{"id":"717b2228-96c0-40cd-bdf4-32ff18550d11","arxiv_id":"1908.06626","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For simply connected simple Chevalley groups over Q, the cuspidal Weyl law for congruence subgroups holds with remainder O(T^{d-δ}) for some δ > 0, where d is the dimension of the symmetric space.","lead":"This paper proves that counts of cuspidal automorphic forms on arithmetic locally symmetric spaces of simple Chevalley type satisfy the Weyl law with a power-saving error term, sharpening the earlier existence theorem of Lindenstrauss and Venkatesh. The proof is a new Hecke-amplification argument inside Arthur's trace formula, and it opens the error-term problem to quantitative applications such as low-lying zeros of L-functions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Power-saving remainder rests on Theorem 5.1, an unproved extension of [13, Thm 3.7] from the constant-term to the full polynomial in T for arbitrary Hecke elements; the footnote asserting 'the proof yields' is the least secure load-bearing point.","rationale":"The paper's central claim is conditional on a powerful geometric-side estimate for Arthur's trace formula with arbitrary Hecke elements. Tracing the logical dependencies gives Theorem 5.1 -> Corollary 5.2 -> Proposition 5.5 -> Theorem 5.6 -> Theorem 5.11 -> Corollary 5.12, whence Theorem 1.1. The internal structure is otherwise coherent: the non-archimedean separation lemma, Proposition 3.4, is an elementary Stone-Weierstrass argument with support and L1 bounds supplied by Lemmas 3.6-3.8; the archimedean localization via Paley-Wiener functions follows Duistermaat-Kolk-Varadarajan; the amplification step in Proposition 5.5 using positivity, Hecke eigenvalue lower bounds, and the choice X = D(µ)^{1/(2A+1)} appears to close correctly. The one point where the paper outsources the core quantitative input is Theorem 5.1, and the footnote explicitly admits that the statement in [13, Theorem 3.7] is weaker, namely the constant-term version. Because the proof needs the full polynomial in T for h = θ*θ and for T with d(T) much larger than log X, the unproved strengthening is not a cosmetic gap: a constant-term-only bound would not control the T-dependent coefficients of the polynomial J_T. Moreover, the asserted uniformity in h, using only ||h||_1 with no support factor, is strong enough to deserve verification rather than an authorial assurance. This is exactly the reader's weakest assumption, and I agree with it. No internal inconsistency or empirical overreach was found; the result is plausible and the proof structure is sound conditional on the external estimate. For these reasons the appropriate verdict remains CONDITIONAL, unchanged from the reader's assessment.","tokens_in":23976,"tokens_out":29370,"duration_ms":302026,"concrete_test":"Independently inspect [13, Theorem 3.7] and its proof. Determine whether it proves a bound for the full polynomial J_T(f ⊗ e_K ⊗ h), i.e., for all coefficients in T, or only for its constant term. If only the constant-term version is proved, re-derive the full statement by differentiating J_T with respect to the coordinates of T and applying the [13] argument to the derivatives; verify that the resulting coefficients are still bounded by ||h||_1 times a constant independent of ms(h) and T. As a quantitative probe, insert h = θ*θ from Proposition 5.5 into Corollary 5.2 and recompute the line 'J_T(F) ≪ B^2 ...' using the constant-term-only version: if an extra factor (1 + ms(h))^{cr} or (1 + ||T||)^{cr} with c > 0 appears, check whether the choice X = D(µ)^{1/(2A+1)} still yields a positive δ in Proposition 5.5.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 5.1, quoted from [13, Theorem 3.7], is the quantitative engine of the paper. In this text it is stated for arbitrary h in H_S and for the full polynomial J_T(f ⊗ e_K ⊗ h), but the footnote in Section 5 concedes that [13] states only the constant-term version and asserts without proof that 'the proof yields the full statement.' Every subsequent estimate, Corollary 5.2, Proposition 5.5, Theorem 5.6, and Theorem 5.11, depends on this stronger form. The proof of Proposition 5.5 applies it with h = θ*θ, where θ is a sum of about X/log X Hecke operators, and at T with d(T) much larger than log X. If [13] only bounds the T-independent part of the polynomial J_T, then the T-dependent coefficients, up to degree r, are uncontrolled; these coefficients are not obviously bounded by ||h||_1 alone, and a missing factor depending on ms(h) or on T could destroy the positive δ in the final bound. Since the authors of the current paper are also authors of [13], this assertion is an insider claim that is not independently checked here. The central theorem would still follow if the extension is valid, but if it fails or requires an additional support factor, the proof of Proposition 5.5 collapses.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":24168,"tokens_out":6511,"duration_ms":63861,"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":[{"comment":"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.","section":"§5, Theorem 5.1 and footnote 3"},{"comment":"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.","section":"§5, Proposition 5.5"},{"comment":"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.","section":"§5, Theorem 5.6 and Corollary 5.7"}],"minor_comments":[{"comment":"The title contains a typo: 'Weyl LA W' should read 'Weyl LAW'.","section":"Title"},{"comment":"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.","section":"§3, Lemma 3.1"},{"comment":"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.","section":"§4, display (7)"},{"comment":"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.","section":"§5, Definition 5.8"},{"comment":"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.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The main unresolved point is the unproved extension of the Finis-Matz geometric-side bound (Theorem 5.1). Since one of the present authors is a coauthor of [13], this is an insider assertion that the refereeing process cannot verify from the submitted text alone. If the extension is valid and documented, the paper is a strong contribution; if not, the central proof collapses. I would encourage the editor to ask for a detailed derivation before acceptance. The other issues (covering factors and exponents) are local and fixable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a substantial paper, not a stunt. The result—power-saving remainder in the cuspidal Weyl law for congruence subgroups of simply connected simple Chevalley groups—genuinely goes beyond Lindenstrauss–Venkatesh, who left the remainder question open. The novelty is real: instead of using a single Hecke operator to kill Eisenstein contributions, the paper amplifies the non-cuspidal part with many operators and then uses positivity in Arthur’s trace formula to show it is negligible. Proposition 3.4 is a nice piece of non-archimedean harmonic analysis; the Stone–Weierstrass separation argument is clean. Theorem 5.13, which handles Hecke twists with an L1 norm and a mild support factor, is a useful strengthening and the advertised applications to low-lying zeros are plausible.\n\nThe proof is assembled from known inputs: Arthur’s basic trace formula, Wallach’s lemma on residual non-temperedness, and the Finis–Matz geometric-side bound. I did not find internal contradictions; the estimates in Lemma 5.3 and Corollary 5.2 are consistent, and the final counting argument in Theorem 5.11 is standard.\n\nThe soft spot is exactly where the reader put it. Theorem 5.1, quoted from [13, Thm 3.7] as a bound on the full polynomial J_T(f ⊗ e_K ⊗ h), is stated in [13] only for the constant term. The footnote says “the proof yields the full statement.” That is an insider claim. It may well be true—the authors are also authors of [13] and are in a position to know—but this paper does not show the argument, and Proposition 5.5, Theorem 5.6, Theorem 5.11, and Theorem 5.13 all lean on it. If the extension fails, or if it requires an extra support factor depending on ms(h) or on T, the power-saving part of Proposition 5.5 collapses. This is a presentation gap, not evidence of an error, but it is load-bearing and the referee should be explicitly asked to check it.\n\nMinor point: the paper makes no attempt to optimize δ; the constant is buried in a Stone–Weierstrass argument. That is fine, but it means the theorem is qualitative. Also, the citation pattern is self-referential but not circular: [13] is a separate paper on Hecke operators and does not contain the Weyl law with remainder.\n\nBottom line: worth a serious referee. I would send it to review and ask for a proof of the extension in Theorem 5.1. I would probably bring it to our reading group; the method is interesting even if the geometric-side gap will need patching.","headline":"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.","tokens_in":24759,"tokens_out":2852,"would_cite":true,"duration_ms":30310,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F72","11F70","22E55"],"pacs":[],"model":"deepseek-v4-flash","headline":"For simply connected simple Chevalley groups, the cuspidal Weyl law holds with a power-saving remainder term.","keywords":["Weyl law","cuspidal spectrum","Arthur trace formula","power-saving remainder","Chevalley groups","congruence subgroups","Hecke operators","locally symmetric spaces"],"falsifier":"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 $p$.","tokens_in":23692,"feed_emoji":"📐","tokens_out":9221,"duration_ms":87177,"temperature":0.7,"pith_summary":"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.","feed_headline":"Power-saving Weyl law for cusp forms on Chevalley groups","feed_subtitle":"Simply connected congruence quotients now come with a power-saving error term, not just a leading asymptotic.","key_machinery":"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 $D(\\mu)^{-1/(2A+1)}$.","core_discovery":"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 $\\|\\tau\\|_1(1+\\mathrm{ms}(\\tau))^r t^{d-\\delta}$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"This reference supplies the geometric-side bound on the trace-formula polynomial $J_T$ (Theorem 5.1) that the whole power-saving argument relies on.","marker":"[13]"},{"why":"This reference establishes the leading-order cuspidal Weyl law that the paper sharpens and whose Hecke-operator idea it extends.","marker":"[25]"},{"why":"This reference supplies the result that all residual spectrum components are non-tempered, which the separation lemma needs to cover residual contributions.","marker":"[40]"},{"why":"This reference establishes the continuity and polynomiality of the geometric side of Arthur's trace formula used to identify and estimate $J_T$.","marker":"[11]"},{"why":"This reference provides the archimedean Paley–Wiener calculus and Plancherel measure computations used to localize spectral parameters and compute main terms.","marker":"[8]"},{"why":"This reference introduces the truncation operator defining the distribution $J^T$, whose positivity and comparison with $J_T$ are essential to the argument.","marker":"[1]"},{"why":"This reference provides Macdonald's formula for the Plancherel density, used to show that the limiting Sato–Tate measure has full support on the tempered torus.","marker":"[26]"}],"fun_headline_variants":["Power-saving Weyl law for cusp forms on congruence quotients","Sharp error term in Weyl law for simply connected Chevalley groups","Power saving for Weyl law remainder on congruence Chevalley quotients","Weyl law remainder improved to power saving for congruence subgroups","Simply connected congruence quotients get power-saving Weyl law"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Power-saving Weyl law for cusp forms on congruence quotients","Sharp error term in Weyl law for simply connected Chevalley groups","Power saving for Weyl law remainder on congruence Chevalley quotients","Weyl law remainder improved to power saving for congruence subgroups","Simply connected congruence quotients get power-saving Weyl law"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000809,"raw_usage":{"total_tokens":3537,"prompt_tokens":917,"completion_tokens":2620,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":533,"completion_tokens_details":{"reasoning_tokens":2530}},"tokens_in":533,"tokens_out":2620,"duration_ms":17824,"temperature":1.0,"reasoning_tokens":2530,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:39:11.206391+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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 $p$.","supporting_citations":[{"cited_title":"On the asymptotics of Hecke operators for reductive groups","cited_arxiv_id":"1905.09078","evidence_quote":"This reference supplies the geometric-side bound on the trace-formula polynomial $J_T$ (Theorem 5.1) that the whole power-saving argument relies on."},{"cited_title":"Lapid, A remark on Eisenstein series , Eisenstein series and applications, 2008, pp","cited_arxiv_id":null,"evidence_quote":"This reference establishes the leading-order cuspidal Weyl law that the paper sharpens and whose Hecke-operator idea it extends."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"This reference supplies the result that all residual spectrum components are non-tempered, which the separation lemma needs to cover residual contributions."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"This reference establishes the continuity and polynomiality of the geometric side of Arthur's trace formula used to identify and estimate $J_T$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"This reference provides the archimedean Paley–Wiener calculus and Plancherel measure computations used to localize spectral parameters and compute main terms."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"This reference introduces the truncation operator defining the distribution $J^T$, whose positivity and comparison with $J_T$ are essential to the argument."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"This reference provides Macdonald's formula for the Plancherel density, used to show that the limiting Sato–Tate measure has full support on the tempered torus."}],"review_version":1}