Panorbital residues and elliptic summability
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916916404158464 |
|---|---|
| author | Arreche, Carlos E. Babbitt, Matthew W. |
| author_facet | Arreche, Carlos E. Babbitt, Matthew W. |
| contents | For $τ$ 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 $f=τ(g)-g$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_18247 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Panorbital residues and elliptic summability Arreche, Carlos E. Babbitt, Matthew W. Algebraic Geometry Commutative Algebra Number Theory For $τ$ 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 $f=τ(g)-g$ 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. |
| title | Panorbital residues and elliptic summability |
| topic | Algebraic Geometry Commutative Algebra Number Theory |
| url | https://arxiv.org/abs/2508.18247 |