Elliptic Multiple Polylogarithms with Arbitrary Arguments in extsc{GiNaC}

We present an algorithm for the numerical evaluation of elliptic multiple polylogarithms for arbitrary arguments and to arbitrary precision. The cornerstone of our approach is a procedure to obtain a convergent $q$-series representation of elliptic multiple polylogarithms. Its coefficients are expressed in terms of ordinary multiple polylogarithms, which can be evaluated efficiently using existing libraries. In a series of preparation steps the elliptic polylogarithms are mapped into a region where the $q$-series converges rapidly. We also present an implementation of our algorithm into the \texttt{GiNaC} framework. This release constitutes the first public package capable of evaluating elliptic multiple polylogarithms to high precision and for arbitrary values of the arguments.

💡 Research Summary

The paper addresses a pressing need in high‑energy theoretical physics: the efficient, high‑precision numerical evaluation of elliptic multiple polylogarithms (eMPLs) for arbitrary complex arguments. While ordinary multiple polylogarithms (MPLs) have become standard tools—supported by mature libraries—eMPLs, which naturally arise in multi‑loop Feynman integrals involving massive particles and elliptic curve geometries, have lacked a robust public implementation. Existing software can only evaluate eMPLs within a limited region of convergence of the q‑series expansion, making it unsuitable for phenomenological applications that require arbitrary kinematic points.

The authors present a complete algorithm and its implementation in the GiNaC computer algebra system (version 1.8.10). The core idea is to rewrite any eMPL as a convergent q‑series whose coefficients are ordinary MPLs and elementary rational functions. The q‑parameter is defined as q_τ = exp(2πi τ), where τ is the modular parameter of the underlying elliptic curve. By applying an SL(2,ℤ) modular transformation, τ can always be mapped into the fundamental domain F, guaranteeing Im τ > √3/2 and thus |q| ≲ 0.004. However, convergence of the double sums that define the kernels g^{(n)}(z,τ) also depends on the imaginary part of the integration variable z; the condition |Im z| < Im τ must be satisfied. To meet this requirement for arbitrary arguments, the algorithm performs two preparatory steps:

1. Modular Normalisation – τ is transformed to τ′ in the fundamental domain, possibly followed by an additional modular transformation to increase Im τ′ if needed.
2. Argument Mapping – each singular point z_i and the endpoint z are shifted and possibly reflected within the lattice Λ_{τ′}=ℤ+τ′ℤ so that their imaginary parts lie inside the strip |Im z| < Im τ′. This is achieved by exploiting the quasi‑periodicity of the integration kernels.

After these transformations, the integration kernels g^{(n)}(z,τ′) admit explicit q‑expansions:

• g^{(1)}(z,τ′) = π cot(πz) + 4π ∑_{m,n≥1} sin(2πmz) q^{mn},
• g^{(k)}(z,τ′) for even k involve cosine series with ζ(k) coefficients,
• g^{(k)}(z,τ′) for odd k involve sine series.

These series converge rapidly because |q| is tiny. The eMPL itself can then be expressed as \