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Summability of Elliptic Functions via Residues

Matthew Babbitt

TL;DR

The dissertation addresses the problem of summability for elliptic functions under a shift by a non-torsion point on an elliptic curve, a key subproblem in computing differential Galois groups of linear difference equations over elliptic curves. It develops a complete obstruction to summability by introducing panororbital residues, augmenting the previously known elliptic orbital residues, and provides both analytic and algebraic formulations. The analytic part constructs a summability criterion via a zeta-expansion, analytic pinnings, and a reduction that isolates constant and zeta-terms—the panororbital residues—whose vanishing characterizes summability. The algebraic part recasts these ideas in genus-1 geometry using divisors and the Riemann–Roch theorem, delivering an equivalent obstruction in the algebraic setting. Together, these results pave the way for algorithmic determination of differential Galois groups for elliptic-difference equations and have potential applications to elliptic hypergeometric functions and walks in the quarter plane.

Abstract

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.

Summability of Elliptic Functions via Residues

TL;DR

The dissertation addresses the problem of summability for elliptic functions under a shift by a non-torsion point on an elliptic curve, a key subproblem in computing differential Galois groups of linear difference equations over elliptic curves. It develops a complete obstruction to summability by introducing panororbital residues, augmenting the previously known elliptic orbital residues, and provides both analytic and algebraic formulations. The analytic part constructs a summability criterion via a zeta-expansion, analytic pinnings, and a reduction that isolates constant and zeta-terms—the panororbital residues—whose vanishing characterizes summability. The algebraic part recasts these ideas in genus-1 geometry using divisors and the Riemann–Roch theorem, delivering an equivalent obstruction in the algebraic setting. Together, these results pave the way for algorithmic determination of differential Galois groups for elliptic-difference equations and have potential applications to elliptic hypergeometric functions and walks in the quarter plane.

Abstract

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.

Paper Structure

This paper contains 40 sections, 42 theorems, 161 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Difference Equations over Elliptic Curves
  3. Recent Applications
  4. Commonalities: Galois Theory
  5. Dissertation Overview
  6. Background on Difference Equations
  7. A History of Picard-Vessiot Theory
  8. Second-Order Linear Difference Equations
  9. The Method of Residues
  10. Residues in the Case of Rational Functions
  11. Discrete Residues
  12. $q$-Discrete Residues
  13. Summability in the Analytic Setting
  14. Review of Elliptic Functions
  15. The Weierstrass Elliptic Functions
  16. ...and 25 more sections

Key Result

Proposition 2.2.1

Let $f(x)\in \mathcal{K}$ be expressed as $a(x)/b(x)$, where the polynomials $a(x),b(x)\in C[x]$ share no linear factors. Let $\mathcal{R}\subset C$ be the set of zeros of $b(x)$. Then there exist constants $c_j(f,\alpha)$ for each $j\in\mathbb{N}$ and $\alpha\in\mathcal{R}$, and a polynomial $g(x)\

Figures (1)

  • Figure 1: A lattice $\Lambda$ with its fundamental parallelogram labeled and shaded

Theorems & Definitions (98)

  • Remark 2.1.1
  • Remark 2.1.2
  • Proposition 2.2.1
  • Remark 2.2.2
  • Definition 2.2.3
  • Theorem 2.2.4
  • Definition 2.2.5
  • Theorem 2.2.6
  • Remark 3.1.1
  • Theorem 3.1.2
  • ...and 88 more