Define K-linear obstructions to summability for the elliptic Mahler operator

Construct a K-linear obstruction to summability for rational functions on an elliptic curve E:y^2=x^3+Ax+B over K when the difference operator σ on M_E=K(x,y) is induced by the multiplication-by-m map [m] with m≥2 under the elliptic group law; that is, define a K-linear map whose kernel equals the image of Δ=σ−id_{M_E} to decide summability in this elliptic Mahler setting.

Background

The paper discusses summability obstructions via residues in several contexts, including the shift, q-dilation, Mahler, and elliptic shift cases. For elliptic shifts (addition-by-t maps), the combination of orbital and panorbital residues provides a complete obstruction to summability.

In contrast, for the operator induced by the multiplication-by-m map on an elliptic curve (an elliptic analogue of Mahler-type dynamics), the authors explicitly note that there is currently no K-linear obstruction to summability defined. Establishing such an obstruction would parallel the successes in other settings and enable effective decision procedures for summability in this elliptic Mahler case.