2025/02/04 by Fresán, Javier, Lam, Yeuk Hay Joshua, Qin, Yichen
#Algebraic Geometry (math.AG) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2502.02147
We answer in the negative Siegel's problem for G-functions, as formulated by Fischler and Rivoal. Roughly, we prove that there are G-functions that cannot be written as polynomial expressions in algebraic pullbacks of hypergeometric functions; our examples satisfy differential equations of order two, which is the smallest possible. In fact, we construct infinitely many non-equivalent rank-two local systems of geometric origin which are not algebraic pullbacks of hypergeometric local systems, thereby providing further counterexamples to Dwork's conjecture and answering a question by Krammer. The main ingredients of the proof are a Lie algebra version of Goursat's lemma, the monodromy computations of hypergeometric local systems due to Beukers and Heckman, as well as results on invariant trace fields of Fuchsian groups.