Panorbital residues and elliptic summability

TL;DR

This paper introduces two new types of residues, panorbital residues, which, along with existing orbital residues, fully characterize the elliptic summability problem across various elliptic curve representations and characteristics.

Contribution

It develops the theory of panorbital residues, completing the obstruction framework for elliptic summability in multiple elliptic curve models and characteristics.

Findings

Complete obstruction characterization for elliptic summability.

Explicit computation examples for summable and non-summable functions.

New results on elliptic functions using orbital and panorbital residues.

Abstract

For $\tau$ the translation automorphism defined by a non-torsion point in an elliptic curve, we consider the elliptic summability problem of deciding whether a given elliptic function $f$ is of the form $f=\tau(g)-g$ for some elliptic function $g$. We introduce two new panorbital residues and show that they, together with the orbital residues introduced in 2018 by Dreyfus, Hardouin, Roques, and Singer, comprise a complete obstruction to the elliptic summability problem. The underlying elliptic curve can be described in any of the usual ways: as a complex torus, as a Tate curve, or as a one-dimensional abelian variety. We develop the necessary results from scratch intrinsically within each setting; in the last two of them, we also work in arbitrary characteristic. We include several basic concrete examples of computation of orbital and panorbital residues for some summable and non-summable functions in each setting. We conclude by applying the technology of orbital and panorbital residues to obtain several new results of independent interest.