt below makes the precise amplification and cancellation tests explicit. A second route uses all power traces of one local parameter; its missing step is a subexponential-in-the-power estimate at that fixed place. Neither route is allowed to discard ramified or infinite places while claiming the full target.

Attempt.

Fix $\pi$ and an unramified finite place $v$ of residue cardinality $q$. In the source's unitary normalization let $\alpha_1,\ldots,\alpha_n\ne0$ be the Satake eigenvalues. Unitary central character gives $\prod_i|\alpha_i|=1$. Set \[ a_r=\sum_{i=1}^n\alpha_i^r,\qquad R=\max_i|\alpha_i|. \]

Lemma 30.1 (cancellation does not hide the spectral radius forever). One has \begin{equation}\label{wsref-0-72-33-30-3a-72-61-64-69-75-73} \limsup_{r\to\infty}|a_r|^{1/r}=R. \end{equation} Proof. The upper bound follows from $|a_r|\le nR^r$. Near zero the absolutely convergent generating function is \[ \sum_{r\ge1}a_r z^{r-1}=\sum_{i=1}^n\frac{\alpha_i}{1-\alpha_i z}. \] Group equal eigenvalues. A distinct eigenvalue $\alpha$ of multiplicity $m$ contributes $m\alpha/(1-\alpha z)$, with nonzero residue $-m$ at $z=\alpha^{-1}$. All other grouped terms are regular there, so none of these poles cancels. The nearest pole to zero has modulus $1/R$. The power-series radius of convergence, or directly the root test and analyticity inside that radius, proves (r30:radius). The shift from $r-1$ to $r$ in the root limit does not change it.$\square$

It follows that the following fixed-place estimate would imply temperedness at this $v$: \begin{equation}\label{wsref-0-72-33-30-3a-70-6f-77-65-72-62-6f-75-6e-64} \text{for every }\varepsilon>0\text{ there is }C_{\pi,v,\varepsilon} \text{ such that }|a_r|\le C_{\pi,v,\varepsilon}q^{\varepsilon r} \quad(r\ge1). \end{equation} Indeed $R\le q^\varepsilon$ for every $\varepsilon>0$, hence $R\le1$; the determinant condition then forces $|\alpha_i|=1$ for all $i$. The constants may depend arbitrarily on $\pi,v,\varepsilon$, but not on $r$. A bound only for one fixed $r$, or one whose constant secretly grows exponentially in $r$, is insufficient.

There is also a direct link with the local logarithmic derivative: \[ L_v(s,\pi)=\prod_i(1-\alpha_iq^{-s})^{-1},\qquad -\frac{L_v'(s,\pi)}{L_v(s,\pi)} =(\log q)\sum_{r\ge1}a_rq^{-rs}, \] where the expansion is initially justified by $q^{-\Re s}R<1$. Extending this expansion into a larger half-plane without proof would assume the bound being sought. Holomorphy of a completed global product does not automatically make each individual Euler factor pole-free in that region, because other factors can compensate.

Lemma 30.2 (a precise conditional amplification). Suppose an unbounded set of positive integers $m$ has the following property: there exists an automorphic representation $\Pi_m$ on $\operatorname{GL}_{\binom{n+m-1}{m}}$, isobaric as a sum of unitary cuspidal representations, unramified at this particular $v$, whose Satake multiset is exactly that of $\operatorname{Sym}^m(\alpha(\pi_v))$. Then $\pi_v$ is tempered.

Proof. For an unramified unitary generic cuspidal component, the coarse estimate $|\beta|<q^{1/2}$ is available independently of its rank; for a $\operatorname{GL}_1$ unitary character it is immediate. Sarnak records this local bound in equation $12$, before the stronger Luo--Rudnick--Sarnak estimate in equation $14$. [S1], pp. 7--8. Taking the union of the Satake multisets of the isobaric summands gives $|\beta|\le q^{1/2}$ for every eigenvalue of $\Pi_{m,v}$. In the symmetric power, $\alpha_i^m$ is one of those eigenvalues. Consequently $|\alpha_i|\le q^{1/(2m)}$ for an unbounded sequence of $m$. Letting $m\to\infty$ and again using the determinant proves the claim. No growth control on the degree of $\Pi_m$ is needed for this limiting argument; the uniform exponent $1/2$ is what matters.$\square$

One must retain the unitary-isobaric hypothesis. General automorphic representations may involve nonunitary twists in residual or induced constructions, and the same estimate cannot simply be assigned to them. Similarly, potential automorphy after a field extension has to be linked back to the original local parameter; it is not identical to the hypothesis as written. The cited recent symmetric-power results provide such deep additional arithmetic structure in their own settings, not a general proof of the hypothesis for arbitrary $\pi$. [E1], [E2].

Stress-testing low-moment and almost-everywhere substitutes. For any real $R>1$, the multiset \[ \{R,-R,R^{-1},-R^{-1}\} \] has determinant one and first power trace zero, yet is not tempered. All its odd power traces vanish, while $a_{2r}=2(R^{2r}+R^{-2r})$. Thus a cancellation argument based on the first Hecke trace, or only odd powers, cannot establish (r30:powerbound). The companion script checks these identities at rational $R$; the displayed formula already proves them for all $R>1$. This is a model parameter, not a claim that an automorphic cuspidal representation with this parameter exists.

The weak functoriality statement in catalogue UnsolvedMath problem 30006675 matches local parameters only outside a finite exceptional set $S_m$ that can depend on the transfer. It does not imply that a fixed $v$ avoids $S_m$ for unboundedly many $m$. For example, if finite places are enumerated as $v_1,v_2,\ldots$, the sets $S_m=\{v_1,\ldots,v_m\}$ are all finite but every fixed place belongs to all sufficiently large $S_m$. Therefore even proving exactly that weak statement would not, by this argument alone, establish the local compatibility in Lemma 30.2. Additional strong transfer or another way to remove the exceptional places is necessary. This is a quantifier obstruction, not a counterexample to functoriality.

Stress test.

The reduction above is deliberately restricted to unramified finite places. There is no single Satake multiset in the same sense for an arbitrary ramified component. Temperedness in Weil--Deligne or Weil--$\operatorname{SL}_2$ language has to retain the monodromy data and normalization; replacing it by unit modulus of every raw Frobenius eigenvalue would be incorrect. At archimedean places the appropriate condition is on real parts of Langlands exponents, not on a finite-place Euler polynomial. The source explains these separate local conventions. [S1], equations $9$--$10$ and the following paragraph.

Averaged estimates over primes do not bound the entire power sequence at a fixed prime. Nor is the subexponential estimate supplied by the unramified bound $|\alpha_i|\le q^{1/2-1/(n^2+1)}$: that gives a fixed positive exponential rate in $r$, precisely the rate amplification is trying to remove. The logarithmic-derivative manipulation is used only in its absolutely convergent region. No unproved interchange or analytic continuation of its Dirichlet series is inserted.

\Needspace{8\baselineskip}

Outcome.

Outcome: Exploratory approach with unresolved gap.

The notebook proves an exact spectral-radius criterion and a conditional symmetric-power amplification at each specified unramified place. It identifies the fixed-place compatibility that weak almost-everywhere functoriality does not provide, and gives an explicit low-moment cancellation obstruction. These are deductions explaining a route and its failure points; their historical novelty is not asserted. Neither the required general family of locally compatible lifts nor the all-place extension has been obtained. Thus the exact universal temperedness statement, especially its ramified and archimedean components, remains unresolved by this attempt.

Sources

• S1 Peter Sarnak, Notes on the Generalized Ramanujan Conjectures, in Harmonic Analysis, the Trace Formula, and Shimura Varieties. https://web.math.princeton.edu/~sarnak/Preprints/SarnakFieldsNotes.pdf. Catalogue formulation source.
• S2 A note on Sarnak's density hypothesis for Sp4. https://ems.press/content/serial-article-files/53056. Catalogue research/status reference; not a proof endorsement.
• S3 Regularized spectral expansion of a Rankin-Selberg period and moments. https://arxiv.org/abs/2609.01358. Catalogue research/status reference; not a proof endorsement.
• E1 George Boxer, Frank Calegari, Toby Gee, James Newton and Jack A. Thorne, The Ramanujan and Sato--Tate Conjectures for Bianchi modular forms, Forum of Mathematics, Pi 13 $2025$, e10, published 21 February 2025. https://doi.org/10.1017/fmp.2024.29. https://www.repository.cam.ac.uk/items/45e39d5a-9515-4073-819e-8962395a7ece. Abstract and theorem scope consulted 27 September 2026; regular algebraic parallel-weight CM case, not arbitrary cuspidal forms.
• E2 James Newton and Jack A. Thorne, Symmetric power functoriality for Hilbert modular forms, Annals of Mathematics 203 $2026$, no. 1, 283--347. https://annals.math.princeton.edu/2026/203-1/p04. Published scope and metadata consulted 27 September 2026. Used for the regular-weight Hilbert modular setting only.