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On the remainder term of the Weyl law for congruence subgroups of Chevalley groups

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Pith's one-line read For simply connected simple Chevalley groups, the cuspidal Weyl law holds with a power-saving remainder term.

desk verdict A serious, likely correct power-saving cuspidal Weyl law for Chevalley-type congruence subgroups; the proof hinges on an unproved extension of a Finis–Matz theorem, and refereeing should focus there. read the letter →

arxiv 1908.06626 v1 pith:NXCXHTJJ submitted 2019-08-19 math.NT math.DGmath.RT

Tobias Finis , Erez Lapid This is my paper · ORCID
classification math.NTmath.DGmath.RT MSC 11F7211F7022E55
keywords WeyllawcuspidalspectrumArthurtraceformulapower-savingremainderChevalleygroupscongruencesubgroupsHeckeoperatorslocallysymmetricspaces
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proves that, for a simply connected simple Chevalley group $G$ over $\mathbb{Q}$ and any congruence subgroup $\Gamma$ of $G(\mathbb{Z})$, the cuspidal spectrum of $X=\Gamma\backslash G(\mathbb{R})/K$ satisfies the Weyl law with a power-saving remainder: the number of cuspidal forms with Laplace eigenvalue at most $T^2$ is $\frac{\mathrm{vol}(X)}{(4\pi)^{d/2}\Gamma(d/2+1)}T^d + O_\Gamma(T^{d-\delta})$ for some $\delta>0$. Earlier work established the leading asymptotic but gave no error term. The paper reaches the remainder by amplifying the non-cuspidal part of Arthur's trace formula with many Hecke operators and showing that it is negligible, rather than by annihilating it. A sympathetic reader would care because power-saving remainders convert a qualitative counting law into a quantitative one, with direct consequences for families of automorphic $L$-functions.

What carries the argument

The engine is a non-archimedean separation lemma (Proposition 3.4). For every prime $p$ and every open $W$-invariant set $U$ in the compact tempered parameter torus $\widehat{T}_0(\mathbb{C})^1$, it produces a self-adjoint Hecke operator $\tau_{U,p}$ with $\tau_{U,p}(e)=0$, $\|\tau_{U,p}\|_1\le B p^A$, $\|\tau_{U,p}\|_2\le B$, support inside $\{x:\|x\|_p\le p^a\}$, and Satake transform at least $1$ on all hermitian parameters outside $U$. Summing such operators over primes $p\equiv 1\pmod N$ gives a test function $\theta$ whose spherical transform is large on every non-cuspidal contribution (proper Levi inductions and residual spectrum), while its norm stays controlled. Positivity of the truncated trace formula then bounds the non-cuspidal spectral measure by the geometric-side estimate of the paper's Theorem 5.1, losing only a power of $T$ via $D(\mu)^{-1/(2A+1)}$.

What would settle it

Read the proof of the cited geometric estimate and test the asserted extension: exhibit a reductive group, a proper parabolic, and a Hecke element $h$ for which $|J_T(f\otimes e_K\otimes h)-v_K h(e)f(e)|$ exceeds the claimed bound. Equivalently, compute the Satake transform of the operators $\tau_{U,p}$ from Proposition 3.4 for $G=\mathrm{SL}(2)$ or $\mathrm{PGL}(2)$ at a non-tempered parameter just outside $U$ and check whether it really stays at least $1$ for all $p$.

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Extended reading notes

Core claim

The paper's central claim is Theorem 1.1: there exists $\delta>0$, depending only on $G$, such that $N_{X,\mathrm{cusp}}(T)=\frac{\mathrm{vol}(X)}{(4\pi)^{d/2}\Gamma(d/2+1)}T^d+O_\Gamma(T^{d-\delta})$ for $T\ge 1$ and every congruence subgroup $\Gamma$ of $G(\mathbb{Z})$, where $X=\Gamma\backslash G(\mathbb{R})/K$. The proof establishes a stronger trace-formula statement (Theorem 5.11): for any open compact subgroup $K$ of $G(\mathbb{A}_{\mathrm{fin}})$ and any bounded $W$-invariant set $D$ in $i\mathfrak{a}_0^*$, the count $m^K_{\mathrm{cusp}}(D)$ of cuspidal representations with archimedean parameter in $D$ differs from $v_K\mu_{\mathrm{pl}}(D)$ by $O_K(\mathrm{vol}(\partial_1 D)(1+\|D\|)^{d-r}+(1+\|D\|)^{d-\delta})$, and for dilating sets with rectifiable boundary this is $O_{K,D}(t^{d-\delta})$. Theorem 5.13 extends the main term to the trace of an arbitrary Hecke operator $\tau$, with error proportional to $\|\tau\|_1(1+\mathrm{ms}(\tau))^r t^{d-\delta}$.

Load-bearing premise

The argument depends on a quoted full-strength geometric-side bound for Arthur's trace formula (Theorem 5.1), which the paper takes from the proof of a cited result that states only a constant-term version; if that stronger bound is not valid for the arbitrary Hecke elements used in Section 5, the power-saving estimates collapse.

Editorial extensions

If this is right

  • For every simply connected simple Chevalley group and every congruence subgroup $\Gamma$, the counting function $N_{X,\mathrm{cusp}}(T)$ has the stated main term with error $O_\Gamma(T^{d-\delta})$, so the cuspidal spectrum is quantitative at Weyl-law scale.
  • The count of cuspidal automorphic forms with archimedean parameter in a dilating family $tD$ is $v_K\mu_{\mathrm{pl}}(tD)+O_{K,D}(t^{d-\delta})$ whenever $\partial D$ is rectifiable.
  • The same bound holds for the trace of an arbitrary Hecke operator $\tau$ on the cuspidal spectrum, with the natural norms of $\tau$ entering the error term.
  • The argument treats all invariant differential operators simultaneously, so the power saving applies to joint spectral parameters, not only the Laplacian.
  • The paper notes these estimates feed into low-lying-zero statistics for families of cuspidal automorphic representations of bounded level spherical at infinity.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • An effective value of the saving exponent $\delta$ is not given; because $\delta$ arises from a Stone–Weierstrass argument, one could try to extract an explicit $\delta$ by quantifying Proposition 3.4, which would be a natural extension.
  • The same amplification should apply to nonsimply connected groups by passing to a finite cover, but the paper only states the simply connected case fully.
  • For groups where the geometric-side estimates are available, the method could yield analogous remainder bounds for Hecke traces and low-lying zeros, which the paper does not work out.
  • If the quoted geometric bound turns out to need modification, the positivity mechanism might still give a weaker but nontrivial remainder by using only the constant-term estimate, a possibility not explored here.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper proves a power-saving remainder term in the Weyl law for the cuspidal spectrum of congruence locally symmetric spaces attached to simply connected, simple Chevalley groups over Q. The proof combines Arthur's trace formula with a new non-archimedean separation lemma (Proposition 3.4), archimedean Paley-Wiener localization, and the authors' earlier geometric-side estimates. The main result, Theorem 1.1 (and its more precise forms Theorem 5.11 and 5.13), asserts that N_{X,cusp}(T) = (vol(X)/(4π)^{d/2}Γ(d/2+1)) T^d + O_Γ(T^{d-δ}) for some δ>0, improving the qualitative Weyl law of Lindenstrauss-Venkatesh.

Significance. If correct, this is a substantial quantitative strengthening of the Lindenstrauss-Venkatesh Weyl law, and the paper also gives a uniform treatment of the full ring of invariant differential operators and a Hecke-equivariant version (Theorem 5.13). The structure is coherent, the positivity argument is elegant, and the separation lemma is elementary but effective. A notable strength is that the paper avoids Arthur's fine spectral expansion and relies only on the basic trace formula, making the argument comparatively accessible. However, the central quantitative estimate is inherited from an external reference in a form that the reference does not explicitly state, and that gap must be closed or documented before the argument is self-contained.

major comments (3)
  1. [§5, Theorem 5.1 and footnote 3] The proof of the main theorem rests on Theorem 5.1, an estimate for the full polynomial J_T(f⊗e_K⊗h), but the cited [13, Theorem 3.7] is stated only for the constant term of that polynomial. The footnote asserts that 'the proof yields the full statement,' yet no derivation is given in this paper. This is load-bearing: Corollary 5.2, Proposition 5.5, Theorem 5.6, and Theorem 5.11 all use the full-strength bound with arbitrary Hecke elements h, including h = θ*θ where θ involves about X/log X Hecke operators and has large L1 norm. If the polynomial coefficients of J_T in T are not controlled by ||h||_1 with only harmless dependence on T and ms(h), the positive power saving δ would not follow. The authors should either prove the extension in an appendix, provide a precise reference to a proof in [13] (with theorem and equation numbers), or state explicitly which weaker input suffices for the subsequent argument.
  2. [§5, Proposition 5.5] The step from the bound for ν_{ncusp}(W B_R(μ)) with R = δ_1/log X to the stated bound for ν_{ncusp}(W B_1(μ)) is not fully justified in the text. Since R can be much smaller than 1, an upper bound on the smaller ball W B_R(μ) does not by itself give an upper bound on the larger ball W B_1(μ); a covering argument introduces a factor of about R^{-r}. Similarly, the removal of the condition d(T) > C_2 log X via the monotonicity (15) should be made explicit, since choosing a larger T can contribute (1+||T||)^r. The claimed final exponent 3r in the log(2+||μ||) factor is plausible, but the text jumps from 'the exponent 2r' to 'the exponent 3r' without spelling out these two mechanisms. This does not threaten the existence of a power saving, since only the value of δ is affected, but it needs to be written precisely.
  3. [§5, Theorem 5.6 and Corollary 5.7] The transition from the spectral-trace estimate in Theorem 5.6 to the counting statement in Corollary 5.7 and then Theorem 5.11 relies on the local bound for ν_{cusp,nt}(W B_k(μ)) from Lemma 5.3. In the k > ‖μ‖/2 regime, the estimate ν_{cusp,nt}(W B_k(μ)) ≪ k^r(k+‖μ‖)^{d-r-1} is used; however, Lemma 5.3 gives this only when the ball W B_1(μ) has d(μ) ≤ 2, and the covering argument passes to centers with possibly small d(λ). The paper does not explicitly verify that the bound extends uniformly to all centers in the covering. I expect this can be fixed by noting that if d(λ) is small, the ball is covered by nearby regular points, but the argument should be stated.
minor comments (5)
  1. [Title] The title contains a typo: 'Weyl LA W' should read 'Weyl LAW'.
  2. [§3, Lemma 3.1] The proof of Lemma 3.1 is extremely brief; a reference for the weak-* limit of the p-adic Plancherel measures, or a one-line justification using Macdonald's formula, would help the reader.
  3. [§4, display (7)] The notation '≪' for β(λ) ≪ β-tilde(λ) is not explicitly declared with respect to which region (presumably λ∈ia_0^*), and this would be clearer if stated.
  4. [§5, Definition 5.8] In the sentence 'Note that for any x∈∂_R A we have B_R(x)⊂∂_{2R} A', the ball should presumably be understood as W B_R(x) or B_R(x) in the quotient; the W-invariance convention is not stated here.
  5. [References] Reference [13] is cited as '2019. arXiv:1905.09078'; it would be useful to indicate whether the full statement asserted in footnote 3 appears in the published version of that paper or only in the arXiv version.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the cuspidal Weyl-law remainder is derived from Arthur's trace formula and an independent geometric-side estimate, not assumed or fitted.

full rationale

The paper's derivation chain is non-circular. It constructs non-archimedean Hecke operators with separation properties (Proposition 3.4), combines Arthur's trace formula with a positivity property, and then applies a geometric-side bound, Theorem 5.1 quoted from [13, Theorem 3.7], to control the total trace of an amplified test function. The target statement, a power-saving Weyl law for the cuspidal spectrum (Theorem 5.11 and Corollary 5.12), is not used as an input anywhere; it emerges as the difference between the spectral and geometric sides. There is no fitted parameter later relabeled as a prediction, no self-referential definition of the main quantity, and no uniqueness theorem imported to force the choice. The only overlapping-author citation is [13] (Finis–Matz), and it is an independent prior estimate on the geometric side of the trace formula with stated assumptions that do not include the cuspidal Weyl law. The footnote to Theorem 5.1, which asserts that the proof of [13] yields the full polynomial statement for arbitrary Hecke elements even though [13] states only the constant-term version, is a real robustness risk and is load-bearing for the final remainder estimate, but it is a missing-proof/correctness concern rather than a circular reduction: no equation of the target result is being quoted back as an input. Accordingly, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The derivation is a chain of deep, previously established theorems rather than an empirical fit. The main external pillars are Arthur's trace formula and the geometric-side bounds of Finis-Matz; both are cited and used, not proved here. No free parameters appear because the theorem is qualitative and the exponents (A, δ) are existential constants.

assumptions (5)
  • domain assumption Arthur's non-invariant trace formula and its truncation theory, including the identity J^T(f)=J_T(f) for large T
    Used throughout Sections 2 and 5 to connect the spectral side to the geometric side; cited to [1,2,4,29] and assumed without proof in this paper.
  • domain assumption Finis-Matz geometric-side bound as stated in Theorem 5.1, including the asserted extension beyond the constant term
    Quoted from [13, Theorem 3.7]; the stronger form used here is asserted to follow from the proof of [13] but is not derived in this paper. It is the main quantitative input for the power saving.
  • domain assumption Wallach's theorem that local components of residual automorphic representations are non-tempered (Lemma 2.1)
    Needed in Proposition 5.5 to make the non-archimedean separation lemma control the residual spectrum; cited to [40] with a brief proof sketch.
  • standard math Satake isomorphism, spherical Paley-Wiener theorem, Harish-Chandra c-function and the p-adic Plancherel density (Macdonald's formula)
    Provides the parameterization of unramified representations, the Plancherel measure, and the bounds used in Sections 3 and 4; standard results cited to [14,15,17,26].
  • domain assumption Strong approximation for simply connected Chevalley groups
    Used in Corollary 5.12 to identify the adelic quotient with the classical quotient Γ\G(R)/K∞; standard arithmetic-geometric input.

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Pith. "Pith review of On the remainder term of the Weyl law for congruence subgroups of Chevalley groups." pith.science (2026). https://pith.science/paper/NXCXHTJJ

@misc{pith2026190806626,
  author       = {Pith},
  title        = {Pith review of: On the remainder term of the Weyl law for congruence subgroups of Chevalley groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NXCXHTJJ}},
  note         = {Machine review of arXiv:1908.06626}
}
abstract

Let $X$ be a locally symmetric space defined by a simple Chevalley group $G$ and a congruence subgroup of $G(\mathbb Q)$. In this generality, the Weyl law for $X$ was proved by Lindenstrauss--Venkatesh. In the case where $G$ is simply connected, we sharpen their result by giving a power saving estimate for the remainder term.

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