An analog of theta-lifting for a curve over dual numbers over a finite field
Abstract
We continue the study of automorphic functions associated with a curve $C$ over the ring $k[\epsilon] /\left(\epsilon^2\right)$, where $k$ is a finite field, begun in [3]. More precisely, we study an example of theta-lifting in this framework and show that it can be understood in terms of the orbit decomposition of the space of automorphic functions $\mathcal{S}\left(\mathrm{SL}_2(F) \backslash \mathrm{SL}_2\left(\mathbb{A}_C\right)\right)$ introduced in [3]. We prove that all strongly cuspidal functions in $\mathcal{S}\left(\mathrm{SL}_2(F) \backslash \mathrm{SL}_2\left(\mathbb{A}_C\right)\right)$ can be constructed using theta-lifting for an appropriate double covering $\widetilde{C} \rightarrow C$.
Fields of Study & Subjects
Algebraic Geometry and Number Theory
advanced mathematical theories
Polynomial and algebraic computation
Publication details
- Journal / source
- Pure and Applied Mathematics Quarterly
- Publisher
- International Press of Boston
- Volume
- 22
- Issue
- 3
- Pages
- 1115–1166
- Publication date
- 1 January 2026
- Language
- EN