Abel's problem, Gauss and Cartier congruences over number fields

Abstract

Abel's problem consists in identifying the conditions under which the diferential equation $y'=ηy$, with $η$ an algebraic function in $\mathbb{C}(x)$, possesses a non-zero algebraic solution $y$. This problem has been algorithmically solved by Risch. In a previous paper, we have presented an alternative solution in the special arithmetic situation where $η$ has a Puiseux expansion with $\textit{rational}$ coefficients at the origin: 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. In this paper, we generalize this criterion to arbitrary $η$ algebraic over $\overline{\mathbb{Q}}(x)$, by means of a natural generalization to number fields of Gauss congruences and of the weaker Cartier congruences recently introduced in this context by Bostan. We then provide applications of this criterion in the hypergeometric setting and for Artin-Mazur zeta functions.