A duality result about special functions for Drinfeld modules of arbitrary rank
In the setting of a Drinfeld module φ over a curve X /Fq, we use a functorial point of view to define Anderson eigenvectors, a generalization of the so-called “special functions” introduced by Anglès, Ngo Dac and Tavares Ribeiro, and prove the existence of a universal object ω φ . We adopt an analo...
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| Dokumenttyp: | Article (Journal) |
| Sprache: | Englisch |
| Veröffentlicht: |
12 March 2025
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Research in the mathematical sciences
Year: 2025, Jahrgang: 12, Heft: 2, Pages: 1-41 |
| ISSN: | 2197-9847 |
| DOI: | 10.1007/s40687-025-00506-w |
| Online-Zugang: | Verlag, kostenfrei, Volltext: https://doi.org/10.1007/s40687-025-00506-w Verlag, kostenfrei, Volltext: https://link.springer.com/article/10.1007/s40687-025-00506-w 
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| Verfasserangaben: | Giacomo Hermes Ferraro |
| Zusammenfassung: | In the setting of a Drinfeld module φ over a curve X /Fq, we use a functorial point of view to define Anderson eigenvectors, a generalization of the so-called “special functions” introduced by Anglès, Ngo Dac and Tavares Ribeiro, and prove the existence of a universal object ω φ . We adopt an analogous approach with the adjoint Drinfeld module φ∗ to define dual Anderson eigenvectors. The universal object of this functor, denoted by ζ φ , is a generalization of Pellarin zeta functions, can be expressed as an Eisenstein-like series over the period lattice, and its coordinates are analytic functions from X (C∞) \ {∞} to C∞. For all integers i, we define dot products ζ φ · ω(i) φ as certain meromorphic differential forms over XC∞ \ {∞} and prove they are actually rational. This amounts to a generalization of Pellarin’s identity for the Carlitz module and is linked to the pairing of the A-motive and the dual A-motive defined by Hartl and Juschka. Finally, we develop an algorithm to compute the forms ζ φ · ω(i) φ when X = P1 and prove a conjecture of Gazda and Maurischat about the invertibility of special functions for Drinfeld modules of rank 1. |
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| Beschreibung: | Gesehen am 07.08.2025 |
| Beschreibung: | Online Resource |
| ISSN: | 2197-9847 |
| DOI: | 10.1007/s40687-025-00506-w |