{"id":"9c9a58f7-ca74-4a5a-a30a-7213d0885cc4","arxiv_id":"2412.01216","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The numerical spectrum of an endomorphism determines the cohomological spectrum: for every even cohomological degree the spectral radii agree, and polarized endomorphisms have all cohomology eigenvalues of the expected absolute value.","lead":"A mathematics paper proves that for surjective endomorphisms of smooth projective varieties, numerical equivalence classes of cycles control the eigenvalues on all l-adic cohomology groups. If correct, this resolves a 1964 Tate conjecture on polarized endomorphisms and extends Deligne's proof of the Weil conjectures beyond the Frobenius case.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.12 proves the polygon equality only for the perturbed c_m = c + m^{-1}Δ; the return to the original c needs a limit argument that is absent.","rationale":"The proof of the main theorem is a single chain, and the decisive step is the perturbation in Theorem 2.12. Everything downstream, including Corollary 2.16 and the application to Tate's conjecture (d), depends on Theorem 2.12. The reader flagged this same point, and I agree it is the weakest link. I also checked whether the perturbed c_m is genuinely needed: for finite c that is not bi-finite, β_i(c) can vanish, e.g. for the constant correspondence X × {p}; in that case NP_c is only finite after adding εΔ, so the perturbation cannot simply be deleted. The continuity needed to return to c is standard but genuinely absent: spectral radii of A + ε I are continuous in ε, and the concave envelope of finitely many point values is continuous in the extended-real topology, so a one-paragraph lemma would repair the proof. I do not see a deeper obstruction. The undefined q in Proposition 2.17 and the typographically confusing derivative interval (likely '−2d^-CP_c(x)' rather than '−2d−CP_c(x)') are presentation issues, not load-bearing. Therefore my review leaves the reader's conditional verdict unchanged.","tokens_in":13975,"tokens_out":29570,"duration_ms":289146,"concrete_test":"Write out the missing continuity lemma for Theorem 2.12: for ε = m^{-1}, show that ρ(c^* + εΔ^* |_{H^j}) → ρ(c^*|_{H^j}) and ρ(c_* + εΔ_* |_{N^i}) → ρ(c_*|_{N^i}), and that the maps ε ↦ NP_{c+εΔ}, ε ↦ CP_{c+εΔ} converge pointwise in the extended-real sense as ε → 0. Then take the equality NP_{c_m} = CP_{c_m} and pass to the limit. As a numerical check, compute this for X = P^1, c = cl(X × {0}), where β_1(c) = 0: compute β_i(c_m) and α_j(c_m) for m = 1, 2, 4, 8, verify the polygon equality for each m, and confirm the limit gives NP_c = CP_c. If the limiting equality holds, the gap is expository; if it fails, Theorem 2.12 has a counterexample.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim NP_c = CP_c (Theorem 1.14 = Theorem 2.12) is not fully established as written. In the proof of Theorem 2.12, after reducing to k = F_q, the author introduces c_m := c + m^{-1}Δ, proves β_i(c_m) > 0, and states that 'after replacing c by c_m, we may assume β_i(c) > 0'. The displayed argument then proves the polygon equality only for such c_m. No m → ∞ limit or continuity statement is supplied. The theorem is claimed for the original finite c, including cases with β_i(c) = 0, e.g. c = cl(X × {p}) on P^1, which is finite but not bi-finite. Passing from c_m back to c requires that NP_{c_m} and CP_{c_m} converge to NP_c and CP_c. This is plausible: α_j(c_m) and β_i(c_m) are spectral radii of A + ε I with ε = m^{-1} on finite-dimensional spaces, hence continuous at ε = 0, and the least-concave-majorant operation is continuous for finitely many points, even with limit values -∞. But the proof neither states nor proves this, so the original c is unsupported at the single load-bearing step. The gap is likely repairable, but it must be written down.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies, for a smooth projective variety X over a field k and a finite cohomological correspondence c, the relation between the numerical spectral radii β_i(c) on N^i(X)⊗R and the l-adic cohomological spectral radii α_j(c) on H^j(X, Q_l)⊗C. The main theorem (Theorem 1.14 = Theorem 2.12) asserts that the least concave majorants of the sequences log β_i(c) and log α_j(c), called the numerical and cohomological polygons, coincide for every finite correspondence. For a surjective endomorphism f, this yields Corollary 2.16 comparing the even-degree cohomological spectral radii with the numerical ones, and for odd degrees gives two-sided bounds in terms of the numerical polygon. Consequences include: for a q-straight endomorphism (in particular, a q-polarized endomorphism), every eigenvalue of f^* on H^j has modulus q^{j/2}, generalizing Deligne's Weil conjecture and proving a conjecture of Tate; estimates for the number of fixed points of int-amplified endomorphisms; and a 'moving target' fixed-point count. The proof reduces to finite fields, introduces a perturbation c_m = c + m^{-1}Δ, verifies the equality for the perturbed correspondences, and then asserts that one may assume β_i(c)>0. The central derivation otherwise relies on Deligne's theorem for Frobenius, Truong's log-concavity theorem, and standard étale cohomology facts, with no free parameters or circular definitions.","tokens_in":14193,"tokens_out":12425,"duration_ms":108168,"significance":"If the main theorem is correct, it is a substantial advance: it provides a general numerical control of cohomological spectra for finite correspondences in arbitrary characteristic, recovers and generalizes known results over C, and proves a long-standing conjecture of Tate for polarized endomorphisms. The fixed-point counting consequences are also of interest to arithmetic dynamics. The paper is transparent about its reliance on Deligne's Weil conjectures, and the structure of the argument is elegant, using the Frobenius correspondence to rotate the polygon and a perturbation to force positivity of numerical radii. The examples of random products and extensions illustrate the scope of the correspondence formalism. However, the proof of the main theorem as written contains a gap at the perturbation step, and an application statement contains an undefined symbol, so the current version does not fully support all of its claims.","major_comments":[{"comment":"The perturbation argument is not closed. The proof introduces c_m := c + m^{-1}Δ, proves β_i(c_m) > 0, and then states 'after replacing c by c_m, we may assume β_i(c) > 0'. The subsequent argument proves the polygon equality for the perturbed c_m, not for the original c. The theorem is stated for every finite c, including cases with β_i(c) = 0 (e.g., c = cl(X × {p}) on P^1). To conclude NP_c = CP_c, one must supply a limiting or continuity argument showing that NP_{c_m} → NP_c and CP_{c_m} → CP_c as m → ∞. This is plausible because the spectral radii α_j(c_m) and β_i(c_m) are spectral radii of matrices depending continuously on m, and the least concave majorant of finitely many points is continuous in the data; but the proof neither states nor proves this. This gap affects the central claim and must be repaired.","section":"§2.5, Theorem 2.12"},{"comment":"The statement of Proposition 2.17 uses the symbol q in the asymptotic formula '#{f^n(x) = h(x)} = qdn + o(...)' without defining q. In context, q appears to denote β_d(f) or deg(f), but this is not said. In the proof, the leading term is written as μ_d^n with μ_d := β_d(f)/β_{d-1}(f), which is inconsistent with the claimed qdn and with Corollary 1.7, where the leading growth is β_d^n. The notation and the leading term must be corrected and clarified.","section":"§2.6, Proposition 2.17"},{"comment":"The invocation of Lemma 2.9 to show β_i(c_m) ≥ β_i(m^{-1}Δ) is not literally valid: Lemma 2.9 assumes both correspondences are bi-finite, while c_m = c + m^{-1}Δ is only known to be finite, not bi-finite, when c is merely finite. The inequality is still likely true by a direct argument using Lemma 2.8 and the effectivity of c_m^n - (m^{-1}Δ)^n, but this needs to be written out explicitly rather than imported from Lemma 2.9.","section":"§2.5, proof of Theorem 2.12"}],"minor_comments":[{"comment":"There is a typo: 'arbitary' should be 'arbitrary'.","section":"Abstract"},{"comment":"The phrase 'spreading our argument' should be 'spreading out argument'.","section":"§2.5, proof of Theorem 2.12"},{"comment":"There are typos: 'int-amplified ampliefied endmorphism' should be 'int-amplified endomorphism'.","section":"§2.6, Proposition 2.17"},{"comment":"The statement 'After replacing c by c_m, we may assume β_i(c) > 0' is misleading because the proof never returns to the original c; it would be clearer to say that the equality is first proved under the additional positivity assumption and then extended by a limiting argument.","section":"§2.5, proof of Theorem 2.12"},{"comment":"The final bound 'by Corollary 2.16, for every i = 0, ..., 2d−1, we get α_i(f) ≤ (β_{d−1}β_d)^{1/2}' uses the concavity of the numerical polygon for the odd-indexed values, not the literal inequalities of Corollary 2.16 for all i; the reasoning should be stated explicitly.","section":"§2.6, proof of Proposition 2.17"}],"recommendation":"major_revision","confidential_remarks":"The central theorem is significant and the approach is promising, but the perturbation gap in Theorem 2.12 is load-bearing. The fix is likely straightforward via a continuity argument, but it must be written down. The undefined q and the leading-term inconsistency in Proposition 2.17 are also issues that need correction. I recommend major revision rather than rejection, as the core ideas appear sound and the gaps are repairable within the manuscript's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The headline result is Theorem 1.14: for every finite cohomological correspondence c, the numerical polygon NP_c equals the cohomological polygon CP_c. That is a genuinely new statement, and it is stronger than what was known. Truong had the equality of the global maxima, Hu handled abelian varieties, and the complex case was known via Hodge theory. The corollary for q-straight endomorphisms, Corollary 1.5, recovers Deligne's Weil Riemann Hypothesis for every straight endomorphism and proves the 1964 Tate conjecture (d) for polarized endomorphisms. The fixed point asymptotics in Corollary 1.7 and the 'moving target' Proposition 2.17 are nice bonuses. The proof is a derivation from Deligne's theorem, Truong's log-concavity, and standard étale cohomology; there is no circularity or tuning.\n\nNow the soft spots. The stress-test note is right: in the proof of Theorem 2.12, after reducing to a finite field, the author introduces c_m = c + m^{-1}Δ, shows that the perturbed c_m has β_i(c_m)>0, and then says 'after replacing c by c_m, we may assume β_i(c)>0.' The displayed argument proves the polygon equality for the perturbed correspondences, but the theorem is claimed for the original c, which may have β_i(c)=0 for some i (for example c = X×{p} on P^1). No continuity argument as m tends to infinity is supplied. This is a real gap in the written proof. It is probably repairable: the α_j and β_i are spectral radii of finite-dimensional linear maps, and the least concave majorant is continuous in finitely many points, so the equality should pass to the limit. But it has to be written down. Importantly, the gap does not affect the endomorphism applications: a surjective endomorphism is bi-finite and all β_i(f) are positive, so the perturbation step is unnecessary there. Thus the main corollaries, including the Tate conjecture application, are on solid ground once the perturbation step is removed or the continuity is added.\n\nA smaller issue: Proposition 2.17 (and its Introduction twin) uses q^{dn} in the conclusion while the q is never defined; in the int-amplified (non-straight) case the main term should be β_d(f)^n. That is a typo, not a mathematical flaw.\n\nOverall, the central idea is good, the proof is mostly coherent, and the missing continuity argument is a repair rather than a refutation. This paper deserves a serious referee; I would send it to review and would bring it to reading group once the proof is cleaned up.","headline":"A major theorem with one missing continuity argument in the general finite-correspondence statement; the endomorphism corollaries and the Tate conjecture application look solid and deserve refereeing.","tokens_in":14762,"tokens_out":4237,"would_cite":true,"duration_ms":37134,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14C25","14F20","14G15"],"pacs":[],"model":"deepseek-v4-flash","headline":"For any surjective self-map of a smooth projective variety, the growth rates on numerical cycle classes and on l-adic cohomology coincide, forcing all eigenvalues of polarized maps to have the predicted size.","keywords":["numerical equivalence","l-adic cohomology","spectral radius","cohomological correspondences","dynamical degrees","endomorphisms","fixed point counting","polarized endomorphisms"],"falsifier":"Take a concrete finite correspondence $c$ on a smooth projective variety over a finite field and compute the characteristic polynomials of $c^*$ on $N^i(X)\\otimes\\mathbb{R}$ and on $H^{2i}(X_{\\bar k},\\mathbb{Q}_l)$ for some $i$; if the spectral radius on $H^{2i}$ is strictly larger than that on $N^i$, the theorem is false. The random-product example $c=(f_1+f_2)/2$ on projective space, with $f_1,f_2$ of distinct algebraic degrees, is a natural test case since the paper shows its numerical radii are not log-concave.","tokens_in":13721,"feed_emoji":"🔢","tokens_out":16164,"duration_ms":134130,"temperature":0.7,"pith_summary":"This paper establishes that, for a smooth projective variety over any field, the numerical spectrum of a surjective endomorphism controls its cohomological spectrum: the spectral radius of $f^*$ on the $i$-th numerical group $N^i(X)$ equals the spectral radius on the $2i$-th $l$-adic cohomology group, and the analogous equality holds for the smallest eigenvalues. The proof works for a broader class of objects—finite cohomological correspondences, meaning effective dimension-$d$ cycles in $X\\times X$ whose projection to the first factor is finite, viewed through their cohomology classes—and the conclusion is that the numerical polygon and the cohomological polygon, the concave hulls of the log spectral radii, are identical. The payoff is a generalization of the Riemann hypothesis over finite fields: if $f$ is $q$-straight, meaning its numerical radii are $1,q,\\ldots,q^d$, then every eigenvalue of $f^*$ on $H^j$ has modulus $q^{j/2}$, settling a conjecture proposed in 1964. This also yields asymptotic formulas for counting fixed points of iterates and for a moving-target variant. The result matters because it derives purely cohomological information from numerical intersection data, which are easier to compute and are defined over any field.","feed_headline":"Numerical spectra pin down cohomology spectra","feed_subtitle":"For straight self-maps, every eigenvalue on degree-j cohomology has size q^(j/2), proving a 1964 conjecture.","key_machinery":"The argument is carried by the two polygons together with a tilting trick. After a standard reduction to the case of a finite ground field, the proof perturbs a finite correspondence $c$ to $c_m=c+m^{-1}\\Delta$, where $\\Delta$ is the diagonal, so that every numerical spectral radius is positive. It then considers twisted correspondences $c_{s,t}=\\Phi_q^s\\circ c^t$, where $\\Phi_q$ is the correspondence induced by the $q$-Frobenius. Because Frobenius eigenvalues have size $q^{j/2}$, the polygons transform linearly: $NP_{c_{s,t}}(x)=\\frac12 s(\\log q)x+t\\,NP_c(x)$ and $CP_{c_{s,t}}(x)=\\frac12 s(\\log q)x+t\\,CP_c(x)$. Choosing the slope $s/t$ makes a prescribed vertex of $CP_c$ the unique maximum of $CP_{c_{s,t}}$; a trace estimate then identifies the maximum of $CP_{c_{s,t}}$ with the maximum of $NP_{c_{s,t}}$, forcing equality at every vertex of the original polygons. The other load-bearing tool is the formula $\\beta_i(c)=\\lim_{n\\to\\infty}((c^n)_*L^i\\cdot L^{d-i})^{1/n}$, which converts numerical spectra into intersection numbers and enables the pseudo-effective cone estimates that control growth.","core_discovery":"The central discovery is the equality of two polygons attached to a cohomological correspondence $c$ on a smooth projective variety $X$ of dimension $d$. The cohomological polygon $CP_c$ is the minimal concave function on $[0,2d]$ that lies above all numbers $\\log\\alpha_j(c)$, where $\\alpha_j(c)$ is the spectral radius of $c^*$ on $H^j(X_{\\bar k},\\mathbb{Q}_l)$; the numerical polygon $NP_c$ is the minimal concave function lying above $\\log\\beta_i(c)$ at the even points $2i$, where $\\beta_i(c)$ is the spectral radius on the numerical group $N^i(X)\\otimes\\mathbb{R}$. Since cohomological equivalence implies numerical equivalence, one always has $NP_c\\le CP_c$. The theorem proves that for every finite correspondence $c$ the two polygons are equal, $NP_c=CP_c$. For an endomorphism $f$, this gives $\\log\\beta_i(f)=\\log\\alpha_{2i}(f)$ for every $i$, together with the corresponding equality for the minimal eigenvalues. Consequently, if $f$ is $q$-straight—its numerical radii are $1,q,\\ldots,q^d$, as happens for $q$-polarized endomorphisms—then every eigenvalue of $f^*$ on $H^j$ has absolute value $q^{j/2}$ for every $j=0,\\ldots,2d$. The paper reads this as a proof of a 1964 conjecture on algebraic cycles, and it derives from the polygon equality asymptotic formulas for counting fixed points, including a moving-target version.","pith_inferences":["Editorial inference: the polygon equality may hold for correspondences that are only generically finite rather than everywhere finite; the perturbation argument via the diagonal suggests finiteness is a convenience rather than the essential condition, but the paper does not prove this.","Editorial inference: in the random-product example, the theorem predicts that the cohomological spectral radii obey the same non-log-concave pattern forced by the numerical radii; computing explicit examples in small dimension would test how sharp the polygon statement is.","Editorial inference: the methods point toward semisimplicity and equidistribution questions for the action of correspondences on cohomology, but the paper does not address those; a natural next step is to ask whether the polygon equality plus log-concavity implies a Hodge-theoretic or motivic refinement."],"forward_implications":["For a surjective endomorphism of a smooth projective variety over any field, the spectral radii on $N^i$ and on $H^{2i}$ coincide for every $i$, and so do the minimal spectral radii; numerical data determine the cohomological spectrum in even degrees.","If $f$ is $q$-straight, then every eigenvalue of $f^*$ on $H^j$ has modulus $q^{j/2}$ for every $j$, extending the classical Riemann hypothesis over finite fields from Frobenius to arbitrary maps with this numerical growth.","For an int-amplified endomorphism (one with $\\beta_d>\\beta_{d-1}$), the number of fixed points of $f^n$ is $\\beta_d^n+O((\\beta_d\\beta_{d-1})^{n/2})$; in the straight case this is $q^{dn}+O(q^{(d-1/2)n})$.","The moving-target estimate holds: if a sequence of maps $h_n$ grows more slowly than $\\beta_d/\\beta_{d-1}$, then the number of solutions to $f^n(x)=h_n(x)$ is $q^{dn}+o((\\beta_d\\beta_{d-1})^{(1+\\epsilon)n})$."],"supporting_citations":[{"why":"It supplies the bounds q^{j/2} for Frobenius eigenvalues, used to compute the spectra of the twisted correspondences c_{s,t}.","marker":"[Del74]"},{"why":"It supplies the result that characteristic polynomials of cohomological correspondences have rational or integral coefficients, used in Fact 2.1.","marker":"[KM74]"},{"why":"It supplies log-concavity of numerical spectral radii, which shows endomorphisms are numerically log-concave.","marker":"[Tru20]"},{"why":"It supplies the pseudo-effective cone structure and intersection estimates behind the formula for numerical spectral radii.","marker":"[Dan20]"},{"why":"It supplies finiteness of surjective endomorphisms and isolation of periodic points for int-amplified maps.","marker":"[Fak03]"}],"fun_headline_variants":["Numerical spectra equal cohomology spectra","Spectral radii match on numerics and cohomology","Polygon equality: numerical forces cohomology","Polarized endomorphisms have q^(j/2) eigenvalues","Every cohomology eigenvalue pinned by numerics"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the polygon equality survives the perturbation limit $c_m=c+m^{-1}\\Delta\\to c$ and that the established Frobenius eigenvalue bounds over finite fields hold; if either gives way, the equality $NP_c=CP_c$ for the original correspondence is unsupported.","fun_headline_variants_meta":{"raw":{"variants":["Numerical spectra equal cohomology spectra","Spectral radii match on numerics and cohomology","Polygon equality: numerical forces cohomology","Polarized endomorphisms have q^(j/2) eigenvalues","Every cohomology eigenvalue pinned by numerics"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000747,"raw_usage":{"total_tokens":3417,"prompt_tokens":1125,"completion_tokens":2292,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":741,"completion_tokens_details":{"reasoning_tokens":2215}},"tokens_in":741,"tokens_out":2292,"duration_ms":18373,"temperature":1.0,"reasoning_tokens":2215,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T04:39:09.176449+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a concrete finite correspondence $c$ on a smooth projective variety over a finite field and compute the characteristic polynomials of $c^*$ on $N^i(X)\\otimes\\mathbb{R}$ and on $H^{2i}(X_{\\bar k},\\mathbb{Q}_l)$ for some $i$; if the spectral radius on $H^{2i}$ is strictly larger than that on $N^i$, the theorem is false. The random-product example $c=(f_1+f_2)/2$ on projective space, with $f_1,f_2$ of distinct algebraic degrees, is a natural test case since the paper shows its numerical radii are not log-concave.","supporting_citations":[],"review_version":1}