Hyperelliptic Integrals to Elliptic Integrals

Abstract

Consider a hyperelliptic integral $I=\int P/(Q\sqrt{S}) dx$, $P,Q,S\in\mathbb{K}[x]$, with $[\mathbb{K}:\mathbb{Q}]<\infty$. When $S$ is of degree $\leq 4$, such integral can be calculated in terms of elementary functions and elliptic integrals of three kinds $\mathcal{F},\mathcal{E},\Pi$. When $S$ is of higher degree, it is typically non elementary, but it is sometimes possible to obtain an expression of $I$ using also elliptic integrals when the Jacobian of $y^2=S(x)$ has elliptic factors. We present an algorithm searching for elliptic factors and a modular criterion for their existence. Then, we present an algorithm for computing an expression of $I$ using elliptic integrals, which always succeed in the completely decomposable Jacobian case.