On Abel's problem and Gauss congruences

Abstract

A classical problem due to Abel is to determine if a differential equation $y'=ηy$ admits a non-trivial solution $y$ algebraic over $\mathbb C(x)$ when $η$ is a given algebraic function over $\mathbb C(x)$. Risch designed an algorithm that, given $η$, determines whether there exists an algebraic solution or not. In this paper, we adopt a different point of view when $η$ admits a Puiseux expansion with rational coefficients at some point in $\mathbb C\cup \{\infty\}$, which can be assumed to be 0 without loss of generality. We prove the following arithmetic characterization: there exists a non-trivial algebraic solution of $y'=ηy$ if and only if the coefficients of the Puiseux expansion of $xη(x)$ at $0$ satisfy Gauss congruences for almost all prime numbers. We then apply our criterion to hypergeometric series: we completely determine the equations $y'=ηy$ with an algebraic solution when $xη(x)$ is an algebraic hypergeometric series with rational parameters, and this enables us to prove a prediction Golyshev made using the theory of motives. We also present three other applications, in particular to diagonals of rational fractions and to directed two-dimensional walks.