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arXiv · 2205.00041

Hyperexponential solutions of elliptic difference equations

Thierry Combot
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Abstract

Consider an elliptic curve $\mathcal{C}$ with coefficients in $\mathbb{K}$ with $[\mathbb{K}:\mathbb{Q}]<\infty$ and $\delta \in \mathcal{C}(\mathbb{K})$ a non torsion point. We consider an elliptic difference equation $\sum_{i=0}^l a_i(p) f(p\oplus i.\delta)=0$ with $\oplus$ the elliptic addition law and $a_i$ polynomials on $\mathcal{C}$. We present an algorithm to compute rational solutions, then an intermediary class we call pseudo-rational solutions, and finally hyperexponential solutions, which are functions $f$ such that $f(p\oplus \delta)/f(p)$ is rational over $\mathcal{C}$.

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Thierry Combot. 2022-04-29. Hyperexponential solutions of elliptic difference equations. https://arxiv.org/abs/2205.00041

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