On Siegel's problem and Dwork's conjecture for $G$-functions

TL;DR

This paper demonstrates that certain $G$-functions cannot be expressed as algebraic pullbacks of hypergeometric functions, providing counterexamples to Siegel's problem and Dwork's conjecture, with implications for the structure of differential equations of geometric origin.

Contribution

It constructs explicit counterexamples of $G$-functions and local systems that defy previous conjectures, advancing understanding of their algebraic and differential properties.

Findings

Existence of $G$-functions not expressible as polynomial algebraic pullbacks.

Construction of infinitely many non-equivalent rank-two local systems of geometric origin.

Counterexamples to Dwork's conjecture and resolution of Krammer's question.

Abstract

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.