Multiple modular L-functions and modular iterated integrals

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.

• The paper establishes an explicit correspondence between modular iterated integrals and multiple modular L-functions, generalizing previous results to include non-cusp forms.
• It introduces rigorous analytic techniques that separate constant terms using binomial and gamma-factor methods, ensuring meromorphic continuation.
• The framework provides new tools for computing special values and studying period integrals, with implications for arithmetic and modular form research.

Detailed Analysis of "Multiple modular L-functions and modular iterated integrals" (2605.05672)

Introduction and Context

The paper develops a comprehensive framework connecting multiple modular $L$-functions, as originally defined by Manin, with modular iterated integrals for general modular forms. The prior results by Choie and Ihara established an explicit correspondence only under the assumption that the modular forms have vanishing constant terms (i.e., are cusp forms), restricting generality and excluding significant classes such as Eisenstein series. The present work removes this assumption, fully generalizing the connection and significantly expanding the analytic and arithmetic landscape of modular iterated integrals and their associated $L$-functions.

Modular Iterated Integrals: Definitions and Analytic Structure

The starting point is the definition of iterated integrals of 1-forms on differentiable manifolds, with fundamental properties paralleling those from multiple zeta value theory. For modular forms $f_1, \ldots, f_n$ (not necessarily cusp forms), the modular iterated integral

$$I_{i\infty}^0\binom{f_1,\dots,f_n}{s,\alpha_2,\dots,\alpha_n}$$

is defined via nested integration over the upper half-plane. The analytic structure is carefully established: these integrals possess meromorphic continuation to $\mathbb{C}^n$ with explicitly determined divisors of poles, controlled by the parameters $s$ and $\alpha_i$, and satisfy a functional equation analogous to classical modular forms' $L$-functions. The key technical advance is the analytic treatment of non-vanishing constant terms in the $q$-expansions, requiring delicate bookkeeping of their contributions.

Multiple Modular $L$-functions: Definition and Functional Relations

The multiple modular $L$0-function associated to modular forms $L$1 is defined as

$L$2

where $L$3 are the Fourier coefficients of $L$4. These series generalize both multiple zeta values and the standard modular $L$5-functions, but involve more complicated summation kernels reflecting the iterated structure.

The main theorems provide explicit, algorithmic linear relations between modular iterated integrals and multiple modular $L$6-functions:

• Every modular iterated integral can be written as a finite linear combination of multiple modular $L$7-functions (and vice versa), with precise combinatorial and gamma-factor weights that explicitly track the influence of constant terms.

Notably, the explicit formulae handle cases where the modular forms have nonzero constant terms, introducing additional terms and combinatorial structures absent in the cusp form case.

Technical Contributions: Handling Nonzero Constant Terms

The primary technical achievement is the decomposition of modular forms $L$8 (separating the cusp and constant parts), and a systematic method for separating and evaluating the integrals involving these constant terms. The combinatorics of how constant terms propagate through the iterated integrals is carefully coordinated, with explicit binomial and gamma-factor sums arising from integration by parts and the binomial theorem.

This careful analysis leads to the meromorphic continuation of the relevant objects and identification of all potential poles, which are shown to be entirely determined by the combinatorics of the exponents and the weights of the modular forms involved.

Mellin Transforms and Explicit Computations

The paper leverages Mellin transform techniques to convert iterated integrals into sums involving modular $L$9-functions and proves converse expressions via repeated application of integration by parts. To validate the theoretical framework, explicit computations (including cases with Eisenstein series) confirm consistency with earlier computations of Brown and extend them within the new generality.

The paper also explains, via pullback techniques on modular curves (notably $f_1, \ldots, f_n$0), how multiple zeta values (MZVs) can be viewed as specializations of modular iterated integrals, establishing deep connections with the arithmetic of modular curves, period integrals, and the theory of modular symbols.

Special Values and Modular Interpretations of MZVs

An explicit modular parametrization allows the pullback of classical polylogarithmic iterated integrals (as in MZVs) to modular iterated integrals on $f_1, \ldots, f_n$1, revealing the modular origin of the combinatorics and arithmetic behind MZVs. The explicit use of Dedekind $f_1, \ldots, f_n$2-products and Eisenstein series in describing these pullbacks highlights the unity of classical and modern approaches to period relations in the theory of automorphic forms.

Implications and Future Directions

This work provides a foundational toolset for the study of modular iterated integrals and multiple modular $f_1, \ldots, f_n$3-functions associated to general modular forms, not just cusp forms. This generalization has consequences for:

• The explicit calculation of special values and functional identities among modular $f_1, \ldots, f_n$4-functions
• Potential applications in the study of periods, Galois actions, and mathematical physics, since modular iterated integrals interpolate between multiple zeta-like and modular-period-like structures
• Generalizations to non-holomorphic settings, e.g., Maass forms, or connections to $f_1, \ldots, f_n$5-adic $f_1, \ldots, f_n$6-functions

Further developments could include extending these methods to automorphic forms on higher rank groups, deeper studies of the Galois and cohomological interpretations, and applications to the $f_1, \ldots, f_n$7-adic and geometric theory of modular forms and iterated Shimura integrals.

Conclusion

The paper accomplishes a complete generalization of the explicit relationship between modular iterated integrals and multiple modular $f_1, \ldots, f_n$8-functions to all modular forms, including those with non-vanishing constant terms. The framework fully describes the meromorphic properties, combinatorics, and functional relations of these objects, opens new directions for the study of special values and their arithmetic, and concretely connects modular, polylogarithmic, and period-theoretic viewpoints. The methodology and explicit computations offer versatile tools for further analytic and arithmetic investigation in the theory of automorphic forms and their periods.