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

TL;DR

This paper generalizes a criterion for the existence of algebraic solutions to differential equations with algebraic coefficients, linking it to generalized Gauss and Cartier congruences over number fields, with applications in hypergeometric functions and zeta functions.

Contribution

It extends previous criteria based on Gauss congruences to arbitrary algebraic functions over number fields, incorporating Cartier congruences and applying to broader mathematical contexts.

Findings

Generalization of Gauss congruences to number fields.

Introduction of Cartier congruences in this context.

Applications to hypergeometric functions and Artin-Mazur zeta functions.

Abstract

Abel's problem consists in identifying the conditions under which the diferential equation $y'=\eta y$, with $\eta$ 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 $\eta$ has a Puiseux expansion with $\textit{rational}$ coefficients at the origin: there exists a non-trivial algebraic solution of $y'=\eta y$ if and only if the coefficients of the Puiseux expansion of $x\eta(x)$ at $0$ satisfy Gauss congruences for almost all prime numbers. In this paper, we generalize this criterion to arbitrary $\eta$ 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.