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

Javier Fresán, Yeuk Hay Joshua Lam, Yichen Qin

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.

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

Abstract

We answer in the negative Siegel's problem for -functions, as formulated by Fischler and Rivoal. Roughly, we prove that there are -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.

Paper Structure

This paper contains 23 sections, 23 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. A sketch of proof
  4. Previous work
  5. Organization
  6. Notation
  7. Preliminaries on hypergeometric connections
  8. G-functions and G-operators
  9. Trace fields and adjoint trace fields
  10. Hypergeometric connections
  11. Differential Galois groups
  12. Tannakian reformulation of Siegel's problem
  13. The category $\mathbf{G}$
  14. The category $\mathbf{H}$
  15. Proofs of main results
  16. ...and 8 more sections

Key Result

Theorem 1.1.3

The answer to siegelproblem is negative. More precisely, there exist $G$-functions of differential order $2$ that cannot be written as a $\overline{\mathbb{Q}}_{}$-polynomial expression in functions of the form eqn:hypgfunction.

Theorems & Definitions (63)

  • Remark 1.1.2
  • Theorem 1.1.3
  • Remark 1.1.4
  • Definition 2.1.1
  • Definition 2.1.2
  • Definition 2.1.3
  • Definition 2.2.1
  • Remark 2.2.2
  • Remark 2.2.3
  • Lemma 2.2.4
  • ...and 53 more