Equivariant primitives of Eisenstein series for congruence subgroups

Claude Duhr; Franca Lippert

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Equivariant primitives of Eisenstein series for congruence subgroups

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2502.04752 v1 pith:QOYJGJ4B submitted 2025-02-07 math.NT hep-thmath-phmath.MP

Claude Duhr , Franca Lippert This is my paper · ORCID

classification math.NThep-thmath-phmath.MP MSC 11F1111F1211F67

keywords Eisensteinseriesprincipalcongruencesubgroupsnon-holomorphicequivariantprimitivesHauptmodulgenuszeroL-functionssingle-valuedlogarithms

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 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.

What carries the argument

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.

What would settle it

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 $N=3,4,5$.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the 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.

What 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.

The 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.

Other 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.

Who 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.

Referee Report

1 major / 5 minor

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.

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 (1)

[Section 6, eqs. (6.15), (6.22), (6.25)-(6.26)] 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.

minor comments (5)

[Section 3, Theorem 1] 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 3, eq. (3.12)] 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 2.2, eq. (2.16)] 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 5, eq. (5.6) and Theorem 6] 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.
[Throughout] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the main results are proven from definitions and standard L-series facts.

full rationale

The central identification of equivariant primitives with non-holomorphic Eisenstein series is derived: Theorem 5 proves modular equivariance and uniqueness from the explicit definition (5.1), and Theorem 6 proves the coefficient formula (5.8) by differentiating the proposed series and using the uniqueness just established. The L-series evaluation in Theorem 1 is a direct computation from the Fourier expansion, with the standard functional equation (3.12) used only for one exceptional case; this is an external standard fact, not an input equivalent to the target result. The cocycle computations in Theorems 3 and 4 are self-contained from the definitions of the regularized integrals. The weight-two genus-zero result in Theorem 8 uses pole/residue arguments on the modular curve plus a constant-determination by asymptotics; those are standard tools and do not assume the conclusion. No fitted parameter is presented as a prediction, and no argument reduces to a self-citation. The cited prior work by the same authors (e.g., ref. [33]) is used only to motivate an alternative spanning set, which is then proved in Proposition 1. There is, however, an internal arithmetic inconsistency in the proof of Theorem 8: eq. (6.22) differs by a factor of 2 from the combination of Theorem 7 with eq. (6.15), and the asymptotics in eqs. (6.25)-(6.26) exhibit the same mismatch. This is a correctness defect in the printed proof, not a circularity, and it does not affect the main equivariant-primitive theorems.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on standard results in modular forms and L-functions (functional equation, regularisation, the equivariant-function correspondence) plus the explicit genus-zero Hauptmodul normalization. No free parameters are fitted to data, and no new entities are postulated.

assumptions (4)

standard math The completed L-function of an Eisenstein series satisfies the functional equation Λ(ξ_v^k, l) = (-1)^l Λ(ξ_{vS}^k, k-l) for k ≥ 3.
Invoked in Section 3, proof of Theorem 1, to evaluate the L-series in the case (b,l)=(0,1). This is a standard property of L-functions of modular forms, but it is not proven in the paper.
domain assumption The regularisation of iterated integrals at tangential base points from Brown (ref. [45]) applies to congruence subgroups.
Used in Section 2.4 to define the regulated integrals from cusps, which underlie the equivariant primitives in Section 5.
standard math Proposition 2 (from refs. [45,61]) states that real-analytic equivariant polynomial-valued functions on H correspond bijectively to modular forms of weights (r,s).
Used in the proof of Theorem 6 to identify the coefficients of the equivariant primitive with non-holomorphic Eisenstein series.
domain assumption For genus-zero modular curves Y(N), a Hauptmodul x(τ) exists with the normalization x(τ)=q_N+O(q_N^2), x_{v0}=0, x_{v∞}=∞.
Used in Section 6, Theorem 8, to express weight-two equivariant primitives as single-valued logarithms. This is standard for genus-zero modular curves.

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Pith. "Pith review of Equivariant primitives of Eisenstein series for congruence subgroups." pith.science (2026). https://pith.science/paper/QOYJGJ4B

@misc{pith2026250204752,
  author       = {Pith},
  title        = {Pith review of: Equivariant primitives of Eisenstein series for congruence subgroups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QOYJGJ4B}},
  note         = {Machine review of arXiv:2502.04752}
}

read the original abstract

We study equivariant primitives of Eisenstein series for principal congruence subgroups and show that they are precisely the corresponding non-holomorphic Eisenstein series. We present closed formulas that naturally generalise existing results for the full modular group. We also focus on Eisenstein series of weight two in the case where the modular curve has genus zero. We show that in those cases the non-holomorphic Eisenstein series of weight two can be written as single-valued logarithms whose argument is a rational function of the Hauptmodul.

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Reviewed August 8, 2026 · model on record in the stance chip above.

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