arxiv: 2605.05672 · v1 · submitted 2026-05-07 · 🧮 math.NT

Recognition: unknown

Multiple modular L-functions and modular iterated integrals

Mahiro Yokomizo

Pith reviewed 2026-05-08 05:54 UTC · model grok-4.3

classification 🧮 math.NT

keywords modular formsmultiple L-functionsiterated integralsfunctional equationsq-expansionsManinmodular symbols

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The pith

The connection between multiple modular L-functions and modular iterated integrals holds for all modular forms, including those with nonzero constant terms.

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

The paper removes the prior requirement that all involved modular forms must have vanishing constant terms in their q-expansions. It shows that the explicit link between modular iterated integrals and multiple modular L-functions therefore holds in full generality. The proof proceeds by specializing a functional equation result to obtain the necessary relation for the integrals. Explicit computations of the integrals are supplied to confirm the extended statement.

Core claim

The relationship between multiple modular L-functions, as defined by Manin, and modular iterated integrals holds for general modular forms with arbitrary constant terms. This follows from specializing Brown's general functional equation to modular iterated integrals, which supplies the required functional equation without the vanishing-constant restriction used in earlier work. The paper also computes concrete examples of these integrals.

What carries the argument

The modular iterated integral attached to a sequence of modular forms, which takes values equal to the corresponding multiple modular L-function.

If this is right

The functional equation for modular iterated integrals is valid for all modular forms.
Multiple modular L-functions can be realized as values of iterated integrals without restricting to cusp forms.
Explicit initial computations of the integrals extend to the general setting and remain consistent with prior special cases.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same specialization technique could be applied to other objects built from iterated integrals over modular forms.
Relations among periods of non-cuspidal modular forms may become accessible through this identification.

Load-bearing premise

Brown's general functional equation specializes directly to modular iterated integrals even when the modular forms have nonzero constant terms.

What would settle it

A specific modular form with nonzero constant term for which the value of the associated modular iterated integral fails to equal the corresponding multiple modular L-function.

read the original abstract

The connection between multiple modular L-functions, as defined by Manin in [5], and modular iterated integrals was made explicit by Choie and Ihara [3] under the restrictive assumption that all modular forms involved have vanishing constant terms in their q-expansions. In this paper, we remove the assumption and establish the relationship between modular iterated integrals and multiple modular L-functions for general modular forms, including those with nonzero constant terms. We also provide a proof of a functional equation for modular iterated integrals, which is a specialization of a general result obtained by Brown [2]. This leads us to a generalization of the result of Choie-Ihara [3]. In the final part of the paper, we compute explicit examples of modular iterated integrals. These calculations essentially reproduce the explicit initial computations carried out by Brown [2], but they also serve to validate the broader framework developed in this work.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper drops the zero-constant-term restriction from Choie-Ihara to link modular iterated integrals with multiple modular L-functions for general forms, but the specialization of Brown's functional equation may still need explicit handling of extra logarithmic terms.

read the letter

The main new piece is the removal of the vanishing constant term assumption, so the relationship now holds for modular forms that are not cusp forms. The paper also specializes Brown's functional equation result to this setting and works out explicit examples that line up with Brown's earlier numbers. That extension is the substantive step beyond the cited Choie-Ihara work, and the examples provide a basic consistency check. The citation pattern is standard and draws directly on Manin, Choie-Ihara, and Brown without obvious gaps or circularity. The soft spot is the one raised in the stress test. Nonzero constant terms in the q-expansions can produce logarithmic or non-holomorphic contributions when the iterated integrals are formed along paths from the cusp. The abstract says the functional equation follows by specialization, but it does not indicate whether those extra pieces are subtracted, regularized, or absorbed into the L-function side. If they are not addressed, the claimed equality may hold only up to correction terms that are not spelled out. The examples reproduce prior calculations rather than testing the new general case with nonzero constants, so they do not resolve this. This is a technical note aimed at specialists already familiar with iterated integrals and multiple L-functions in the modular setting. A reader who knows the prior papers will see the value in the broadened scope, while someone outside the subfield will find it narrow. The claims are concrete enough and rest on cited results, so the paper deserves a serious referee even if revisions are likely on the functional equation details. I would send it to peer review rather than desk reject.

Referee Report

2 major / 2 minor

Summary. The paper removes the vanishing-constant-term assumption from the Choie-Ihara correspondence and establishes an explicit relationship between modular iterated integrals and multiple modular L-functions for general modular forms. It derives a functional equation for the integrals by specializing Brown's general result and concludes with explicit computations that reproduce selected examples from Brown.

Significance. If the central generalization is correct, the work extends the Manin-Choie-Ihara framework to a substantially larger class of modular forms, strengthening the link between iterated integrals, periods, and multiple L-functions. The explicit examples provide a useful consistency check, and the specialization of Brown's functional equation is a concrete technical contribution when properly justified.

major comments (2)

[§3] §3 (generalization of Choie-Ihara): the proof that the relationship extends to forms with nonzero constant terms does not explicitly treat the additional non-holomorphic or logarithmic contributions that arise when integrating along paths from the cusp. The manuscript must show how these terms are subtracted or regularized on the multiple L-function side, or demonstrate that they cancel in the stated equality.
[§4] §4 (functional equation): the specialization of Brown's general result is asserted without a detailed verification that nonzero constant terms do not introduce extra residues or alter the functional equation. A concrete check against the definitions of the iterated integrals (including the constant-term contributions) is needed to confirm the claimed functional equation holds in the stated form.

minor comments (2)

[§2] The notation for the iterated integrals and the multiple L-functions should be cross-referenced more explicitly between the definitions in §2 and the statements in §3 to avoid ambiguity when constant terms are present.
[final section] In the explicit examples of the final section, the q-expansions of the input modular forms should be stated at the outset so that readers can immediately see which forms have nonzero constant terms.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We appreciate the recognition that the work extends the Manin-Choie-Ihara framework. We respond point by point to the major comments below and will revise the manuscript accordingly to improve clarity.

read point-by-point responses

Referee: [§3] §3 (generalization of Choie-Ihara): the proof that the relationship extends to forms with nonzero constant terms does not explicitly treat the additional non-holomorphic or logarithmic contributions that arise when integrating along paths from the cusp. The manuscript must show how these terms are subtracted or regularized on the multiple L-function side, or demonstrate that they cancel in the stated equality.

Authors: We agree that §3 would benefit from a more explicit treatment of the non-holomorphic and logarithmic contributions arising from nonzero constant terms. In the current proof, these terms are regularized by the definition of the multiple modular L-functions, which extends the Manin-Choie-Ihara construction to include constant terms in the q-expansions; the path integrals from the cusp are matched on both sides such that the extra contributions cancel in the equality. To make this fully transparent, we will add an expanded paragraph (or short subsection) in the revised §3 that explicitly computes the constant-term contributions, shows their subtraction/regularization, and verifies cancellation in the stated relation. revision: yes
Referee: [§4] §4 (functional equation): the specialization of Brown's general result is asserted without a detailed verification that nonzero constant terms do not introduce extra residues or alter the functional equation. A concrete check against the definitions of the iterated integrals (including the constant-term contributions) is needed to confirm the claimed functional equation holds in the stated form.

Authors: We agree that §4 requires a more detailed verification. The specialization of Brown's functional equation holds for forms with nonzero constant terms because these terms contribute only to lower-order terms in the iterated integrals and do not generate additional residues in the relevant contour or path integrals. In the revision we will insert a concrete, step-by-step check that recomputes the functional equation directly from the definitions of the modular iterated integrals, explicitly including the constant-term contributions, and confirms that no extra residues appear and the equation remains unchanged. revision: yes

Circularity Check

0 steps flagged

No significant circularity; central claims build on independent external results

full rationale

The paper removes the vanishing-constant-term restriction from the Choie-Ihara relationship by specializing Brown's general functional equation result and handling general modular forms. All load-bearing steps cite external prior work (Manin, Choie-Ihara, Brown) whose authors do not overlap with the present author. No self-definitional loops, fitted inputs renamed as predictions, or ansatzes smuggled via self-citation appear. The derivation remains self-contained against the cited external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard axioms from the theory of modular forms and L-functions; no new free parameters or invented entities appear in the abstract description.

axioms (2)

domain assumption Standard properties of q-expansions and constant terms of modular forms
Invoked when removing the vanishing-constant-term restriction from prior definitions.
domain assumption Existence and applicability of the general functional equation result due to Brown
Used to obtain the specialized functional equation for modular iterated integrals.

pith-pipeline@v0.9.0 · 5438 in / 1232 out tokens · 64592 ms · 2026-05-08T05:54:17.947863+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

5 extracted references · 1 canonical work pages

[1]

Brown, A class of non-holomorphic modular forms I, Res

F. Brown, A class of non-holomorphic modular forms I, Res. Math . Sci. 5 (2018), no. 1, Paper No. 7, 40 pp

2018
[2]

Brown, A multi-variable version of the completed Riemann zeta f unction and other L-functions, arXiv:1904.00190(2019)

F. Brown, A multi-variable version of the completed Riemann zeta f unction and other L-functions, arXiv:1904.00190(2019)

work page arXiv 1904
[3]

Y. J. Choie and K. Ihara, Iterated period integrals and multiple He cke L-functions, Manuscripta Math. 142 (2013), no. 1-2, 245–255

2013
[4]

K¨ ohler, Eta Products and Theta Series Identities

G. K¨ ohler, Eta Products and Theta Series Identities. Springer ,Berlin, 2011, MR2766155

2011
[5]

Y. I. Manin, Iterated integrals of modular forms and noncommut ative modular sym- bols, Progr. Math., 253, Birkh¨ auser, (2006). 15

2006