Hyperexponential solutions of elliptic difference equations

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 $\mathcal{C}$.

19 pages