Elliptic hyperlogarithms

TL;DR

The paper advances the theory of elliptic polylogarithms by proving that the tildeΓ functions of BDDT form an $\mathcal{O}(\mathcal{E}_S)$-basis of the minimal stable subalgebra $A_{\mathcal{E}_S}$ on an elliptic curve with punctures, providing an integration-stable alternative to the elliptic HL basis. It develops two compatible regularisation schemes for tildeΓ, establishes their relation to elliptic HLs, and shows that the algebra they generate coincides with $A_{\mathcal{E}_S}$. Furthermore, it connects tildeΓ with the $\mathrm{E_3}$ constructions and supplies an independent proof of BD ext{DT}'s stability result via elliptic uniformisation. The results yield concrete, computable bases for multivalued elliptic functions with controlled monodromy and growth, with potential applications in number theory and string-theory contexts. Overall, the work provides a robust, alternative framework for elliptic hyperlogarithms and strengthens the integration-stability paradigm in genus one.

Abstract

Let $\mathcal E$ be a complex elliptic curve and $S$ be a non-empty finite subset of $\mathcal E$. We show that the functions $\tildeΓ$ introduced in arXiv:1712.07089 out of string theory motivations give rise to a basis of the minimal algebra $A_{\mathcal E\smallsetminus S}$ of holomorphic multivalued functions on $\mathcal E\smallsetminus S$ which is stable under integration, introduced in arXiv:2212.03119; this basis is alternative to the basis of $A_{\mathcal E\smallsetminus S}$ constructed in loc. cit. using elliptic analogues of the hyperlogarithm functions.

Theorems & Definitions (116)

• Definition 2.1
• Theorem 2.3: see EZ2
• proof
• Lemma 2.4: EZ2
• proof
• Lemma 2.5
• proof
• Definition 2.6
• Lemma 2.7
• proof