{"id":"183f9cd0-aa2c-4ec4-8ac4-5e45e7dcb6fc","arxiv_id":"2502.04752","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Equivariant primitives of Eisenstein series for principal congruence subgroups are shown to equal the corresponding non-holomorphic Eisenstein series, including new weight-two cases expressed via single-valued logarithms.","lead":"This paper shows that the natural real-analytic integrals (equivariant primitives) of Eisenstein series for congruence subgroups are exactly the known non-holomorphic Eisenstein series. This gives number theorists and physicists explicit formulas for functions that appear in scattering amplitudes and in the theory of modular forms.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Factor-of-two inconsistency in §6 breaks the printed proof of Theorem 8: eq (6.22) is incompatible with eqs. (6.15), (6.25), and (6.26).","rationale":"The reader's flagged assumption, the unproved functional equation (3.12), is not the most serious issue: eq. (3.12) is a standard consequence of the S-transformation of the completed L-function, and substituting τ ↦ −1/τ in the defining integral reproduces the stated sign and the replacement v ↦ vS. A dedicated check of the N = 3 exceptional case matches the formula, so I do not regard this as a load-bearing defect. The genuine problem is in §6, where the proof of the genus-zero single-valued-logarithm formula contains an internal factor-of-two inconsistency. Eq. (6.22) cannot be correct as printed if eqs. (6.15), (6.25), and (6.26) are also correct; the four statements cannot all hold simultaneously. Since Theorem 8 is one of the two advertised headline results, the paper should not be accepted as-is without this discrepancy being resolved. The mismatch is localized and appears fixable, so conditional acceptance is the appropriate verdict rather than rejection. Theorems 5–7 and the central identification of equivariant primitives with non-holomorphic Eisenstein series seem well supported and are not called into question by this concern.","tokens_in":22220,"tokens_out":35222,"duration_ms":311923,"concrete_test":"Re-derive eq. (6.22) from the Fourier expansion near i∞. Compute Gv∞_2(τ) = Re∫ gv∞_2(z) and show directly that Gv∞_2(τ) = 2π·ζ₂¹·Im τ + O(q_N), with ζ₂¹ = ∑_{n≡1 mod N, n≠0} n⁻² = π²/(N² sin²(π/N)). Then use log|x(τ)|² = −4π·Im τ/N to determine the coefficient in front of log|x(τ)|². If the coefficient is −π²/(2N sin²(π/N)), eq. (6.22) needs the missing factor 1/2 and eq. (6.26) is consistent; if the coefficient is −π²/(N sin²(π/N)), then eq. (6.15) and Theorem 7 must be re-examined. As an additional numerical check, evaluate both sides of eq. (6.15) for N = 3 using the expanded Hauptmodul to verify the constants.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The most load-bearing weak point is in the proof of the advertised genus-zero, weight-two result. In §6, eq. (6.22) claims Gv∞_2(τ) = −π²/(N sin²(π/N))·log|x(τ)|². But Theorem 7 gives Gv∞_2 = −2π·gv∞_{0,0}, and eq. (6.15) gives gv∞_{0,0} = π/(4N sin²(π/N))·log|x(τ)|². Combining these yields Gv∞_2 = −π²/(2N sin²(π/N))·log|x(τ)|², a factor of 2 smaller than eq. (6.22). The same factor of 2 appears inside the proof: eq. (6.25) obtains Gv∞_2 ~ 2π·a₀(G^{v₀}_2)·Im τ, while substituting log|x|² ~ −4π·Im τ/N into eq. (6.22) would give 4π·a₀(G^{v₀}_2)·Im τ. The displayed identity in eq. (6.26) is also off by the same factor: its right-hand side is −2π·a₀(G^{v₀}_2)·Im τ, whereas the left-hand side with the printed coefficient is −4π·a₀(G^{v₀}_2)·Im τ. Thus the written proof of Theorem 8 is internally inconsistent. The final formulas (6.15) appear to be salvageable with the corrected factor, and the main equivariant-primitive theorems (5–7) are not affected, but as printed the genus-zero single-valued-logarithm claim is not established by the given argument.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper develops the theory of equivariant primitives of Eisenstein series for principal congruence subgroups Gamma(N). The authors compute the L-series of the Eisenstein series H^v_k in closed form (Theorem 1), evaluate the Eisenstein cocycles on the generators S and T (Theorem 3), and show that the real part of the cocycle is a coboundary (Theorem 4). This allows them to construct an equivariant primitive G^v_k for each Eisenstein series of weight k >= 3, and to prove that its polynomial coefficients are exactly the non-holomorphic Eisenstein series (Theorems 5 and 6). For weight two, Theorem 7 identifies the equivariant primitive with -2 pi times a difference of non-holomorphic Eisenstein series of weight (0,0). Finally, in genus-zero cases, Theorem 8 expresses those weight-two series as single-valued logarithms of rational functions of the Hauptmodul. The paper is clearly written and the main derivations in Sections 5 and 6 are detailed, but the printed proof of Theorem 8 contains a localized factor-of-two inconsistency.","tokens_in":22546,"tokens_out":19544,"duration_ms":162376,"significance":"If correct, the paper gives a complete and explicit description of equivariant primitives of Eisenstein series for all principal congruence subgroups, extending Brown's results for SL2(Z). The derivations are parameter-free and derived from definitions, and the final formulas are concrete and checkable. Theorems 5-7 provide a clean identification with non-holomorphic Eisenstein series, and Theorem 8 offers a single-valued-logarithm representation in genus-zero cases that will be useful in both number theory and physics. The main caveat is the factor-of-two error in the proof of Theorem 8; however, it is localized and correctable, and it does not affect the central equivariant-primitive theorems. Overall, this is a solid contribution that advances the study of equivariant iterated Eisenstein integrals for congruence subgroups.","major_comments":[{"comment":"The coefficient in eq. (6.22) is a factor of 2 too large. Combining Theorem 7 (G^v_2 = -2 pi g^v_{0,0}) with the v_infinity case of eq. (6.15), namely g^{v_infinity}_{0,0} = [pi/(4N sin^2(pi/N))] log|x(tau)|^2, gives G^{v_infinity}_2 = -[pi^2/(2N sin^2(pi/N))] log|x(tau)|^2, not the coefficient printed in (6.22). The asymptotic analysis in (6.25)-(6.26) also selects the halved coefficient: (6.25) gives G^{v_infinity}_2 ~ 2 pi a_0(G^{v_0}_2) Im tau, while log|x(tau)|^2 ~ -4 pi Im tau/N, so with the halved coefficient the two terms in (6.24) cancel, whereas with the printed coefficient eq. (6.26) would equate -4 pi a_0 Im tau with -2 pi a_0 Im tau. Note also that the factor N/2 a_0(G^{v_0}_2) in (6.26) equals pi^2/(2N sin^2(pi/N)), so the left-hand side of (6.26) should read pi^2/(2N sin^2(pi/N)) log|x(tau)|^2. As printed, the proof of Theorem 8 is internally inconsistent, although the stated formulas in (6.15) appear to be correct after this factor is fixed.","section":"Section 6, eqs. (6.15), (6.22), (6.25)-(6.26)"}],"minor_comments":[{"comment":"The proof of Theorem 1 is only given for k > 3; the case k = 2 is dismissed with the sentence 'The case k = 2 is similar.' Since Theorem 1 is used for weight-two Eisenstein cocycles, the missing case should be either proved or explicitly referenced.","section":"Section 3, Theorem 1"},{"comment":"The functional equation Lambda(xi^v_k, l) = (-1)^l Lambda(xi^{vS}_k, k-l) is stated without proof or citation. It is used in the proof of Theorem 1 for the exceptional case (b,l) = (0,1), and therefore it underpins the cocycle values in Theorem 3. Please add a proof or a standard reference.","section":"Section 3, eq. (3.12)"},{"comment":"The summation in eq. (2.16) is written as 'sum_{v in Z_N} a_v G^v_2', but the vectors v are elements of Z_N^2, not Z_N. The range of summation should be v in Z_N^2, or the notation should be clarified.","section":"Section 2.2, eq. (2.16)"},{"comment":"The non-holomorphic Eisenstein series G^v_{r,s} are defined for 'r, s positive integers', but Theorem 6 uses indices with r = 0 or s = 0, since r + s = k - 2. The definition should be extended to non-negative integers with r + s > 0, with the convergence behavior stated for that range.","section":"Section 5, eq. (5.6) and Theorem 6"},{"comment":"There are a few typographical and notation issues: 'Hautmodul' in the Introduction should be 'Hauptmodul'; 'bahaviour' in the proof of Theorem 7 should be 'behaviour'; and the visual similarity between the holomorphic Eisenstein series G^v_k and the equivariant primitive G^v_k might be confusing, so a clear declaration of the font/notation difference would help.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"This is a well-written paper with correct main ideas, but the proof of Theorem 8 has a localized factor-of-two error that must be fixed. The missing justifications for the k = 2 case in Theorem 1 and for the functional equation (3.12) should also be addressed. I recommend major revision rather than rejection, since the error is easily correctable and the central equivariant-primitive theorems are unaffected."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the main result is good, and accepting it is about right, but the printed proof of Theorem 8 has a real factor-of-two inconsistency that must be corrected.\n\nWhat is new: the paper gives the first complete description of equivariant primitives of Eisenstein series for principal congruence subgroups, proving they are precisely the non-holomorphic Eisenstein series. The L-series evaluation in terms of Clausen values and the cocycle computation are solid. Extending Brown's theorem from SL2(Z) and Drewitt–Pimm's from Γ0(N) to all Γ(N) is a natural but non-trivial step, and the appearance of odd weight and weight two is genuinely new. The core argument in Section 5 (Theorems 5–7) is clean; I did not find circularity or fitted parameters.\n\nThe soft spot is real. In Section 6, the proof of Theorem 8 is internally inconsistent. Eq. (6.22) claims G^{v∞}_2 = −π²/(N sin²(π/N)) log|x|², but Theorem 7 plus eq. (6.15) gives G^{v∞}_2 = −2π g^{v∞}_{0,0} = −π²/(2N sin²(π/N)) log|x|²—a factor of two. The same factor appears if you put log|x|² ~ −4π Im τ/N into (6.22) and compare with (6.25): you get 4π a0 Im τ instead of 2π a0 Im τ. Eq. (6.26) has the same off-by-two. The final formulas (6.15) appear to be the correct ones—integrating the residue-derived differential (6.20) gives the 1/4 coefficient—so the theorem statement is probably right, but the written argument does not establish it.\n\nOther gaps are minor: Theorem 1 skips k=2 with a hand-wave, and the functional equation (3.12) is used without proof or reference. Both are standard and likely fixable. The citation pattern looks fair, crediting Brown and Drewitt–Pimm appropriately.\n\nWho this is for: people working on iterated integrals of modular forms, especially with QFT or string-theory motivation. A serious referee should see it. My advice to an editor: send it out, but expect a revision that fixes the factor-of-two and fills the two small gaps. Once that is done, this is a solid contribution.","headline":"The main equivariant-primitive result is sound, but the genus-zero weight-two section has a factor-of-two slip in the printed proof that needs fixing before the paper is citable.","tokens_in":23066,"tokens_out":6199,"would_cite":true,"duration_ms":49479,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F11","11F12","11F67"],"pacs":[],"model":"deepseek-v4-flash","headline":"The equivariant primitives of Eisenstein series for principal congruence subgroups are exactly the corresponding non-holomorphic Eisenstein series, with closed formulas and, in the weight-two genus-zero case, single-valued logarithms of a…","keywords":["Eisenstein series","principal congruence subgroups","non-holomorphic Eisenstein series","equivariant primitives","Hauptmodul","genus zero","L-functions","single-valued logarithms"],"falsifier":"Take a fixed level $N>1$, weight $k\\geq 3$, and vector $v$, and numerically evaluate both sides of the identity in Theorem 6 at several random points in a fundamental domain for $\\Gamma(N)$; any disagreement would falsify the central identification. For the genus-zero weight-two case, the same direct check can be run on the logarithm formula of Theorem 8 at $\\tau=i$, where the Hauptmodul is explicitly known for small levels such as $N=3,4,5$.","tokens_in":22005,"feed_emoji":"📐","tokens_out":11859,"duration_ms":103075,"temperature":0.7,"pith_summary":"This paper proves that, for principal congruence subgroups $\\Gamma(N)$, the equivariant primitives of holomorphic Eisenstein series—the real-analytic functions whose differential reproduces the relevant Eisenstein series and that transform equivariantly under $\\Gamma(N)$—are precisely the corresponding non-holomorphic Eisenstein series. This extends to all congruence subgroups a result that had been established only for the full modular group, and it gives explicit closed formulas for the primitives. A novel feature at level $N>1$ is weight two, where the paper shows that for genus-zero modular curves the non-holomorphic Eisenstein series can be written as single-valued logarithms of rational functions of the Hauptmodul. The interest is that these functions are the natural congruence-subgroup analogues of single-valued polylogarithms, so the paper provides a concrete bridge between iterated Eisenstein integrals, modular geometry, and the transcendental functions that appear in perturbative scattering amplitudes.","feed_headline":"Eisenstein primitives are non-holomorphic Eisenstein series","feed_subtitle":"Proved for every principal congruence subgroup, with closed formulas and genus-zero log representations.","key_machinery":"The load-bearing object is the equivariant primitive itself, defined with a regulated integral from the cusp $i\\infty$: $\\mathbb{G}_v^k(\\tau)=P_{G_v^k}+\\operatorname{Re}\\int_{\\vec 1_\\infty}^{\\tau} G_v^k(z)$. The polynomial correction $P_{G_v^k}$ is chosen so that the real part of the Eisenstein cocycle $C_v^k$, which measures the failure of the integral to transform covariantly, becomes a coboundary; Theorem 4 computes this real part explicitly. The cocycle values on the generators $S$ and $T$ are evaluated using the completed $L$-functions $\\Lambda(\\xi_v^k,l)$, whose closed forms are given in Theorem 1 in terms of Bernoulli polynomials and Clausen values. The final identification with non-holomorphic Eisenstein series is driven by a telescoping differentiation identity, eq. (5.13), which shows that the candidate built from non-holomorphic Eisenstein series satisfies the same differential equation as the equivariant primitive; uniqueness then forces equality. In the genus-zero weight-two case the mechanism shifts to residues at cusps: the differential forms $g_v^2$ have nonzero residues, fixing the constants in the relation to $d\\log$ of translations of the Hauptmodul, which integrates to the single-valued logarithms of Theorem 8.","core_discovery":"On the paper's own terms, the central discovery is an exact identification: the equivariant primitive of an Eisenstein series for $\\Gamma(N)$ is the non-holomorphic Eisenstein series of the same index. Concretely, Theorem 5 constructs, for each weight-$k$ Eisenstein series $G_v^k$ with $k\\geq 3$, the unique modular-equivariant solution of $d\\mathbb{G}_v^k = \\operatorname{Re} G_v^k$, and Theorem 6 shows this solution equals $-\\frac{2\\pi}{k-1}$ times the sum over $r+s=k-2$ of $G_{r,s}^v(\\tau)(X-\\tau Y)^r(X-\\bar\\tau Y)^s$, so its coefficients are non-holomorphic Eisenstein series of weights $(r,s)$. In weight two the primitive of $g_v^2$ is $-2\\pi$ times the difference $g_{0,0}^v$ of non-holomorphic Eisenstein series of weights $(0,0)$, and when the modular curve $\\Gamma(N)\\backslash\\mathbb{H}$ has genus zero, Theorem 8 writes these differences as single-valued logarithms of rational functions of the Hauptmodul, i.e. of a generator of the function field of the modular curve. The paper presents these results as a natural generalization of the full-modular-group statement, with the novelty that odd weights and weight two now occur.","pith_inferences":["The authors' closing remark suggests that higher-genus weight-two equivariant primitives should be expressible as single-valued combinations of abelian integrals of the third kind on the modular curve; that statement is a conjecture in the paper, not a proved result.","The appearance of Clausen values rather than only zeta values suggests that higher-length equivariant iterated Eisenstein integrals for congruence subgroups will involve iterated cyclotomic polylogarithms, mirroring the single-valued structure familiar from physics computations.","One could test Theorem 8 numerically for small levels such as $N=3,4,5$ by expanding both sides near each cusp; agreement at all cusps would be evidence that the logarithmic representation is canonical, while any discrepancy would indicate a missed constant."],"forward_implications":["For every principal congruence subgroup $\\Gamma(N)$ and every Eisenstein series of weight $k\\geq 3$, the equivariant primitive is a finite linear combination of non-holomorphic Eisenstein series with coefficients given explicitly by Theorem 6.","Because the principal congruence subgroup results can be descended by coset sums, the same description holds for every congruence subgroup, covering cases such as $\\Gamma_0(N)$ that were previously handled separately.","For genus-zero modular curves, the weight-two equivariant primitives are not merely modular functions but single-valued logarithms of rational functions of the Hauptmodul, giving closed transcendental expressions.","The completed $L$-series of these Eisenstein series are evaluated in closed form by Bernoulli polynomials and Clausen values, making the cocycles that control modular transformation fully explicit.","This establishes the length-one case of equivariant iterated Eisenstein integrals for all principal congruence subgroups, the base step needed for higher-length generalizations."],"supporting_citations":[{"why":"Supplies the regulated-integral construction of equivariant primitives for the full modular group that this paper generalizes to congruence subgroups.","marker":"[45]"},{"why":"Defines the broader framework of equivariant iterated Eisenstein integrals in which the length-one primitives studied here sit.","marker":"[44]"},{"why":"Provides the values of Eisenstein cocycles on generators for principal congruence subgroups, which Theorem 3 re-derives in the paper's conventions.","marker":"[58]"},{"why":"Gives the prior generalization to congruence subgroups of the form $\\Gamma_0(N)$, the closest existing result that this paper extends to all principal congruence subgroups.","marker":"[56]"},{"why":"Supplies the Eichler-Shimura isomorphism that attaches a cocycle to each modular form and underlies the cocycle computations.","marker":"[59]"},{"why":"Provides the decomposition of real-analytic equivariant functions into modular components of weights $(r,s)$ used in the proof of Theorem 6.","marker":"[61]"},{"why":"Introduces the alternative spanning set of Eisenstein series $H_v^k$ and its Fourier expansion, which the paper uses throughout.","marker":"[33]"},{"why":"Supplies standard facts on modular curves, cusps, and non-holomorphic Eisenstein series used in the weight-two genus-zero argument.","marker":"[60]"}],"fun_headline_variants":["Equivariant primitives equal non-holomorphic Eisenstein series","Closed formulas: Eisenstein primitives are non-holomorphic series","Genus-zero modular curves yield single-valued log Hauptmodul","Weight-two Eisenstein: primitive as single-valued logarithms","For all congruence subgroups: primitives match non-holomorphic"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof invokes, without derivation, the functional equation $\\Lambda(\\xi_v^k,l)=(-1)^l\\Lambda(\\xi_{vS}^k,k-l)$ for the completed $L$-function of the Eisenstein series and uses it to evaluate the one exceptional $L$-series value needed in the cocycle computation; the identification of equivariant primitives with non-holomorphic Eisenstein series depends on that identity.","fun_headline_variants_meta":{"raw":{"variants":["Equivariant primitives equal non-holomorphic Eisenstein series","Closed formulas: Eisenstein primitives are non-holomorphic series","Genus-zero modular curves yield single-valued log Hauptmodul","Weight-two Eisenstein: primitive as single-valued logarithms","For all congruence subgroups: primitives match non-holomorphic"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000129,"raw_usage":{"total_tokens":1102,"prompt_tokens":905,"completion_tokens":197,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":521,"completion_tokens_details":{"reasoning_tokens":111}},"tokens_in":521,"tokens_out":197,"duration_ms":2451,"temperature":1.0,"reasoning_tokens":111,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T21:36:25.510297+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a fixed level $N>1$, weight $k\\geq 3$, and vector $v$, and numerically evaluate both sides of the identity in Theorem 6 at several random points in a fundamental domain for $\\Gamma(N)$; any disagreement would falsify the central identification. For the genus-zero weight-two case, the same direct check can be run on the logarithm formula of Theorem 8 at $\\tau=i$, where the Hauptmodul is explicitly known for small levels such as $N=3,4,5$.","supporting_citations":[{"cited_title":"A class of non-holomorphic modular forms I","cited_arxiv_id":null,"evidence_quote":"Supplies the regulated-integral construction of equivariant primitives for the full modular group that this paper generalizes to congruence subgroups."},{"cited_title":"A class of non-holomorphic modular forms II: Equivariant i terated Eisenstein integrals","cited_arxiv_id":null,"evidence_quote":"Defines the broader framework of equivariant iterated Eisenstein integrals in which the length-one primitives studied here sit."},{"cited_title":"Modular symbols, Eisenstein series, and congruences","cited_arxiv_id":null,"evidence_quote":"Provides the values of Eisenstein cocycles on generators for principal congruence subgroups, which Theorem 3 re-derives in the paper's conventions."},{"cited_title":"Real-analytic modular forms for Γ 0(N ) and their L-series","cited_arxiv_id":null,"evidence_quote":"Gives the prior generalization to congruence subgroups of the form $\\Gamma_0(N)$, the closest existing result that this paper extends to all principal congruence subgroups."},{"cited_title":"Elementary theory of L-functions and Eisenstein series","cited_arxiv_id":null,"evidence_quote":"Supplies the Eichler-Shimura isomorphism that attaches a cocycle to each modular form and underlies the cocycle computations."},{"cited_title":"On quasi-modular forms, almost holomorphic modular forms , and the vector-valued modular forms of Shimura","cited_arxiv_id":null,"evidence_quote":"Provides the decomposition of real-analytic equivariant functions into modular components of weights $(r,s)$ used in the proof of Theorem 6."},{"cited_title":"A First Course in Modular Forms","cited_arxiv_id":null,"evidence_quote":"Supplies standard facts on modular curves, cusps, and non-holomorphic Eisenstein series used in the weight-two genus-zero argument."}],"review_version":1}