Summability of Elliptic Functions via Residues
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866913765652430848 |
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| author | Babbitt, Matthew |
| author_facet | Babbitt, Matthew |
| contents | Summability has been a central object of study in difference algebra over the past half-century. It serves as a cornerstone of algebraic methods to study linear recurrences over various fields of coefficients and with respect to various kinds of difference operators. Recently, Dreyfus, Hardouin, Roques, and Singer introduced a notion of elliptic orbital residues, which altogether serve as a partial obstruction to summability for elliptic functions with respect to the shift by a non-torsion point over an elliptic curve. We explain how to refine this into a complete obstruction, which promises to be useful in applications of difference equations over elliptic curves, such as elliptic hypergeometric functions and the combinatorics of walks in the quarter plane. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_22770 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Summability of Elliptic Functions via Residues Babbitt, Matthew Number Theory Algebraic Geometry 39A06 (Primary) 14H52, 12H10, 40C15 (Secondary) Summability has been a central object of study in difference algebra over the past half-century. It serves as a cornerstone of algebraic methods to study linear recurrences over various fields of coefficients and with respect to various kinds of difference operators. Recently, Dreyfus, Hardouin, Roques, and Singer introduced a notion of elliptic orbital residues, which altogether serve as a partial obstruction to summability for elliptic functions with respect to the shift by a non-torsion point over an elliptic curve. We explain how to refine this into a complete obstruction, which promises to be useful in applications of difference equations over elliptic curves, such as elliptic hypergeometric functions and the combinatorics of walks in the quarter plane. |
| title | Summability of Elliptic Functions via Residues |
| topic | Number Theory Algebraic Geometry 39A06 (Primary) 14H52, 12H10, 40C15 (Secondary) |
| url | https://arxiv.org/abs/2503.22770 |