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arXiv · 2504.09429

Galois groups of reductions modulo p of D-finite series

Xavier Caruso·Florian Fürnsinn·Daniel Vargas-Montoya
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Abstract

The aim of this paper is to investigate the algebraicity behavior of reductions of $D$-finite power series modulo prime numbers. For many classes of D-finite functions, such as diagonals of multivariate algebraic series or hypergeometric functions, it is known that their reductions modulo prime numbers, when defined, are algebraic. We formulate a conjecture that uniformizes the Galois groups of these reductions across different prime numbers. We then focus on hypergeometric functions, which serves as a test case for our conjecture. Refining the construction of an annihilating polynomial for the reduction of a hypergeometric function modulo a prime number p, we extract information on the respective Galois groups and show that they behave nicely as p varies.

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Xavier Caruso, Florian Fürnsinn, Daniel Vargas-Montoya. 2025-04-13. Galois groups of reductions modulo p of D-finite series. https://arxiv.org/abs/2504.09429

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