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This research reveals a method for constructing strongly cuspidal functions in finite fields, suggesting new insights into automorphic forms.
We continue the study of automorphic functions associated with a curve C over the ring k[ε]/(ε²), where k is a finite field, begun in arXiv:2303.16259. Namely, 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 S(SL₂(F) SL₂(AC)) introduced in loc.cit. We prove that all strongly cuspidal functions in S(SL₂(F) SL₂(AC)) can be constructed using theta-lifting for an appropriate double covering C̃→ C.
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Kazhdan et al. (2025) studied this question.
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