On Siegel's problem and Dwork's conjecture for $G$-functions
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866915137090224128 |
|---|---|
| author | Fresán, Javier Lam, Yeuk Hay Joshua Qin, Yichen |
| author_facet | Fresán, Javier Lam, Yeuk Hay Joshua Qin, Yichen |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_02147 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Siegel's problem and Dwork's conjecture for $G$-functions Fresán, Javier Lam, Yeuk Hay Joshua Qin, Yichen Number Theory Algebraic Geometry Classical Analysis and ODEs 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. |
| title | On Siegel's problem and Dwork's conjecture for $G$-functions |
| topic | Number Theory Algebraic Geometry Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2502.02147 |